3.4.36 \(\int \frac {(b x+c x^2)^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=242 \[ \frac {6 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac {2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac {2 d^3 (c d-b e)^3}{e^7 \sqrt {d+e x}}-\frac {6 d^2 \sqrt {d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \]

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Rubi [A]  time = 0.10, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \begin {gather*} \frac {6 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac {2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac {6 d^2 \sqrt {d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}-\frac {2 d^3 (c d-b e)^3}{e^7 \sqrt {d+e x}}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^3*(c*d - b*e)^3)/(e^7*Sqrt[d + e*x]) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*Sqrt[d + e*x])/e^7 + (2*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3/2))/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b
^2*e^2)*(d + e*x)^(5/2))/(5*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) - (2*c^2*(2
*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7)

Rule 698

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

Rubi steps

\begin {align*} \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^{3/2}}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 \sqrt {d+e x}}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \sqrt {d+e x}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{3/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{7/2}}{e^6}+\frac {c^3 (d+e x)^{9/2}}{e^6}\right ) \, dx\\ &=-\frac {2 d^3 (c d-b e)^3}{e^7 \sqrt {d+e x}}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) \sqrt {d+e x}}{e^7}+\frac {2 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 206, normalized size = 0.85 \begin {gather*} \frac {2 \left (495 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-231 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+1155 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-385 c^2 (d+e x)^5 (2 c d-b e)-1155 d^3 (c d-b e)^3-3465 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+105 c^3 (d+e x)^6\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*d^3*(c*d - b*e)^3 - 3465*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 1155*d*(c*d - b*e)*(5*c^2*d^2 -
5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 231*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 495*c*
(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 385*c^2*(2*c*d - b*e)*(d + e*x)^5 + 105*c^3*(d + e*x)^6))/(115
5*e^7*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.11, size = 335, normalized size = 1.38 \begin {gather*} \frac {2 \left (1155 b^3 d^3 e^3+3465 b^3 d^2 e^3 (d+e x)-1155 b^3 d e^3 (d+e x)^2+231 b^3 e^3 (d+e x)^3-3465 b^2 c d^4 e^2-13860 b^2 c d^3 e^2 (d+e x)+6930 b^2 c d^2 e^2 (d+e x)^2-2772 b^2 c d e^2 (d+e x)^3+495 b^2 c e^2 (d+e x)^4+3465 b c^2 d^5 e+17325 b c^2 d^4 e (d+e x)-11550 b c^2 d^3 e (d+e x)^2+6930 b c^2 d^2 e (d+e x)^3-2475 b c^2 d e (d+e x)^4+385 b c^2 e (d+e x)^5-1155 c^3 d^6-6930 c^3 d^5 (d+e x)+5775 c^3 d^4 (d+e x)^2-4620 c^3 d^3 (d+e x)^3+2475 c^3 d^2 (d+e x)^4-770 c^3 d (d+e x)^5+105 c^3 (d+e x)^6\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^3/(d + e*x)^(3/2),x]

[Out]

(2*(-1155*c^3*d^6 + 3465*b*c^2*d^5*e - 3465*b^2*c*d^4*e^2 + 1155*b^3*d^3*e^3 - 6930*c^3*d^5*(d + e*x) + 17325*
b*c^2*d^4*e*(d + e*x) - 13860*b^2*c*d^3*e^2*(d + e*x) + 3465*b^3*d^2*e^3*(d + e*x) + 5775*c^3*d^4*(d + e*x)^2
- 11550*b*c^2*d^3*e*(d + e*x)^2 + 6930*b^2*c*d^2*e^2*(d + e*x)^2 - 1155*b^3*d*e^3*(d + e*x)^2 - 4620*c^3*d^3*(
d + e*x)^3 + 6930*b*c^2*d^2*e*(d + e*x)^3 - 2772*b^2*c*d*e^2*(d + e*x)^3 + 231*b^3*e^3*(d + e*x)^3 + 2475*c^3*
d^2*(d + e*x)^4 - 2475*b*c^2*d*e*(d + e*x)^4 + 495*b^2*c*e^2*(d + e*x)^4 - 770*c^3*d*(d + e*x)^5 + 385*b*c^2*e
*(d + e*x)^5 + 105*c^3*(d + e*x)^6))/(1155*e^7*Sqrt[d + e*x])

