3.4.38 \(\int \frac {(b x+c x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=240 \[ \frac {2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac {6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}-\frac {2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]

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Rubi [A]  time = 0.11, antiderivative size = 240, 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 {2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac {6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt {d+e x}}-\frac {6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac {2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}-\frac {2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

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

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Mathematica [A]  time = 0.13, size = 206, normalized size = 0.86 \begin {gather*} \frac {2 \left (35 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-35 (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 )-105 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 )-21 c^2 (d+e x)^5 (2 c d-b e)-7 d^3 (c d-b e)^3+35 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+5 c^3 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(2*(-7*d^3*(c*d - b*e)^3 + 35*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) - 105*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c
*d*e + b^2*e^2)*(d + e*x)^2 - 35*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^3 + 35*c*(5*c^2*d
^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 21*c^2*(2*c*d - b*e)*(d + e*x)^5 + 5*c^3*(d + e*x)^6))/(35*e^7*(d + e*
x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.11, size = 335, normalized size = 1.40 \begin {gather*} \frac {2 \left (7 b^3 d^3 e^3-35 b^3 d^2 e^3 (d+e x)+105 b^3 d e^3 (d+e x)^2+35 b^3 e^3 (d+e x)^3-21 b^2 c d^4 e^2+140 b^2 c d^3 e^2 (d+e x)-630 b^2 c d^2 e^2 (d+e x)^2-420 b^2 c d e^2 (d+e x)^3+35 b^2 c e^2 (d+e x)^4+21 b c^2 d^5 e-175 b c^2 d^4 e (d+e x)+1050 b c^2 d^3 e (d+e x)^2+1050 b c^2 d^2 e (d+e x)^3-175 b c^2 d e (d+e x)^4+21 b c^2 e (d+e x)^5-7 c^3 d^6+70 c^3 d^5 (d+e x)-525 c^3 d^4 (d+e x)^2-700 c^3 d^3 (d+e x)^3+175 c^3 d^2 (d+e x)^4-42 c^3 d (d+e x)^5+5 c^3 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(2*(-7*c^3*d^6 + 21*b*c^2*d^5*e - 21*b^2*c*d^4*e^2 + 7*b^3*d^3*e^3 + 70*c^3*d^5*(d + e*x) - 175*b*c^2*d^4*e*(d
+ e*x) + 140*b^2*c*d^3*e^2*(d + e*x) - 35*b^3*d^2*e^3*(d + e*x) - 525*c^3*d^4*(d + e*x)^2 + 1050*b*c^2*d^3*e*
(d + e*x)^2 - 630*b^2*c*d^2*e^2*(d + e*x)^2 + 105*b^3*d*e^3*(d + e*x)^2 - 700*c^3*d^3*(d + e*x)^3 + 1050*b*c^2
*d^2*e*(d + e*x)^3 - 420*b^2*c*d*e^2*(d + e*x)^3 + 35*b^3*e^3*(d + e*x)^3 + 175*c^3*d^2*(d + e*x)^4 - 175*b*c^
2*d*e*(d + e*x)^4 + 35*b^2*c*e^2*(d + e*x)^4 - 42*c^3*d*(d + e*x)^5 + 21*b*c^2*e*(d + e*x)^5 + 5*c^3*(d + e*x)
^6))/(35*e^7*(d + e*x)^(5/2))

