\(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [1920]

Optimal result

Integrand size = 37, antiderivative size = 341 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=-\frac {7 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac {7 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}} \]

[Out]

7/192*(-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)^(3/2)/c^3/d^3/e+7/60*(-a*e^2+
c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2+1/6*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d
+7/1024*(-a*e^2+c*d^2)^6*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^(9/2)/d^(9/2)/e^(5/2)-7/512*(-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)^(1/2)/c^4/d^4/e^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {7 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}-\frac {7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac {7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d} \]

[In]

Int[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-7*(c*d^2 - a*e^2)^4*(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])/(512*c^4*d^4*e^
2) + (7*(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)^(3/2))/(192*c^3*
d^3*e) + (7*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*c^2*d^2) + ((d + e*x)*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(6*c*d) + (7*(c*d^2 - a*e^2)^6*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sq
rt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(1024*c^(9/2)*d^(9/2)*e^(5/2))

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 )^{5/2}}{6 c d}+\frac {\left (7 \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 )^{3/2} \, dx}{12 d} \\ & = \frac {7 \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 )^{5/2}}{60 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 )^{5/2}}{6 c d}+\frac {\left (7 \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 )^{3/2} \, dx}{24 d^2} \\ & = \frac {7 \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 )^{3/2}}{192 c^3 d^3 e}+\frac {7 \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 )^{5/2}}{60 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 )^{5/2}}{6 c d}-\frac {\left (7 \left (c d^2-a e^2\right )^4\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^3 d^3 e} \\ & = -\frac {7 \left (c d^2-a e^2\right )^4 \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}}{512 c^4 d^4 e^2}+\frac {7 \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 )^{3/2}}{192 c^3 d^3 e}+\frac {7 \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 )^{5/2}}{60 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 )^{5/2}}{6 c d}+\frac {\left (7 \left (c d^2-a e^2\right )^6\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^4 d^4 e^2} \\ & = -\frac {7 \left (c d^2-a e^2\right )^4 \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}}{512 c^4 d^4 e^2}+\frac {7 \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 )^{3/2}}{192 c^3 d^3 e}+\frac {7 \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 )^{5/2}}{60 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 )^{5/2}}{6 c d}+\frac {\left (7 \left (c d^2-a e^2\right )^6\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 )}{512 c^4 d^4 e^2} \\ & = -\frac {7 \left (c d^2-a e^2\right )^4 \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}}{512 c^4 d^4 e^2}+\frac {7 \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 )^{3/2}}{192 c^3 d^3 e}+\frac {7 \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 )^{5/2}}{60 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 )^{5/2}}{6 c d}+\frac {7 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{9/2} d^{9/2} e^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.06 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10}+35 a^4 c d e^8 (17 d+2 e x)-14 a^3 c^2 d^2 e^6 \left (99 d^2+28 d e x+4 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (281 d^3+150 d^2 e x+52 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (595 d^4+5752 d^3 e x+9528 d^2 e^2 x^2+6560 d e^3 x^3+1664 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x+3016 d^3 e^2 x^2+6192 d^2 e^3 x^3+4736 d e^4 x^4+1280 e^5 x^5\right )\right )}{(a e+c d x) (d+e x)}+\frac {105 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 c^{9/2} d^{9/2} e^{5/2}} \]

[In]

Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10 + 35*a^4*c*d*e^8*(17*d + 2*e*x) - 14
*a^3*c^2*d^2*e^6*(99*d^2 + 28*d*e*x + 4*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(281*d^3 + 150*d^2*e*x + 52*d*e^2*x^2 + 8
*e^3*x^3) + a*c^4*d^4*e^2*(595*d^4 + 5752*d^3*e*x + 9528*d^2*e^2*x^2 + 6560*d*e^3*x^3 + 1664*e^4*x^4) + c^5*d^
5*(-105*d^5 + 70*d^4*e*x + 3016*d^3*e^2*x^2 + 6192*d^2*e^3*x^3 + 4736*d*e^4*x^4 + 1280*e^5*x^5)))/((a*e + c*d*
x)*(d + e*x)) + (105*(c*d^2 - a*e^2)^6*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(
(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(7680*c^(9/2)*d^(9/2)*e^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1187\) vs. \(2(307)=614\).

