\(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1933]

Optimal result

Integrand size = 37, antiderivative size = 414 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}} \]

[Out]

-15/2048*(-a*e^2+c*d^2)^4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4/e^2+3/128*(-
a*e^2+c*d^2)^2*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/e+9/112*(-a*e^2+c*d^2)*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2+1/8*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d-45/327
68*(-a*e^2+c*d^2)^8*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2))/c^(11/2)/d^(11/2)/e^(7/2)+45/16384*(-a*e^2+c*d^2)^6*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)/c^5/d^5/e^3

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=-\frac {45 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac {45 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d} \]

[In]

Int[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(45*(c*d^2 - a*e^2)^6*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16384*c^5*d^5*
e^3) - (15*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2048*
c^4*d^4*e^2) + (3*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))
/(128*c^3*d^3*e) + (9*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(112*c^2*d^2) + ((d + e*x
)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(8*c*d) - (45*(c*d^2 - a*e^2)^8*ArcTanh[(c*d^2 + a*e^2 + 2*c*
d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(32768*c^(11/2)*d^(11/2)*e^(7
/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac {\left (9 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{16 d} \\ & = \frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac {\left (9 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{32 d^2} \\ & = \frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {\left (15 \left (c d^2-a e^2\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{256 c^3 d^3 e} \\ & = -\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac {\left (45 \left (c d^2-a e^2\right )^6\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^4 d^4 e^2} \\ & = \frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {\left (45 \left (c d^2-a e^2\right )^8\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32768 c^5 d^5 e^3} \\ & = \frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {\left (45 \left (c d^2-a e^2\right )^8\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16384 c^5 d^5 e^3} \\ & = \frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.27 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (315 a^7 e^{14}-105 a^6 c d e^{12} (23 d+2 e x)+21 a^5 c^2 d^2 e^{10} \left (383 d^2+76 d e x+8 e^2 x^2\right )-3 a^4 c^3 d^3 e^8 \left (5053 d^3+1754 d^2 e x+424 d e^2 x^2+48 e^3 x^3\right )+a^3 c^4 d^4 e^6 \left (17609 d^4+9800 d^3 e x+4176 d^2 e^2 x^2+1088 d e^3 x^3+128 e^4 x^4\right )+3 a^2 c^5 d^5 e^4 \left (2681 d^5+31014 d^4 e x+66928 d^3 e^2 x^2+68320 d^2 e^3 x^3+34432 d e^4 x^4+6912 e^5 x^5\right )+3 a c^6 d^6 e^2 \left (-805 d^6+532 d^5 e x+32344 d^4 e^2 x^2+87744 d^3 e^3 x^3+99968 d^2 e^4 x^4+53760 d e^5 x^5+11264 e^6 x^6\right )+c^7 d^7 \left (315 d^7-210 d^6 e x+168 d^5 e^2 x^2+32624 d^4 e^3 x^3+98432 d^3 e^4 x^4+119040 d^2 e^5 x^5+66560 d e^6 x^6+14336 e^7 x^7\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {315 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{114688 c^{11/2} d^{11/2} e^{7/2}} \]

[In]

Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(315*a^7*e^14 - 105*a^6*c*d*e^12*(23*d + 2*e*x) + 2
1*a^5*c^2*d^2*e^10*(383*d^2 + 76*d*e*x + 8*e^2*x^2) - 3*a^4*c^3*d^3*e^8*(5053*d^3 + 1754*d^2*e*x + 424*d*e^2*x
^2 + 48*e^3*x^3) + a^3*c^4*d^4*e^6*(17609*d^4 + 9800*d^3*e*x + 4176*d^2*e^2*x^2 + 1088*d*e^3*x^3 + 128*e^4*x^4
) + 3*a^2*c^5*d^5*e^4*(2681*d^5 + 31014*d^4*e*x + 66928*d^3*e^2*x^2 + 68320*d^2*e^3*x^3 + 34432*d*e^4*x^4 + 69
12*e^5*x^5) + 3*a*c^6*d^6*e^2*(-805*d^6 + 532*d^5*e*x + 32344*d^4*e^2*x^2 + 87744*d^3*e^3*x^3 + 99968*d^2*e^4*
x^4 + 53760*d*e^5*x^5 + 11264*e^6*x^6) + c^7*d^7*(315*d^7 - 210*d^6*e*x + 168*d^5*e^2*x^2 + 32624*d^4*e^3*x^3
+ 98432*d^3*e^4*x^4 + 119040*d^2*e^5*x^5 + 66560*d*e^6*x^6 + 14336*e^7*x^7)))/((a*e + c*d*x)^2*(d + e*x)^2) -
(315*(c*d^2 - a*e^2)^8*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5
/2)*(d + e*x)^(5/2))))/(114688*c^(11/2)*d^(11/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1559\) vs. \(2(376)=752\).

