### 3.251 $$\int \frac{(b x+c x^2)^3}{(d+e x)^3} \, dx$$

Optimal. Leaf size=200 $-\frac{x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}+\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac{3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac{c^3 x^4}{4 e^3}$

[Out]

-(((c*d - b*e)*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*x)/e^6) + (3*c*(c*d - b*e)*(2*c*d - b*e)*x^2)/(2*e^5) - (c^2
*(c*d - b*e)*x^3)/e^4 + (c^3*x^4)/(4*e^3) - (d^3*(c*d - b*e)^3)/(2*e^7*(d + e*x)^2) + (3*d^2*(c*d - b*e)^2*(2*
c*d - b*e))/(e^7*(d + e*x)) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*Log[d + e*x])/e^7

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Rubi [A]  time = 0.21873, antiderivative size = 200, 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} $-\frac{x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}+\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}-\frac{d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac{3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac{c^3 x^4}{4 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.131485, size = 207, normalized size = 1.03 $\frac{6 c e^2 x^2 \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )+4 e x \left (-9 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e-10 c^3 d^3\right )+12 d \left (6 b^2 c d e^2-b^3 e^3-10 b c^2 d^2 e+5 c^3 d^3\right ) \log (d+e x)-4 c^2 e^3 x^3 (c d-b e)+\frac{12 d^2 (c d-b e)^2 (2 c d-b e)}{d+e x}-\frac{2 d^3 (c d-b e)^3}{(d+e x)^2}+c^3 e^4 x^4}{4 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*e*(-10*c^3*d^3 + 18*b*c^2*d^2*e - 9*b^2*c*d*e^2 + b^3*e^3)*x + 6*c*e^2*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*x^
2 - 4*c^2*e^3*(c*d - b*e)*x^3 + c^3*e^4*x^4 - (2*d^3*(c*d - b*e)^3)/(d + e*x)^2 + (12*d^2*(c*d - b*e)^2*(2*c*d
- b*e))/(d + e*x) + 12*d*(5*c^3*d^3 - 10*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*Log[d + e*x])/(4*e^7)

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Maple [A]  time = 0.051, size = 335, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}{x}^{4}}{4\,{e}^{3}}}+{\frac{b{x}^{3}{c}^{2}}{{e}^{3}}}-{\frac{{c}^{3}d{x}^{3}}{{e}^{4}}}+{\frac{3\,{b}^{2}{x}^{2}c}{2\,{e}^{3}}}-{\frac{9\,b{x}^{2}{c}^{2}d}{2\,{e}^{4}}}+3\,{\frac{{x}^{2}{c}^{3}{d}^{2}}{{e}^{5}}}+{\frac{{b}^{3}x}{{e}^{3}}}-9\,{\frac{{b}^{2}cdx}{{e}^{4}}}+18\,{\frac{b{c}^{2}{d}^{2}x}{{e}^{5}}}-10\,{\frac{{c}^{3}{d}^{3}x}{{e}^{6}}}+{\frac{{d}^{3}{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{d}^{4}{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{3\,{d}^{5}b{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-3\,{\frac{d\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}+18\,{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{2}c}{{e}^{5}}}-30\,{\frac{{d}^{3}\ln \left ( ex+d \right ) b{c}^{2}}{{e}^{6}}}+15\,{\frac{{d}^{4}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}-3\,{\frac{{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+12\,{\frac{{b}^{2}c{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{b{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^3*x^4/e^3+1/e^3*x^3*b*c^2-c^3*d*x^3/e^4+3/2/e^3*x^2*b^2*c-9/2/e^4*x^2*b*c^2*d+3/e^5*x^2*c^3*d^2+1/e^3*b^
3*x-9/e^4*b^2*c*d*x+18/e^5*b*c^2*d^2*x-10/e^6*c^3*d^3*x+1/2*d^3/e^4/(e*x+d)^2*b^3-3/2*d^4/e^5/(e*x+d)^2*b^2*c+
3/2*d^5/e^6/(e*x+d)^2*b*c^2-1/2*d^6/e^7/(e*x+d)^2*c^3-3*d/e^4*ln(e*x+d)*b^3+18*d^2/e^5*ln(e*x+d)*b^2*c-30*d^3/
e^6*ln(e*x+d)*b*c^2+15*d^4/e^7*ln(e*x+d)*c^3-3*d^2/e^4/(e*x+d)*b^3+12*d^3/e^5/(e*x+d)*b^2*c-15*d^4/e^6/(e*x+d)
*b*c^2+6*d^5/e^7/(e*x+d)*c^3