Time = 2.85 (sec) , antiderivative size = 1188, normalized size of antiderivative = 3.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1188\)

[In]

int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

d^2*(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*(1/6*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-7/12*(a*e^2+c*d^2)/c/d/e*(1/5*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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))))-1/6*a/c*(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*(1/5*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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 [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 1042, normalized size of antiderivative = 3.06 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\left [\frac {105 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d e} \log \left (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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} - 105 \, c^{6} d^{11} e + 595 \, a c^{5} d^{9} e^{3} + 1686 \, a^{2} c^{4} d^{7} e^{5} - 1386 \, a^{3} c^{3} d^{5} e^{7} + 595 \, a^{4} c^{2} d^{3} e^{9} - 105 \, a^{5} c d e^{11} + 128 \, {\left (37 \, c^{6} d^{7} e^{5} + 13 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (387 \, c^{6} d^{8} e^{4} + 410 \, a c^{5} d^{6} e^{6} + 3 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (377 \, c^{6} d^{9} e^{3} + 1191 \, a c^{5} d^{7} e^{5} + 39 \, a^{2} c^{4} d^{5} e^{7} - 7 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} + 2876 \, a c^{5} d^{8} e^{4} + 450 \, a^{2} c^{4} d^{6} e^{6} - 196 \, a^{3} c^{3} d^{4} e^{8} + 35 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, c^{5} d^{5} e^{3}}, -\frac {105 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} - 105 \, c^{6} d^{11} e + 595 \, a c^{5} d^{9} e^{3} + 1686 \, a^{2} c^{4} d^{7} e^{5} - 1386 \, a^{3} c^{3} d^{5} e^{7} + 595 \, a^{4} c^{2} d^{3} e^{9} - 105 \, a^{5} c d e^{11} + 128 \, {\left (37 \, c^{6} d^{7} e^{5} + 13 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (387 \, c^{6} d^{8} e^{4} + 410 \, a c^{5} d^{6} e^{6} + 3 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (377 \, c^{6} d^{9} e^{3} + 1191 \, a c^{5} d^{7} e^{5} + 39 \, a^{2} c^{4} d^{5} e^{7} - 7 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} + 2876 \, a c^{5} d^{8} e^{4} + 450 \, a^{2} c^{4} d^{6} e^{6} - 196 \, a^{3} c^{3} d^{4} e^{8} + 35 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, c^{5} d^{5} e^{3}}\right ] \]

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/30720*(105*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*
a^5*c*d^2*e^10 + a^6*e^12)*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*
(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 595*a*c^5*d^9*e^3 + 1686*a^2*c^4*d^7*e^5 - 1386*a^3*c^3*d^5*e^7 + 595
*a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(37*c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 + 16*(387*c^6*d^8*e^4 + 410*
a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^3 + 8*(377*c^6*d^9*e^3 + 1191*a*c^5*d^7*e^5 + 39*a^2*c^4*d^5*e^7 - 7*a^3*
c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 + 2876*a*c^5*d^8*e^4 + 450*a^2*c^4*d^6*e^6 - 196*a^3*c^3*d^4*e^8 + 35*a^
4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^3), -1/15360*(105*(c^6*d^12 - 6*a*c
^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 595*a*c^5
*d^9*e^3 + 1686*a^2*c^4*d^7*e^5 - 1386*a^3*c^3*d^5*e^7 + 595*a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(37*c^6*
d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 + 16*(387*c^6*d^8*e^4 + 410*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^3 + 8*(377*c^
6*d^9*e^3 + 1191*a*c^5*d^7*e^5 + 39*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 + 2876*a*c^5
*d^8*e^4 + 450*a^2*c^4*d^6*e^6 - 196*a^3*c^3*d^4*e^8 + 35*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x))/(c^5*d^5*e^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2909 vs. \(2 (333) = 666\).