Time = 2.58 (sec) , antiderivative size = 1560, normalized size of antiderivative = 3.77

method	result	size
default	\(\text {Expression too large to display}\)	\(1560\)

[In]

int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

d^2*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d
^2)^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a
*e^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2
*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2))/(c*d*e)^(1/2))))+e^2*(1/8*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-9/16*(a*e^2+c*d^2)/c/d/e*(1/7*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c
*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/
2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))-1/8*a/c*(1/12*(2*c*d
*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/
8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c
/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^
2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1
/2)))))+2*d*e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(2*c*d*e*x+a*e^
2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(2*c*d*
e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4
*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d
/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (376) = 752\).

Time = 0.50 (sec) , antiderivative size = 1520, normalized size of antiderivative = 3.67 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/458752*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 -
56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 +
c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*
sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(14336*c^8*d^8*e^8*x^7 + 315*c^8*d^15*e - 2415*a*c^7*d^13*e^3 +
8043*a^2*c^6*d^11*e^5 + 17609*a^3*c^5*d^9*e^7 - 15159*a^4*c^4*d^7*e^9 + 8043*a^5*c^3*d^5*e^11 - 2415*a^6*c^2*
d^3*e^13 + 315*a^7*c*d*e^15 + 1024*(65*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 768*(155*c^8*d^10*e^6 + 210*a*c^7
*d^8*e^8 + 27*a^2*c^6*d^6*e^10)*x^5 + 128*(769*c^8*d^11*e^5 + 2343*a*c^7*d^9*e^7 + 807*a^2*c^6*d^7*e^9 + a^3*c
^5*d^5*e^11)*x^4 + 16*(2039*c^8*d^12*e^4 + 16452*a*c^7*d^10*e^6 + 12810*a^2*c^6*d^8*e^8 + 68*a^3*c^5*d^6*e^10
- 9*a^4*c^4*d^4*e^12)*x^3 + 24*(7*c^8*d^13*e^3 + 4043*a*c^7*d^11*e^5 + 8366*a^2*c^6*d^9*e^7 + 174*a^3*c^5*d^7*
e^9 - 53*a^4*c^4*d^5*e^11 + 7*a^5*c^3*d^3*e^13)*x^2 - 2*(105*c^8*d^14*e^2 - 798*a*c^7*d^12*e^4 - 46521*a^2*c^6
*d^10*e^6 - 4900*a^3*c^5*d^8*e^8 + 2631*a^4*c^4*d^6*e^10 - 798*a^5*c^3*d^4*e^12 + 105*a^6*c^2*d^2*e^14)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^4), 1/229376*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c
^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d
^2*e^14 + a^8*e^16)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a
*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(14336*c^8*d^8*e^8*x^7 + 3
15*c^8*d^15*e - 2415*a*c^7*d^13*e^3 + 8043*a^2*c^6*d^11*e^5 + 17609*a^3*c^5*d^9*e^7 - 15159*a^4*c^4*d^7*e^9 +
8043*a^5*c^3*d^5*e^11 - 2415*a^6*c^2*d^3*e^13 + 315*a^7*c*d*e^15 + 1024*(65*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^
6 + 768*(155*c^8*d^10*e^6 + 210*a*c^7*d^8*e^8 + 27*a^2*c^6*d^6*e^10)*x^5 + 128*(769*c^8*d^11*e^5 + 2343*a*c^7*
d^9*e^7 + 807*a^2*c^6*d^7*e^9 + a^3*c^5*d^5*e^11)*x^4 + 16*(2039*c^8*d^12*e^4 + 16452*a*c^7*d^10*e^6 + 12810*a
^2*c^6*d^8*e^8 + 68*a^3*c^5*d^6*e^10 - 9*a^4*c^4*d^4*e^12)*x^3 + 24*(7*c^8*d^13*e^3 + 4043*a*c^7*d^11*e^5 + 83
66*a^2*c^6*d^9*e^7 + 174*a^3*c^5*d^7*e^9 - 53*a^4*c^4*d^5*e^11 + 7*a^5*c^3*d^3*e^13)*x^2 - 2*(105*c^8*d^14*e^2
- 798*a*c^7*d^12*e^4 - 46521*a^2*c^6*d^10*e^6 - 4900*a^3*c^5*d^8*e^8 + 2631*a^4*c^4*d^6*e^10 - 798*a^5*c^3*d^
4*e^12 + 105*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8918 vs. \(2 (406) = 812\).