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fricas [A]  time = 0.40, size = 302, normalized size = 1.26 \begin {gather*} \frac {2 \, {\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 896 \, b^{2} c d^{4} e^{2} + 112 \, b^{3} d^{3} e^{3} - 3 \, {\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \, b^{2} c e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \, b^{2} c d e^{5} - 7 \, b^{3} e^{6}\right )} x^{3} - 30 \, {\left (64 \, c^{3} d^{4} e^{2} - 112 \, b c^{2} d^{3} e^{3} + 56 \, b^{2} c d^{2} e^{4} - 7 \, b^{3} d e^{5}\right )} x^{2} - 40 \, {\left (64 \, c^{3} d^{5} e - 112 \, b c^{2} d^{4} e^{2} + 56 \, b^{2} c d^{3} e^{3} - 7 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*e^6*x^6 - 1024*c^3*d^6 + 1792*b*c^2*d^5*e - 896*b^2*c*d^4*e^2 + 112*b^3*d^3*e^3 - 3*(4*c^3*d*e^5 -
7*b*c^2*e^6)*x^5 + 5*(8*c^3*d^2*e^4 - 14*b*c^2*d*e^5 + 7*b^2*c*e^6)*x^4 - 5*(64*c^3*d^3*e^3 - 112*b*c^2*d^2*e
^4 + 56*b^2*c*d*e^5 - 7*b^3*e^6)*x^3 - 30*(64*c^3*d^4*e^2 - 112*b*c^2*d^3*e^3 + 56*b^2*c*d^2*e^4 - 7*b^3*d*e^5
)*x^2 - 40*(64*c^3*d^5*e - 112*b*c^2*d^4*e^2 + 56*b^2*c*d^3*e^3 - 7*b^3*d^2*e^4)*x)*sqrt(e*x + d)/(e^10*x^3 +
3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [A]  time = 0.21, size = 359, normalized size = 1.50 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {x e + d} c^{3} d^{3} e^{42} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} e^{43} - 175 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt {x e + d} b c^{2} d^{2} e^{43} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c e^{44} - 420 \, \sqrt {x e + d} b^{2} c d e^{44} + 35 \, \sqrt {x e + d} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \, {\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \, {\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \, {\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} - 20 \, {\left (x e + d\right )} b^{2} c d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} - 15 \, {\left (x e + d\right )}^{2} b^{3} d e^{3} + 5 \, {\left (x e + d\right )} b^{3} d^{2} e^{3} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{5 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*e^42 - 42*(x*e + d)^(5/2)*c^3*d*e^42 + 175*(x*e + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt
(x*e + d)*c^3*d^3*e^42 + 21*(x*e + d)^(5/2)*b*c^2*e^43 - 175*(x*e + d)^(3/2)*b*c^2*d*e^43 + 1050*sqrt(x*e + d)
*b*c^2*d^2*e^43 + 35*(x*e + d)^(3/2)*b^2*c*e^44 - 420*sqrt(x*e + d)*b^2*c*d*e^44 + 35*sqrt(x*e + d)*b^3*e^45)*
e^(-49) - 2/5*(75*(x*e + d)^2*c^3*d^4 - 10*(x*e + d)*c^3*d^5 + c^3*d^6 - 150*(x*e + d)^2*b*c^2*d^3*e + 25*(x*e
+ d)*b*c^2*d^4*e - 3*b*c^2*d^5*e + 90*(x*e + d)^2*b^2*c*d^2*e^2 - 20*(x*e + d)*b^2*c*d^3*e^2 + 3*b^2*c*d^4*e^
2 - 15*(x*e + d)^2*b^3*d*e^3 + 5*(x*e + d)*b^3*d^2*e^3 - b^3*d^3*e^3)*e^(-7)/(x*e + d)^(5/2)