Time = 1.76 (sec) , antiderivative size = 2909, normalized size of antiderivative = 8.53 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c*d*e**3*x**5/6 + x**4*(2*a*c*d*e**5 + 4*c**2*d**3*
e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e) + x**3*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*
e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5
*c*d*e))/(4*c*d*e) + x**2*(4*a**2*d*e**5 + 12*a*c*d**3*e**3 - 4*a*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*
(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c) + 4*c**2*d**5*e - (7*a*e**2/2 + 7*c*d**2/2)*(a**2*e**6 + 43*a*c*d**2*e**4
/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 1
1*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e) + x*(6*a**2*d**2*e**4 + 8*a*c*d**4*e**2 - 3*a*(a**2*e**6 + 43*a
*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*
a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c) + c**2*d**6 - (5*a*e**2/2 + 5*c*d**2/2)*(4*a**2*d*e**5 + 12*a*c*d*
*3*e**3 - 4*a*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c) + 4*c**2*d**5*e
- (7*a*e**2/2 + 7*c*d**2/2)*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2
*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c
*d*e) + (4*a**2*d**3*e**3 + 2*a*c*d**5*e - 2*a*(4*a**2*d*e**5 + 12*a*c*d**3*e**3 - 4*a*(2*a*c*d*e**5 + 4*c**2*
d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c) + 4*c**2*d**5*e - (7*a*e**2/2 + 7*c*d**2/2)*(a**2*e*
*6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*
e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(6*a**2*d**2*e**4
+ 8*a*c*d**4*e**2 - 3*a*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*
d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c) + c**2*d**6 - (5*a*e**2/2
+ 5*c*d**2/2)*(4*a**2*d*e**5 + 12*a*c*d**3*e**3 - 4*a*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/
2 + 11*c*d**2/2)/6)/(5*c) + 4*c**2*d**5*e - (7*a*e**2/2 + 7*c*d**2/2)*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2
*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)
/6)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(c*d*e)) + (a**2*d**4*e**2 - a*(6*a**2*d**2*e**4 + 8*a*c*d**4*
e**2 - 3*a*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c*
*2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c) + c**2*d**6 - (5*a*e**2/2 + 5*c*d**2/2
)*(4*a**2*d*e**5 + 12*a*c*d**3*e**3 - 4*a*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2
/2)/6)/(5*c) + 4*c**2*d**5*e - (7*a*e**2/2 + 7*c*d**2/2)*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 -
(9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e)
)/(4*c*d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(4*a**2*d**3*e**3 + 2*a*c*d**5*e - 2*a*(4*a**2*d*e**5 + 12*a
*c*d**3*e**3 - 4*a*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c) + 4*c**2*d
**5*e - (7*a*e**2/2 + 7*c*d**2/2)*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/
2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c) - (3
*a*e**2/2 + 3*c*d**2/2)*(6*a**2*d**2*e**4 + 8*a*c*d**4*e**2 - 3*a*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**
4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/
(5*c*d*e))/(4*c) + c**2*d**6 - (5*a*e**2/2 + 5*c*d**2/2)*(4*a**2*d*e**5 + 12*a*c*d**3*e**3 - 4*a*(2*a*c*d*e**5
+ 4*c**2*d**3*e**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c) + 4*c**2*d**5*e - (7*a*e**2/2 + 7*c*d**2/2
)*(a**2*e**6 + 43*a*c*d**2*e**4/6 + 6*c**2*d**4*e**2 - (9*a*e**2/2 + 9*c*d**2/2)*(2*a*c*d*e**5 + 4*c**2*d**3*e
**3 - c*d*e**3*(11*a*e**2/2 + 11*c*d**2/2)/6)/(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*sqrt(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))**(5/2)/(5*(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))**(7/2)/(7*(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))**(9/2)/(9*(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)**(3/2)*Piecewise
((d**2*x, Eq(e, 0)), ((d + e*x)**3/(3*e), True)), True))

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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 [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.52 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c d e^{3} x + \frac {37 \, c^{6} d^{7} e^{7} + 13 \, a c^{5} d^{5} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {387 \, c^{6} d^{8} e^{6} + 410 \, a c^{5} d^{6} e^{8} + 3 \, a^{2} c^{4} d^{4} e^{10}}{c^{5} d^{5} e^{5}}\right )} x + \frac {377 \, c^{6} d^{9} e^{5} + 1191 \, a c^{5} d^{7} e^{7} + 39 \, a^{2} c^{4} d^{5} e^{9} - 7 \, a^{3} c^{3} d^{3} e^{11}}{c^{5} d^{5} e^{5}}\right )} x + \frac {35 \, c^{6} d^{10} e^{4} + 2876 \, a c^{5} d^{8} e^{6} + 450 \, a^{2} c^{4} d^{6} e^{8} - 196 \, a^{3} c^{3} d^{4} e^{10} + 35 \, a^{4} c^{2} d^{2} e^{12}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{6} d^{11} e^{3} - 595 \, a c^{5} d^{9} e^{5} - 1686 \, a^{2} c^{4} d^{7} e^{7} + 1386 \, a^{3} c^{3} d^{5} e^{9} - 595 \, a^{4} c^{2} d^{3} e^{11} + 105 \, a^{5} c d e^{13}}{c^{5} d^{5} e^{5}}\right )} - \frac {7 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{4} d^{4} e^{2}} \]

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*c*d*e^3*x + (37*c^6*d^7*e^7 + 13*a*c^5*d^5*
e^9)/(c^5*d^5*e^5))*x + (387*c^6*d^8*e^6 + 410*a*c^5*d^6*e^8 + 3*a^2*c^4*d^4*e^10)/(c^5*d^5*e^5))*x + (377*c^6
*d^9*e^5 + 1191*a*c^5*d^7*e^7 + 39*a^2*c^4*d^5*e^9 - 7*a^3*c^3*d^3*e^11)/(c^5*d^5*e^5))*x + (35*c^6*d^10*e^4 +
2876*a*c^5*d^8*e^6 + 450*a^2*c^4*d^6*e^8 - 196*a^3*c^3*d^4*e^10 + 35*a^4*c^2*d^2*e^12)/(c^5*d^5*e^5))*x - (10
5*c^6*d^11*e^3 - 595*a*c^5*d^9*e^5 - 1686*a^2*c^4*d^7*e^7 + 1386*a^3*c^3*d^5*e^9 - 595*a^4*c^2*d^3*e^11 + 105*
a^5*c*d*e^13)/(c^5*d^5*e^5)) - 7/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 +
15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqr
t(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^4*d^4*e^2)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2} \,d x \]

[In]

int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)

[Out]

int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2), x)