Time = 8.45 (sec) , antiderivative size = 8918, normalized size of antiderivative = 21.54 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**4*x**7/8 + x**6*(3*a*c**2*d**2*e**6 +
5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e) + x**5*(3*a**2*c*d*e**7 + 113*a*c**
2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*
d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e) + x**4*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c
**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*
c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*
e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c
*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e) + x**3*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3
- 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d*
*2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e -
(9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5
*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*
c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**
2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e)
)/(4*c*d*e) + x**2*(10*a**3*d**2*e**6 + 30*a**2*c*d**4*e**4 + 15*a*c**2*d**6*e**2 - 4*a*(a**3*e**8 + 15*a**2*c
*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 +
15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**
5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*
(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)*(5*a**3*d*e
**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5
*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*
c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 3
0*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8
)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*
d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 +
15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e) + x*(10*a**3*d**3*e**5 + 15*a**2*c*d**5
*e**3 + 3*a*c**2*d**7*e - 3*a*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**
7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4
*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**
2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**
2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d
*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*
d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2
+ 5*c*d**2/2)*(10*a**3*d**2*e**6 + 30*a**2*c*d**4*e**4 + 15*a*c**2*d**6*e**2 - 4*a*(a**3*e**8 + 15*a**2*c*d**
2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c
*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8
+ 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*
a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)*(5*a**3*d*e**7
+ 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**
3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d*
*2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*
c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7
*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5
*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*
c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e) + (5*a**3*d**4*e**4 + 3*a**2*c*d
**6*e**2 - 2*a*(10*a**3*d**2*e**6 + 30*a**2*c*d**4*e**4 + 15*a*c**2*d**6*e**2 - 4*a*(a**3*e**8 + 15*a**2*c*d**
2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c
*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8
+ 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*
a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)*(5*a**3*d*e**7
+ 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**
3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d*
*2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*
c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7
*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5
*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*
c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(10*a**3*d**3*e**5
+ 15*a**2*c*d**5*e**3 + 3*a*c**2*d**7*e - 3*a*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a
*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e*
*6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*
e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3
*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**
2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**
2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*
c) - (5*a*e**2/2 + 5*c*d**2/2)*(10*a**3*d**2*e**6 + 30*a**2*c*d**4*e**4 + 15*a*c**2*d**6*e**2 - 4*a*(a**3*e**8
+ 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15
*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c
**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**
2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)
*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 +
10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*
e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d
**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15
*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/
8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(1
5*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(c*d*e)) + (a**
3*d**5*e**3 - a*(10*a**3*d**3*e**5 + 15*a**2*c*d**5*e**3 + 3*a*c**2*d**7*e - 3*a*(5*a**3*d*e**7 + 30*a**2*c*d*
*3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2
/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*
d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4
- 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*
d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*
e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(
7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(10*a**3*d**2*e**6 + 30*a**2*c*d**4*e**4 + 1
5*a*c**2*d**6*e**2 - 4*a*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*
c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c
*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2
*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c) + c*
*3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c
*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**
3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9
*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4
- c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*
*2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5
*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3
*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(5*a**3*d**4*e**4 + 3*a**2*c*d**6*e**2 - 2*a*(10*a**3*d**2*e**6 + 30*a**2*c
*d**4*e**4 + 15*a*c**2*d**6*e**2 - 4*a*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*
d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a
*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**
2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*
e))/(5*c) + c**3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 -
5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**
2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (
9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*
c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c
*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2
*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))
/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(10*a**3*d**3*e**5 + 15*a**2*c*d**5*e**3 + 3*a*c**2*d**7*e - 3*a
*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 +
10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*
e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d
**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15
*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/
8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(1
5*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(10*a**3*d**2*
e**6 + 30*a**2*c*d**4*e**4 + 15*a*c**2*d**6*e**2 - 4*a*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4
- 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d
**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e
**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7
*c*d*e))/(6*c*d*e))/(5*c) + c**3*d**8 - (7*a*e**2/2 + 7*c*d**2/2)*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*
c**2*d**5*e**3 - 5*a*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/
2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5
*c**3*d**7*e - (9*a*e**2/2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6*a*(3*a*c**
2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11
*a*e**2/2 + 11*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d
**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*
d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sq
rt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*
e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 -
c*d**2)/(2*c*d*e))**2), True)), Ne(c*d*e, 0)), (2*(c**2*d**6*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/(7*(a**2*e**
4 + 2*a*c*d**2*e**2 + c**2*d**4)) + 2*c*d**3*e*(a*d*e + x*(a*e**2 + c*d**2))**(9/2)/(9*(a**2*e**4 + 2*a*c*d**2
*e**2 + c**2*d**4)) + e**2*(a*d*e + x*(a*e**2 + c*d**2))**(11/2)/(11*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)
))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), ((a*d*e)**(5/2)*Piecewise((d**2*x, Eq(e, 0)), ((d + e*x)**3/(3*
e), True)), True))

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (376) = 752\).