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fricas [A]  time = 0.39, size = 280, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e - 12672 \, b^{2} c d^{4} e^{2} + 3696 \, b^{3} d^{3} e^{3} - 35 \, {\left (4 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (40 \, c^{3} d^{2} e^{4} - 110 \, b c^{2} d e^{5} + 99 \, b^{2} c e^{6}\right )} x^{4} - {\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \, b^{2} c d e^{5} - 231 \, b^{3} e^{6}\right )} x^{3} + 2 \, {\left (320 \, c^{3} d^{4} e^{2} - 880 \, b c^{2} d^{3} e^{3} + 792 \, b^{2} c d^{2} e^{4} - 231 \, b^{3} d e^{5}\right )} x^{2} - 8 \, {\left (320 \, c^{3} d^{5} e - 880 \, b c^{2} d^{4} e^{2} + 792 \, b^{2} c d^{3} e^{3} - 231 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{1155 \, {\left (e^{8} x + d e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*c^3*e^6*x^6 - 5120*c^3*d^6 + 14080*b*c^2*d^5*e - 12672*b^2*c*d^4*e^2 + 3696*b^3*d^3*e^3 - 35*(4*c^
3*d*e^5 - 11*b*c^2*e^6)*x^5 + 5*(40*c^3*d^2*e^4 - 110*b*c^2*d*e^5 + 99*b^2*c*e^6)*x^4 - (320*c^3*d^3*e^3 - 880
*b*c^2*d^2*e^4 + 792*b^2*c*d*e^5 - 231*b^3*e^6)*x^3 + 2*(320*c^3*d^4*e^2 - 880*b*c^2*d^3*e^3 + 792*b^2*c*d^2*e
^4 - 231*b^3*d*e^5)*x^2 - 8*(320*c^3*d^5*e - 880*b*c^2*d^4*e^2 + 792*b^2*c*d^3*e^3 - 231*b^3*d^2*e^4)*x)*sqrt(
e*x + d)/(e^8*x + d*e^7)

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giac [A]  time = 0.19, size = 371, normalized size = 1.53 \begin {gather*} \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} e^{70} - 770 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d e^{70} + 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{2} e^{70} - 4620 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{70} + 5775 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt {x e + d} c^{3} d^{5} e^{70} + 385 \, {\left (x e + d\right )}^{\frac {9}{2}} b c^{2} e^{71} - 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} d e^{71} + 6930 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d^{2} e^{71} - 11550 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{3} e^{71} + 17325 \, \sqrt {x e + d} b c^{2} d^{4} e^{71} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c e^{72} - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c d e^{72} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d^{2} e^{72} - 13860 \, \sqrt {x e + d} b^{2} c d^{3} e^{72} + 231 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{73} - 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{73} + 3465 \, \sqrt {x e + d} b^{3} d^{2} e^{73}\right )} e^{\left (-77\right )} - \frac {2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*e^70 - 770*(x*e + d)^(9/2)*c^3*d*e^70 + 2475*(x*e + d)^(7/2)*c^3*d^2*e^70 - 4
620*(x*e + d)^(5/2)*c^3*d^3*e^70 + 5775*(x*e + d)^(3/2)*c^3*d^4*e^70 - 6930*sqrt(x*e + d)*c^3*d^5*e^70 + 385*(
x*e + d)^(9/2)*b*c^2*e^71 - 2475*(x*e + d)^(7/2)*b*c^2*d*e^71 + 6930*(x*e + d)^(5/2)*b*c^2*d^2*e^71 - 11550*(x
*e + d)^(3/2)*b*c^2*d^3*e^71 + 17325*sqrt(x*e + d)*b*c^2*d^4*e^71 + 495*(x*e + d)^(7/2)*b^2*c*e^72 - 2772*(x*e
+ d)^(5/2)*b^2*c*d*e^72 + 6930*(x*e + d)^(3/2)*b^2*c*d^2*e^72 - 13860*sqrt(x*e + d)*b^2*c*d^3*e^72 + 231*(x*e
+ d)^(5/2)*b^3*e^73 - 1155*(x*e + d)^(3/2)*b^3*d*e^73 + 3465*sqrt(x*e + d)*b^3*d^2*e^73)*e^(-77) - 2*(c^3*d^6
- 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*e^(-7)/sqrt(x*e + d)

