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

Optimal. Leaf size=414 \[ \frac {45 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}} \]

[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

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Rubi [A]
time = 0.23, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \begin {gather*} -\frac {45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {(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*}

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Mathematica [A]
time = 1.63, size = 351, normalized size = 0.85 \begin {gather*} \frac {\left (c d^2-a e^2\right )^8 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^5 \left (315 e^7-\frac {2415 c d e^6 (d+e x)}{a e+c d x}+\frac {8043 c^2 d^2 e^5 (d+e x)^2}{(a e+c d x)^2}-\frac {15159 c^3 d^3 e^4 (d+e x)^3}{(a e+c d x)^3}+\frac {17609 c^4 d^4 e^3 (d+e x)^4}{(a e+c d x)^4}+\frac {8043 c^5 d^5 e^2 (d+e x)^5}{(a e+c d x)^5}-\frac {2415 c^6 d^6 e (d+e x)^6}{(a e+c d x)^6}+\frac {315 c^7 d^7 (d+e x)^7}{(a e+c d x)^7}\right )}{\left (c d^2-a e^2\right )^8 (d+e x)^2}-\frac {315 \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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]

((c*d^2 - a*e^2)^8*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)^5*(315*e^7 - (2415*
c*d*e^6*(d + e*x))/(a*e + c*d*x) + (8043*c^2*d^2*e^5*(d + e*x)^2)/(a*e + c*d*x)^2 - (15159*c^3*d^3*e^4*(d + e*
x)^3)/(a*e + c*d*x)^3 + (17609*c^4*d^4*e^3*(d + e*x)^4)/(a*e + c*d*x)^4 + (8043*c^5*d^5*e^2*(d + e*x)^5)/(a*e
+ c*d*x)^5 - (2415*c^6*d^6*e*(d + e*x)^6)/(a*e + c*d*x)^6 + (315*c^7*d^7*(d + e*x)^7)/(a*e + c*d*x)^7))/((c*d^
2 - a*e^2)^8*(d + e*x)^2) - (315*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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1559\) vs. \(2(376)=752\).
time = 0.70, size = 1560, normalized size = 3.77