Time = 0.37 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.90 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {1}{114688} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, c^{2} d^{2} e^{4} x + \frac {65 \, c^{9} d^{10} e^{10} + 33 \, a c^{8} d^{8} e^{12}}{c^{7} d^{7} e^{7}}\right )} x + \frac {3 \, {\left (155 \, c^{9} d^{11} e^{9} + 210 \, a c^{8} d^{9} e^{11} + 27 \, a^{2} c^{7} d^{7} e^{13}\right )}}{c^{7} d^{7} e^{7}}\right )} x + \frac {769 \, c^{9} d^{12} e^{8} + 2343 \, a c^{8} d^{10} e^{10} + 807 \, a^{2} c^{7} d^{8} e^{12} + a^{3} c^{6} d^{6} e^{14}}{c^{7} d^{7} e^{7}}\right )} x + \frac {2039 \, c^{9} d^{13} e^{7} + 16452 \, a c^{8} d^{11} e^{9} + 12810 \, a^{2} c^{7} d^{9} e^{11} + 68 \, a^{3} c^{6} d^{7} e^{13} - 9 \, a^{4} c^{5} d^{5} e^{15}}{c^{7} d^{7} e^{7}}\right )} x + \frac {3 \, {\left (7 \, c^{9} d^{14} e^{6} + 4043 \, a c^{8} d^{12} e^{8} + 8366 \, a^{2} c^{7} d^{10} e^{10} + 174 \, a^{3} c^{6} d^{8} e^{12} - 53 \, a^{4} c^{5} d^{6} e^{14} + 7 \, a^{5} c^{4} d^{4} e^{16}\right )}}{c^{7} d^{7} e^{7}}\right )} x - \frac {105 \, c^{9} d^{15} e^{5} - 798 \, a c^{8} d^{13} e^{7} - 46521 \, a^{2} c^{7} d^{11} e^{9} - 4900 \, a^{3} c^{6} d^{9} e^{11} + 2631 \, a^{4} c^{5} d^{7} e^{13} - 798 \, a^{5} c^{4} d^{5} e^{15} + 105 \, a^{6} c^{3} d^{3} e^{17}}{c^{7} d^{7} e^{7}}\right )} x + \frac {315 \, c^{9} d^{16} e^{4} - 2415 \, a c^{8} d^{14} e^{6} + 8043 \, a^{2} c^{7} d^{12} e^{8} + 17609 \, a^{3} c^{6} d^{10} e^{10} - 15159 \, a^{4} c^{5} d^{8} e^{12} + 8043 \, a^{5} c^{4} d^{6} e^{14} - 2415 \, a^{6} c^{3} d^{4} e^{16} + 315 \, a^{7} c^{2} d^{2} e^{18}}{c^{7} d^{7} e^{7}}\right )} + \frac {45 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{32768 \, \sqrt {c d e} c^{5} d^{5} e^{3}} \]

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/114688*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(4*(14*c^2*d^2*e^4*x + (65*c^9*d^10*e^10 +
33*a*c^8*d^8*e^12)/(c^7*d^7*e^7))*x + 3*(155*c^9*d^11*e^9 + 210*a*c^8*d^9*e^11 + 27*a^2*c^7*d^7*e^13)/(c^7*d^
7*e^7))*x + (769*c^9*d^12*e^8 + 2343*a*c^8*d^10*e^10 + 807*a^2*c^7*d^8*e^12 + a^3*c^6*d^6*e^14)/(c^7*d^7*e^7))
*x + (2039*c^9*d^13*e^7 + 16452*a*c^8*d^11*e^9 + 12810*a^2*c^7*d^9*e^11 + 68*a^3*c^6*d^7*e^13 - 9*a^4*c^5*d^5*
e^15)/(c^7*d^7*e^7))*x + 3*(7*c^9*d^14*e^6 + 4043*a*c^8*d^12*e^8 + 8366*a^2*c^7*d^10*e^10 + 174*a^3*c^6*d^8*e^
12 - 53*a^4*c^5*d^6*e^14 + 7*a^5*c^4*d^4*e^16)/(c^7*d^7*e^7))*x - (105*c^9*d^15*e^5 - 798*a*c^8*d^13*e^7 - 465
21*a^2*c^7*d^11*e^9 - 4900*a^3*c^6*d^9*e^11 + 2631*a^4*c^5*d^7*e^13 - 798*a^5*c^4*d^5*e^15 + 105*a^6*c^3*d^3*e
^17)/(c^7*d^7*e^7))*x + (315*c^9*d^16*e^4 - 2415*a*c^8*d^14*e^6 + 8043*a^2*c^7*d^12*e^8 + 17609*a^3*c^6*d^10*e
^10 - 15159*a^4*c^5*d^8*e^12 + 8043*a^5*c^4*d^6*e^14 - 2415*a^6*c^3*d^4*e^16 + 315*a^7*c^2*d^2*e^18)/(c^7*d^7*
e^7)) + 45/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8
- 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*
d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^5*d^5*e^3)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \]

[In]

int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)