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maple [A]  time = 0.07, size = 286, normalized size = 1.18 \begin {gather*} \frac {\frac {2}{11} c^{3} x^{6} e^{6}+\frac {2}{3} b \,c^{2} e^{6} x^{5}-\frac {8}{33} c^{3} d \,e^{5} x^{5}+\frac {6}{7} b^{2} c \,e^{6} x^{4}-\frac {20}{21} b \,c^{2} d \,e^{5} x^{4}+\frac {80}{231} c^{3} d^{2} e^{4} x^{4}+\frac {2}{5} b^{3} e^{6} x^{3}-\frac {48}{35} b^{2} c d \,e^{5} x^{3}+\frac {32}{21} b \,c^{2} d^{2} e^{4} x^{3}-\frac {128}{231} c^{3} d^{3} e^{3} x^{3}-\frac {4}{5} b^{3} d \,e^{5} x^{2}+\frac {96}{35} b^{2} c \,d^{2} e^{4} x^{2}-\frac {64}{21} b \,c^{2} d^{3} e^{3} x^{2}+\frac {256}{231} c^{3} d^{4} e^{2} x^{2}+\frac {16}{5} b^{3} d^{2} e^{4} x -\frac {384}{35} b^{2} c \,d^{3} e^{3} x +\frac {256}{21} b \,c^{2} d^{4} e^{2} x -\frac {1024}{231} c^{3} d^{5} e x +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {768}{35} b^{2} c \,d^{4} e^{2}+\frac {512}{21} b \,c^{2} d^{5} e -\frac {2048}{231} c^{3} d^{6}}{\sqrt {e x +d}\, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*c^3*e^6*x^6+385*b*c^2*e^6*x^5-140*c^3*d*e^5*x^5+495*b^2*c*e^6*x^4-550*b*c^2*d*e^5*x^4+200*c^3*d^2*
e^4*x^4+231*b^3*e^6*x^3-792*b^2*c*d*e^5*x^3+880*b*c^2*d^2*e^4*x^3-320*c^3*d^3*e^3*x^3-462*b^3*d*e^5*x^2+1584*b
^2*c*d^2*e^4*x^2-1760*b*c^2*d^3*e^3*x^2+640*c^3*d^4*e^2*x^2+1848*b^3*d^2*e^4*x-6336*b^2*c*d^3*e^3*x+7040*b*c^2
*d^4*e^2*x-2560*c^3*d^5*e*x+3696*b^3*d^3*e^3-12672*b^2*c*d^4*e^2+14080*b*c^2*d^5*e-5120*c^3*d^6)/(e*x+d)^(1/2)
/e^7

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maxima [A]  time = 1.39, size = 279, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (\frac {105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} - 385 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 231 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \sqrt {e x + d}}{e^{6}} - \frac {1155 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{1155 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*((105*(e*x + d)^(11/2)*c^3 - 385*(2*c^3*d - b*c^2*e)*(e*x + d)^(9/2) + 495*(5*c^3*d^2 - 5*b*c^2*d*e + b
^2*c*e^2)*(e*x + d)^(7/2) - 231*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(5/2) + 115
5*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(3/2) - 3465*(2*c^3*d^5 - 5*b*c^2*d^4*e
+ 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*sqrt(e*x + d))/e^6 - 1155*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d
^3*e^3)/(sqrt(e*x + d)*e^6))/e

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mupad [B]  time = 0.21, size = 268, normalized size = 1.11 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{5\,e^7}-\frac {-2\,b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2-6\,b\,c^2\,d^5\,e+2\,c^3\,d^6}{e^7\,\sqrt {d+e\,x}}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{7\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{3\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(5/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(5*e^7) - (2*c^3*d^6 - 2*b^3*d^3*
e^3 + 6*b^2*c*d^4*e^2 - 6*b*c^2*d^5*e)/(e^7*(d + e*x)^(1/2)) + (2*c^3*(d + e*x)^(11/2))/(11*e^7) - ((12*c^3*d
- 6*b*c^2*e)*(d + e*x)^(9/2))/(9*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(7*e^7) +
((d + e*x)^(3/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e))/(3*e^7) + (6*d^2*(b*e - c*d)^
2*(b*e - 2*c*d)*(d + e*x)^(1/2))/e^7

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sympy [A]  time = 46.53, size = 284, normalized size = 1.17 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {2 d^{3} \left (b e - c d\right )^{3}}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 6 b^{3} d e^{3} + 36 b^{2} c d^{2} e^{2} - 60 b c^{2} d^{3} e + 30 c^{3} d^{4}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (6 b^{3} d^{2} e^{3} - 24 b^{2} c d^{3} e^{2} + 30 b c^{2} d^{4} e - 12 c^{3} d^{5}\right )}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**3*(d + e*x)**(11/2)/(11*e**7) + 2*d**3*(b*e - c*d)**3/(e**7*sqrt(d + e*x)) + (d + e*x)**(9/2)*(6*b*c**2*e
- 12*c**3*d)/(9*e**7) + (d + e*x)**(7/2)*(6*b**2*c*e**2 - 30*b*c**2*d*e + 30*c**3*d**2)/(7*e**7) + (d + e*x)*
*(5/2)*(2*b**3*e**3 - 24*b**2*c*d*e**2 + 60*b*c**2*d**2*e - 40*c**3*d**3)/(5*e**7) + (d + e*x)**(3/2)*(-6*b**3
*d*e**3 + 36*b**2*c*d**2*e**2 - 60*b*c**2*d**3*e + 30*c**3*d**4)/(3*e**7) + sqrt(d + e*x)*(6*b**3*d**2*e**3 -
24*b**2*c*d**3*e**2 + 30*b*c**2*d**4*e - 12*c**3*d**5)/e**7

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