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

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+c*d*e*x)/(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+c*d*e*x)/(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+c*d*e*x)/(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)))))+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+c*d*e*x)/(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))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [A]
time = 2.88, size = 1483, normalized size = 3.58 \begin {gather*} \left [\frac {{\left (315 \, {\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 )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} - 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (210 \, c^{8} d^{14} x e^{2} - 315 \, c^{8} d^{15} e + 210 \, a^{6} c^{2} d^{2} x e^{14} - 315 \, a^{7} c d e^{15} - 21 \, {\left (8 \, a^{5} c^{3} d^{3} x^{2} - 115 \, a^{6} c^{2} d^{3}\right )} e^{13} + 12 \, {\left (12 \, a^{4} c^{4} d^{4} x^{3} - 133 \, a^{5} c^{3} d^{4} x\right )} e^{12} - {\left (128 \, a^{3} c^{5} d^{5} x^{4} - 1272 \, a^{4} c^{4} d^{5} x^{2} + 8043 \, a^{5} c^{3} d^{5}\right )} e^{11} - 2 \, {\left (10368 \, a^{2} c^{6} d^{6} x^{5} + 544 \, a^{3} c^{5} d^{6} x^{3} - 2631 \, a^{4} c^{4} d^{6} x\right )} e^{10} - 3 \, {\left (11264 \, a c^{7} d^{7} x^{6} + 34432 \, a^{2} c^{6} d^{7} x^{4} + 1392 \, a^{3} c^{5} d^{7} x^{2} - 5053 \, a^{4} c^{4} d^{7}\right )} e^{9} - 56 \, {\left (256 \, c^{8} d^{8} x^{7} + 2880 \, a c^{7} d^{8} x^{5} + 3660 \, a^{2} c^{6} d^{8} x^{3} + 175 \, a^{3} c^{5} d^{8} x\right )} e^{8} - {\left (66560 \, c^{8} d^{9} x^{6} + 299904 \, a c^{7} d^{9} x^{4} + 200784 \, a^{2} c^{6} d^{9} x^{2} + 17609 \, a^{3} c^{5} d^{9}\right )} e^{7} - 6 \, {\left (19840 \, c^{8} d^{10} x^{5} + 43872 \, a c^{7} d^{10} x^{3} + 15507 \, a^{2} c^{6} d^{10} x\right )} e^{6} - {\left (98432 \, c^{8} d^{11} x^{4} + 97032 \, a c^{7} d^{11} x^{2} + 8043 \, a^{2} c^{6} d^{11}\right )} e^{5} - 4 \, {\left (8156 \, c^{8} d^{12} x^{3} + 399 \, a c^{7} d^{12} x\right )} e^{4} - 21 \, {\left (8 \, c^{8} d^{13} x^{2} - 115 \, a c^{7} d^{13}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{458752 \, c^{6} d^{6}}, \frac {{\left (315 \, {\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 )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (210 \, c^{8} d^{14} x e^{2} - 315 \, c^{8} d^{15} e + 210 \, a^{6} c^{2} d^{2} x e^{14} - 315 \, a^{7} c d e^{15} - 21 \, {\left (8 \, a^{5} c^{3} d^{3} x^{2} - 115 \, a^{6} c^{2} d^{3}\right )} e^{13} + 12 \, {\left (12 \, a^{4} c^{4} d^{4} x^{3} - 133 \, a^{5} c^{3} d^{4} x\right )} e^{12} - {\left (128 \, a^{3} c^{5} d^{5} x^{4} - 1272 \, a^{4} c^{4} d^{5} x^{2} + 8043 \, a^{5} c^{3} d^{5}\right )} e^{11} - 2 \, {\left (10368 \, a^{2} c^{6} d^{6} x^{5} + 544 \, a^{3} c^{5} d^{6} x^{3} - 2631 \, a^{4} c^{4} d^{6} x\right )} e^{10} - 3 \, {\left (11264 \, a c^{7} d^{7} x^{6} + 34432 \, a^{2} c^{6} d^{7} x^{4} + 1392 \, a^{3} c^{5} d^{7} x^{2} - 5053 \, a^{4} c^{4} d^{7}\right )} e^{9} - 56 \, {\left (256 \, c^{8} d^{8} x^{7} + 2880 \, a c^{7} d^{8} x^{5} + 3660 \, a^{2} c^{6} d^{8} x^{3} + 175 \, a^{3} c^{5} d^{8} x\right )} e^{8} - {\left (66560 \, c^{8} d^{9} x^{6} + 299904 \, a c^{7} d^{9} x^{4} + 200784 \, a^{2} c^{6} d^{9} x^{2} + 17609 \, a^{3} c^{5} d^{9}\right )} e^{7} - 6 \, {\left (19840 \, c^{8} d^{10} x^{5} + 43872 \, a c^{7} d^{10} x^{3} + 15507 \, a^{2} c^{6} d^{10} x\right )} e^{6} - {\left (98432 \, c^{8} d^{11} x^{4} + 97032 \, a c^{7} d^{11} x^{2} + 8043 \, a^{2} c^{6} d^{11}\right )} e^{5} - 4 \, {\left (8156 \, c^{8} d^{12} x^{3} + 399 \, a c^{7} d^{12} x\right )} e^{4} - 21 \, {\left (8 \, c^{8} d^{13} x^{2} - 115 \, a c^{7} d^{13}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{229376 \, c^{6} d^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(1/2)*log(8*c^2*d^3*x*e
+ c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 - 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2
)*sqrt(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) - 4*(210*c^8*d^14*x*e^2 - 315*c^8*d^15*e + 210*a^6*c^