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Maxima [A]  time = 1.156, size = 378, normalized size = 1.89 \begin{align*} \frac{11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{c^{3} e^{3} x^{4} - 4 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} - 4 \,{\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac{3 \,{\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 )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c*d^4*e^2 - 5*b^3*d^3*e^3 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2
*c*d^3*e^3 - b^3*d^2*e^4)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*(c^3*d*e^2 - b*c^2*e^3)*x^
3 + 6*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*x^2 - 4*(10*c^3*d^3 - 18*b*c^2*d^2*e + 9*b^2*c*d*e^2 - b^3*e^3
)*x)/e^6 + 3*(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)*log(e*x + d)/e^7

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Fricas [B]  time = 1.66007, size = 868, normalized size = 4.34 \begin{align*} \frac{c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e + 42 \, b^{2} c d^{4} e^{2} - 10 \, b^{3} d^{3} e^{3} - 2 \,{\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} +{\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \, b^{2} c e^{6}\right )} x^{4} - 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} - 2 \,{\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \, b^{2} c d^{2} e^{4} - 4 \, b^{3} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, b^{2} c d^{3} e^{3} + 2 \, b^{3} d^{2} e^{4}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} +{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 2 \,{\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/4*(c^3*e^6*x^6 + 22*c^3*d^6 - 54*b*c^2*d^5*e + 42*b^2*c*d^4*e^2 - 10*b^3*d^3*e^3 - 2*(c^3*d*e^5 - 2*b*c^2*e^
6)*x^5 + (5*c^3*d^2*e^4 - 10*b*c^2*d*e^5 + 6*b^2*c*e^6)*x^4 - 4*(5*c^3*d^3*e^3 - 10*b*c^2*d^2*e^4 + 6*b^2*c*d*
e^5 - b^3*e^6)*x^3 - 2*(34*c^3*d^4*e^2 - 63*b*c^2*d^3*e^3 + 33*b^2*c*d^2*e^4 - 4*b^3*d*e^5)*x^2 - 4*(4*c^3*d^5
*e - 3*b*c^2*d^4*e^2 - 3*b^2*c*d^3*e^3 + 2*b^3*d^2*e^4)*x + 12*(5*c^3*d^6 - 10*b*c^2*d^5*e + 6*b^2*c*d^4*e^2 -
b^3*d^3*e^3 + (5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 2*(5*c^3*d^5*e - 10*b*c^
2*d^4*e^2 + 6*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 5.79209, size = 277, normalized size = 1.38 \begin{align*} \frac{c^{3} x^{4}}{4 e^{3}} - \frac{3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{5 b^{3} d^{3} e^{3} - 21 b^{2} c d^{4} e^{2} + 27 b c^{2} d^{5} e - 11 c^{3} d^{6} + x \left (6 b^{3} d^{2} e^{4} - 24 b^{2} c d^{3} e^{3} + 30 b c^{2} d^{4} e^{2} - 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (b c^{2} e - c^{3} d\right )}{e^{4}} + \frac{x^{2} \left (3 b^{2} c e^{2} - 9 b c^{2} d e + 6 c^{3} d^{2}\right )}{2 e^{5}} + \frac{x \left (b^{3} e^{3} - 9 b^{2} c d e^{2} + 18 b c^{2} d^{2} e - 10 c^{3} d^{3}\right )}{e^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*x**4/(4*e**3) - 3*d*(b*e - c*d)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7 - (5*b**3*d**3*e*
*3 - 21*b**2*c*d**4*e**2 + 27*b*c**2*d**5*e - 11*c**3*d**6 + x*(6*b**3*d**2*e**4 - 24*b**2*c*d**3*e**3 + 30*b*
c**2*d**4*e**2 - 12*c**3*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + x**3*(b*c**2*e - c**3*d)/e**4 + x
**2*(3*b**2*c*e**2 - 9*b*c**2*d*e + 6*c**3*d**2)/(2*e**5) + x*(b**3*e**3 - 9*b**2*c*d*e**2 + 18*b*c**2*d**2*e
- 10*c**3*d**3)/e**6

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Giac [A]  time = 1.3611, size = 356, normalized size = 1.78 \begin{align*} 3 \,{\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 )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} - 36 \, b^{2} c d x e^{8} + 4 \, b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(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^(-7)*log(abs(x*e + d)) + 1/4*(c^3*x^4*e^9 - 4*c
^3*d*x^3*e^8 + 12*c^3*d^2*x^2*e^7 - 40*c^3*d^3*x*e^6 + 4*b*c^2*x^3*e^9 - 18*b*c^2*d*x^2*e^8 + 72*b*c^2*d^2*x*e
^7 + 6*b^2*c*x^2*e^9 - 36*b^2*c*d*x*e^8 + 4*b^3*x*e^9)*e^(-12) + 1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c*d
^4*e^2 - 5*b^3*d^3*e^3 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*e^(-7)/(x*e + d)
^2