### 3.279 $$\int \frac{(d+e x)^3}{(b x+c x^2)^3} \, dx$$

Optimal. Leaf size=137 $\frac{3 d^2 (c d-b e)}{b^4 x}+\frac{3 d (c d-b e)^2}{b^4 (b+c x)}+\frac{(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac{3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac{3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}-\frac{d^3}{2 b^3 x^2}$

[Out]

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

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Rubi [A]  time = 0.147119, antiderivative size = 137, 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} $\frac{3 d^2 (c d-b e)}{b^4 x}+\frac{3 d (c d-b e)^2}{b^4 (b+c x)}+\frac{(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac{3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac{3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}-\frac{d^3}{2 b^3 x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/(b*x + c*x^2)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.15537, size = 138, normalized size = 1.01 $-\frac{-6 d \log (x) \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )+6 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \log (b+c x)+\frac{b^2 (b e-c d)^3}{c (b+c x)^2}+\frac{b^2 d^3}{x^2}+\frac{6 b d^2 (b e-c d)}{x}-\frac{6 b d (c d-b e)^2}{b+c x}}{2 b^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.058, size = 238, normalized size = 1.7 \begin{align*} -{\frac{{d}^{3}}{2\,{b}^{3}{x}^{2}}}+3\,{\frac{d\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-9\,{\frac{{d}^{2}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{3}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-3\,{\frac{{d}^{2}e}{{b}^{3}x}}+3\,{\frac{{d}^{3}c}{{b}^{4}x}}-{\frac{{e}^{3}}{2\,c \left ( cx+b \right ) ^{2}}}+{\frac{3\,d{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{3\,{d}^{2}ec}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{3}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-3\,{\frac{d\ln \left ( cx+b \right ){e}^{2}}{{b}^{3}}}+9\,{\frac{{d}^{2}\ln \left ( cx+b \right ) ce}{{b}^{4}}}-6\,{\frac{{d}^{3}\ln \left ( cx+b \right ){c}^{2}}{{b}^{5}}}+3\,{\frac{d{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-6\,{\frac{{d}^{2}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{3}}{{b}^{4} \left ( cx+b \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d^3/b^3/x^2+3*d/b^3*ln(x)*e^2-9*d^2/b^4*ln(x)*c*e+6*d^3/b^5*ln(x)*c^2-3*d^2/b^3/x*e+3*d^3/b^4/x*c-1/2/c/(
c*x+b)^2*e^3+3/2/b/(c*x+b)^2*d*e^2-3/2/b^2*c/(c*x+b)^2*d^2*e+1/2/b^3*c^2/(c*x+b)^2*d^3-3*d/b^3*ln(c*x+b)*e^2+9
*d^2/b^4*ln(c*x+b)*c*e-6*d^3/b^5*ln(c*x+b)*c^2+3*d/b^2/(c*x+b)*e^2-6*d^2/b^3/(c*x+b)*c*e+3*d^3/b^4/(c*x+b)*c^2

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Maxima [A]  time = 1.08297, size = 293, normalized size = 2.14 \begin{align*} -\frac{b^{3} c d^{3} - 6 \,{\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{2} c^{2} d e^{2}\right )} x^{3} -{\left (18 \, b c^{3} d^{3} - 27 \, b^{2} c^{2} d^{2} e + 9 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{2} d^{3} - 3 \, b^{3} c d^{2} e\right )} x}{2 \,{\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac{3 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac{3 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (x\right )}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.93519, size = 770, normalized size = 5.62 \begin{align*} -\frac{b^{4} c d^{3} - 6 \,{\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} -{\left (18 \, b^{2} c^{3} d^{3} - 27 \, b^{3} c^{2} d^{2} e + 9 \, b^{4} c d e^{2} - b^{5} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{2} d^{3} - 3 \, b^{4} c d^{2} e\right )} x + 6 \,{\left ({\left (2 \, c^{5} d^{3} - 3 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{4} c d e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \,{\left ({\left (2 \, c^{5} d^{3} - 3 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2}\right )} x^{4} + 2 \,{\left (2 \, b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2}\right )} x^{3} +{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{4} c d e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 4.0019, size = 371, normalized size = 2.71 \begin{align*} \frac{- b^{3} c d^{3} + x^{3} \left (6 b^{2} c^{2} d e^{2} - 18 b c^{3} d^{2} e + 12 c^{4} d^{3}\right ) + x^{2} \left (- b^{4} e^{3} + 9 b^{3} c d e^{2} - 27 b^{2} c^{2} d^{2} e + 18 b c^{3} d^{3}\right ) + x \left (- 6 b^{3} c d^{2} e + 4 b^{2} c^{2} d^{3}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac{3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (x + \frac{3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} - 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} - \frac{3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log{\left (x + \frac{3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} + 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b**3*c*d**3 + x**3*(6*b**2*c**2*d*e**2 - 18*b*c**3*d**2*e + 12*c**4*d**3) + x**2*(-b**4*e**3 + 9*b**3*c*d*e*
*2 - 27*b**2*c**2*d**2*e + 18*b*c**3*d**3) + x*(-6*b**3*c*d**2*e + 4*b**2*c**2*d**3))/(2*b**6*c*x**2 + 4*b**5*
c**2*x**3 + 2*b**4*c**3*x**4) + 3*d*(b*e - 2*c*d)*(b*e - c*d)*log(x + (3*b**3*d*e**2 - 9*b**2*c*d**2*e + 6*b*c
**2*d**3 - 3*b*d*(b*e - 2*c*d)*(b*e - c*d))/(6*b**2*c*d*e**2 - 18*b*c**2*d**2*e + 12*c**3*d**3))/b**5 - 3*d*(b
*e - 2*c*d)*(b*e - c*d)*log(x + (3*b**3*d*e**2 - 9*b**2*c*d**2*e + 6*b*c**2*d**3 + 3*b*d*(b*e - 2*c*d)*(b*e -
c*d))/(6*b**2*c*d*e**2 - 18*b*c**2*d**2*e + 12*c**3*d**3))/b**5

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Giac [A]  time = 1.26788, size = 296, normalized size = 2.16 \begin{align*} \frac{3 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{3 \,{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} x^{3} e + 18 \, b c^{3} d^{3} x^{2} + 6 \, b^{2} c^{2} d x^{3} e^{2} - 27 \, b^{2} c^{2} d^{2} x^{2} e + 4 \, b^{2} c^{2} d^{3} x + 9 \, b^{3} c d x^{2} e^{2} - 6 \, b^{3} c d^{2} x e - b^{3} c d^{3} - b^{4} x^{2} e^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*log(abs(x))/b^5 - 3*(2*c^3*d^3 - 3*b*c^2*d^2*e + b^2*c*d*e^2)*log(abs(
c*x + b))/(b^5*c) + 1/2*(12*c^4*d^3*x^3 - 18*b*c^3*d^2*x^3*e + 18*b*c^3*d^3*x^2 + 6*b^2*c^2*d*x^3*e^2 - 27*b^2
*c^2*d^2*x^2*e + 4*b^2*c^2*d^3*x + 9*b^3*c*d*x^2*e^2 - 6*b^3*c*d^2*x*e - b^3*c*d^3 - b^4*x^2*e^3)/((c*x^2 + b*
x)^2*b^4*c)