2*d^2*x*e^14 - 315*a^7*c*d*e^15 - 21*(8*a^5*c^3*d^3*x^2 - 115*a^6*c^2*d^3)*e^13 + 12*(12*a^4*c^4*d^4*x^3 - 133
*a^5*c^3*d^4*x)*e^12 - (128*a^3*c^5*d^5*x^4 - 1272*a^4*c^4*d^5*x^2 + 8043*a^5*c^3*d^5)*e^11 - 2*(10368*a^2*c^6
*d^6*x^5 + 544*a^3*c^5*d^6*x^3 - 2631*a^4*c^4*d^6*x)*e^10 - 3*(11264*a*c^7*d^7*x^6 + 34432*a^2*c^6*d^7*x^4 + 1
392*a^3*c^5*d^7*x^2 - 5053*a^4*c^4*d^7)*e^9 - 56*(256*c^8*d^8*x^7 + 2880*a*c^7*d^8*x^5 + 3660*a^2*c^6*d^8*x^3
+ 175*a^3*c^5*d^8*x)*e^8 - (66560*c^8*d^9*x^6 + 299904*a*c^7*d^9*x^4 + 200784*a^2*c^6*d^9*x^2 + 17609*a^3*c^5*
d^9)*e^7 - 6*(19840*c^8*d^10*x^5 + 43872*a*c^7*d^10*x^3 + 15507*a^2*c^6*d^10*x)*e^6 - (98432*c^8*d^11*x^4 + 97
032*a*c^7*d^11*x^2 + 8043*a^2*c^6*d^11)*e^5 - 4*(8156*c^8*d^12*x^3 + 399*a*c^7*d^12*x)*e^4 - 21*(8*c^8*d^13*x^
2 - 115*a*c^7*d^13)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^6*d^6), 1/229376*(315*(c^8*d^1
6 - 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^2*x + a*x*e^2 + (c*d*x^2 +
a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 + (c^2*d^2*x^2 + a*c*d^2)*e^2)) -
2*(210*c^8*d^14*x*e^2 - 315*c^8*d^15*e + 210*a^6*c^2*d^2*x*e^14 - 315*a^7*c*d*e^15 - 21*(8*a^5*c^3*d^3*x^2 -
115*a^6*c^2*d^3)*e^13 + 12*(12*a^4*c^4*d^4*x^3 - 133*a^5*c^3*d^4*x)*e^12 - (128*a^3*c^5*d^5*x^4 - 1272*a^4*c^4
*d^5*x^2 + 8043*a^5*c^3*d^5)*e^11 - 2*(10368*a^2*c^6*d^6*x^5 + 544*a^3*c^5*d^6*x^3 - 2631*a^4*c^4*d^6*x)*e^10
- 3*(11264*a*c^7*d^7*x^6 + 34432*a^2*c^6*d^7*x^4 + 1392*a^3*c^5*d^7*x^2 - 5053*a^4*c^4*d^7)*e^9 - 56*(256*c^8*
d^8*x^7 + 2880*a*c^7*d^8*x^5 + 3660*a^2*c^6*d^8*x^3 + 175*a^3*c^5*d^8*x)*e^8 - (66560*c^8*d^9*x^6 + 299904*a*c
^7*d^9*x^4 + 200784*a^2*c^6*d^9*x^2 + 17609*a^3*c^5*d^9)*e^7 - 6*(19840*c^8*d^10*x^5 + 43872*a*c^7*d^10*x^3 +
15507*a^2*c^6*d^10*x)*e^6 - (98432*c^8*d^11*x^4 + 97032*a*c^7*d^11*x^2 + 8043*a^2*c^6*d^11)*e^5 - 4*(8156*c^8*
d^12*x^3 + 399*a*c^7*d^12*x)*e^4 - 21*(8*c^8*d^13*x^2 - 115*a*c^7*d^13)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2
+ a*d)*e))*e^(-4)/(c^6*d^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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Giac [A]
time = 0.79, size = 736, normalized size = 1.78 \begin {gather*} \frac {1}{114688} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, c^{2} d^{2} x e^{4} + \frac {{\left (65 \, c^{9} d^{10} e^{10} + 33 \, a c^{8} d^{8} e^{12}\right )} e^{\left (-7\right )}}{c^{7} d^{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 )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (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}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (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}\right )} e^{\left (-7\right )}}{c^{7} d^{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 )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (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}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (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}\right )} e^{\left (-7\right )}}{c^{7} d^{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 )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{32768 \, \sqrt {c d} c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(4*(14*c^2*d^2*x*e^4 + (65*c^9*d^10*e^10 +
33*a*c^8*d^8*e^12)*e^(-7)/(c^7*d^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)*e^(-
7)/(c^7*d^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)*e^(-7)/(
c^7*d^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)*e^(-7)/(c^7*d^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)*e^(-7)/(c^7*d^7))*x - (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 + 10
5*a^6*c^3*d^3*e^17)*e^(-7)/(c^7*d^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 + 17
609*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)*e^(-7)/(c^7*d^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)*e^(-7/2)*log(
abs(-c*d^2 - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqrt(c*d)*e^(1/2) - a*e^2))
/(sqrt(c*d)*c^5*d^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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