3.256 \(\int \frac{(b x+c x^2)^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=230 \[ -\frac{c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac{d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.159425, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac{d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}-\frac{c^3}{e^7 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

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

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)^8} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^8}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^7}+\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)^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^5}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^4}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac{c^3}{e^6 (d+e x)^2}\right ) \, dx\\ &=-\frac{d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}+\frac{d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}-\frac{c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)^3}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0711211, size = 221, normalized size = 0.96 \[ -\frac{4 b^2 c e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+b^3 e^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+10 b c^2 e \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+20 c^3 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )}{140 e^7 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

-(b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*b^2*c*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d
*e^3*x^3 + 35*e^4*x^4) + 10*b*c^2*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5
*x^5) + 20*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)
)/(140*e^7*(d + e*x)^7)

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Maple [A]  time = 0.048, size = 274, normalized size = 1.2 \begin{align*} -{\frac{3\,{c}^{2} \left ( be-2\,cd \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3} \left ({b}^{3}{e}^{3}-3\,{b}^{2}cd{e}^{2}+3\,b{c}^{2}{d}^{2}e-{c}^{3}{d}^{3} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}+{\frac{3\,d \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+10\,b{c}^{2}{d}^{2}e-5\,{c}^{3}{d}^{3} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{c \left ({b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{2} \left ({b}^{3}{e}^{3}-4\,{b}^{2}cd{e}^{2}+5\,b{c}^{2}{d}^{2}e-2\,{c}^{3}{d}^{3} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^8,x)

[Out]

-3/2*c^2*(b*e-2*c*d)/e^7/(e*x+d)^2+1/7*d^3*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/e^7/(e*x+d)^7+3/5*d*(
b^3*e^3-6*b^2*c*d*e^2+10*b*c^2*d^2*e-5*c^3*d^3)/e^7/(e*x+d)^5-c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^3-1/
2*d^2*(b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)/e^7/(e*x+d)^6-c^3/e^7/(e*x+d)-1/4*(b^3*e^3-12*b^2*c*d*e^
2+30*b*c^2*d^2*e-20*c^3*d^3)/e^7/(e*x+d)^4

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Maxima [A]  time = 1.17091, size = 451, normalized size = 1.96 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 210 \,{\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6}\right )} x^{4} + 35 \,{\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 21 \,{\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 7 \,{\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{140 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/140*(140*c^3*e^6*x^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 4*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 210*(2*c^3*d*e^5 + b*c
^2*e^6)*x^5 + 70*(10*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 2*b^2*c*e^6)*x^4 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 +
4*b^2*c*d*e^5 + b^3*e^6)*x^3 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 7*(2
0*c^3*d^5*e + 10*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Fricas [A]  time = 1.75796, size = 701, normalized size = 3.05 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 210 \,{\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6}\right )} x^{4} + 35 \,{\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 21 \,{\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 7 \,{\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{140 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/140*(140*c^3*e^6*x^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 4*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 210*(2*c^3*d*e^5 + b*c
^2*e^6)*x^5 + 70*(10*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 2*b^2*c*e^6)*x^4 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 +
4*b^2*c*d*e^5 + b^3*e^6)*x^3 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 7*(2
0*c^3*d^5*e + 10*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.23886, size = 360, normalized size = 1.57 \begin{align*} -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 420 \, c^{3} d x^{5} e^{5} + 700 \, c^{3} d^{2} x^{4} e^{4} + 700 \, c^{3} d^{3} x^{3} e^{3} + 420 \, c^{3} d^{4} x^{2} e^{2} + 140 \, c^{3} d^{5} x e + 20 \, c^{3} d^{6} + 210 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 350 \, b c^{2} d^{2} x^{3} e^{4} + 210 \, b c^{2} d^{3} x^{2} e^{3} + 70 \, b c^{2} d^{4} x e^{2} + 10 \, b c^{2} d^{5} e + 140 \, b^{2} c x^{4} e^{6} + 140 \, b^{2} c d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 28 \, b^{2} c d^{3} x e^{3} + 4 \, b^{2} c d^{4} e^{2} + 35 \, b^{3} x^{3} e^{6} + 21 \, b^{3} d x^{2} e^{5} + 7 \, b^{3} d^{2} x e^{4} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{140 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/140*(140*c^3*x^6*e^6 + 420*c^3*d*x^5*e^5 + 700*c^3*d^2*x^4*e^4 + 700*c^3*d^3*x^3*e^3 + 420*c^3*d^4*x^2*e^2
+ 140*c^3*d^5*x*e + 20*c^3*d^6 + 210*b*c^2*x^5*e^6 + 350*b*c^2*d*x^4*e^5 + 350*b*c^2*d^2*x^3*e^4 + 210*b*c^2*d
^3*x^2*e^3 + 70*b*c^2*d^4*x*e^2 + 10*b*c^2*d^5*e + 140*b^2*c*x^4*e^6 + 140*b^2*c*d*x^3*e^5 + 84*b^2*c*d^2*x^2*
e^4 + 28*b^2*c*d^3*x*e^3 + 4*b^2*c*d^4*e^2 + 35*b^3*x^3*e^6 + 21*b^3*d*x^2*e^5 + 7*b^3*d^2*x*e^4 + b^3*d^3*e^3
)*e^(-7)/(x*e + d)^7