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maple [A]  time = 0.05, size = 286, normalized size = 1.19 \begin {gather*} \frac {\frac {2}{7} c^{3} x^{6} e^{6}+\frac {6}{5} b \,c^{2} e^{6} x^{5}-\frac {24}{35} c^{3} d \,e^{5} x^{5}+2 b^{2} c \,e^{6} x^{4}-4 b \,c^{2} d \,e^{5} x^{4}+\frac {16}{7} c^{3} d^{2} e^{4} x^{4}+2 b^{3} e^{6} x^{3}-16 b^{2} c d \,e^{5} x^{3}+32 b \,c^{2} d^{2} e^{4} x^{3}-\frac {128}{7} c^{3} d^{3} e^{3} x^{3}+12 b^{3} d \,e^{5} x^{2}-96 b^{2} c \,d^{2} e^{4} x^{2}+192 b \,c^{2} d^{3} e^{3} x^{2}-\frac {768}{7} c^{3} d^{4} e^{2} x^{2}+16 b^{3} d^{2} e^{4} x -128 b^{2} c \,d^{3} e^{3} x +256 b \,c^{2} d^{4} e^{2} x -\frac {1024}{7} c^{3} d^{5} e x +\frac {32}{5} b^{3} d^{3} e^{3}-\frac {256}{5} b^{2} c \,d^{4} e^{2}+\frac {512}{5} b \,c^{2} d^{5} e -\frac {2048}{35} c^{3} d^{6}}{\left (e x +d \right )^{\frac {5}{2}} 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)^(7/2),x)

[Out]

2/35*(5*c^3*e^6*x^6+21*b*c^2*e^6*x^5-12*c^3*d*e^5*x^5+35*b^2*c*e^6*x^4-70*b*c^2*d*e^5*x^4+40*c^3*d^2*e^4*x^4+3
5*b^3*e^6*x^3-280*b^2*c*d*e^5*x^3+560*b*c^2*d^2*e^4*x^3-320*c^3*d^3*e^3*x^3+210*b^3*d*e^5*x^2-1680*b^2*c*d^2*e
^4*x^2+3360*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2+280*b^3*d^2*e^4*x-2240*b^2*c*d^3*e^3*x+4480*b*c^2*d^4*e^2*x
-2560*c^3*d^5*e*x+112*b^3*d^3*e^3-896*b^2*c*d^4*e^2+1792*b*c^2*d^5*e-1024*c^3*d^6)/(e*x+d)^(5/2)/e^7

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maxima [A]  time = 1.38, size = 277, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\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 )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\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} + 15 \, {\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 )}^{2} - 5 \, {\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 )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{6}}\right )}}{35 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 35*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^
2)*(e*x + d)^(3/2) - 35*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*sqrt(e*x + d))/e^6 - 7*(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 + 15*(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)^2 - 5*(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)*(e*x + d))/((e*x + d)^(5/2)*e
^6))/e

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mupad [B]  time = 0.08, size = 278, normalized size = 1.16 \begin {gather*} \frac {\sqrt {d+e\,x}\,\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 )}{e^7}+\frac {\left (d+e\,x\right )\,\left (-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5\right )-{\left (d+e\,x\right )}^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 )-\frac {2\,c^3\,d^6}{5}+\frac {2\,b^3\,d^3\,e^3}{5}-\frac {6\,b^2\,c\,d^4\,e^2}{5}+\frac {6\,b\,c^2\,d^5\,e}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{3\,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)^(7/2),x)

[Out]

((d + e*x)^(1/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/e^7 + ((d + e*x)*(4*c^3*d^5 - 2*b
^3*d^2*e^3 + 8*b^2*c*d^3*e^2 - 10*b*c^2*d^4*e) - (d + e*x)^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) - (2*c^3*d^6)/5 + (2*b^3*d^3*e^3)/5 - (6*b^2*c*d^4*e^2)/5 + (6*b*c^2*d^5*e)/5)/(e^7*(d + e*x)^(5
/2)) + (2*c^3*(d + e*x)^(7/2))/(7*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(5/2))/(5*e^7) + ((d + e*x)^(3/2)*(
30*c^3*d^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(3*e^7)

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sympy [A]  time = 82.25, size = 248, normalized size = 1.03 \begin {gather*} \frac {2 c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{7}} + \frac {2 d^{3} \left (b e - c d\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}} - \frac {2 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7} \left (d + e x\right )^{\frac {3}{2}}} + \frac {6 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \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 )}{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)**(7/2),x)

[Out]

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

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