### 3.274 $$\int \frac{1}{(d+e x)^2 (b x+c x^2)^2} \, dx$$

Optimal. Leaf size=144 $-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}+\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{1}{b^2 d^2 x}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}$

[Out]

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

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Rubi [A]  time = 0.167268, antiderivative size = 144, 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} $-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}+\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{1}{b^2 d^2 x}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^2*(b*x + c*x^2)^2),x]

[Out]

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

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{1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{1}{b^2 d^2 x^2}-\frac{2 (c d+b e)}{b^3 d^3 x}+\frac{c^4}{b^2 (-c d+b e)^2 (b+c x)^2}+\frac{2 c^4 (-c d+2 b e)}{b^3 (-c d+b e)^3 (b+c x)}+\frac{e^4}{d^2 (c d-b e)^2 (d+e x)^2}+\frac{2 e^4 (2 c d-b e)}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{1}{b^2 d^2 x}-\frac{c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac{e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac{2 (c d+b e) \log (x)}{b^3 d^3}+\frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end{align*}

Mathematica [A]  time = 0.178529, size = 145, normalized size = 1.01 $-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}+\frac{2 c^3 (2 b e-c d) \log (b+c x)}{b^3 (b e-c d)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{1}{b^2 d^2 x}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^2*(b*x + c*x^2)^2),x]

[Out]

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

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Maple [A]  time = 0.065, size = 185, normalized size = 1.3 \begin{align*} -{\frac{1}{{b}^{2}{d}^{2}x}}-2\,{\frac{\ln \left ( x \right ) e}{{d}^{3}{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{d}^{2}{b}^{3}}}-{\frac{{c}^{3}}{ \left ( be-cd \right ) ^{2}{b}^{2} \left ( cx+b \right ) }}+4\,{\frac{{c}^{3}\ln \left ( cx+b \right ) e}{ \left ( be-cd \right ) ^{3}{b}^{2}}}-2\,{\frac{{c}^{4}\ln \left ( cx+b \right ) d}{ \left ( be-cd \right ) ^{3}{b}^{3}}}-{\frac{{e}^{3}}{{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}+2\,{\frac{{e}^{4}\ln \left ( ex+d \right ) b}{{d}^{3} \left ( be-cd \right ) ^{3}}}-4\,{\frac{{e}^{3}\ln \left ( ex+d \right ) c}{{d}^{2} \left ( be-cd \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [B]  time = 1.2827, size = 504, normalized size = 3.5 \begin{align*} \frac{2 \,{\left (c^{4} d - 2 \, b c^{3} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} + \frac{2 \,{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (e x + d\right )}{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}} - \frac{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + 2 \,{\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} +{\left (2 \, c^{3} d^{3} - b c^{2} d^{2} e - b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} +{\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} +{\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac{2 \,{\left (c d + b e\right )} \log \left (x\right )}{b^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 85.7863, size = 1262, normalized size = 8.76 \begin{align*} -\frac{b^{2} c^{3} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - b^{5} d^{2} e^{3} + 2 \,{\left (b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4}\right )} x^{2} +{\left (2 \, b c^{4} d^{5} - 3 \, b^{2} c^{3} d^{4} e + 3 \, b^{4} c d^{2} e^{3} - 2 \, b^{5} d e^{4}\right )} x - 2 \,{\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2}\right )} x^{3} +{\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2}\right )} x^{2} +{\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e\right )} x\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} +{\left (2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} +{\left (2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \,{\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} +{\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} +{\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e + 2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} +{\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} +{\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [B]  time = 1.84037, size = 749, normalized size = 5.2 \begin{align*} \frac{{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 2 \, b^{3} c d e^{5} - b^{4} e^{6}\right )} e^{\left (-2\right )} \log \left (\frac{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} -{\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} +{\left | b \right |} e^{2} \right |}}\right )}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )}{\left | b \right |}} - \frac{{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left ({\left | -c + \frac{2 \, c d}{x e + d} - \frac{c d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{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}} - \frac{e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )}{\left (x e + d\right )}} - \frac{\frac{2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}}{c d^{2} - b d e} - \frac{{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )}{\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2}{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 2*b^3*c*d*e^5 - b^4*e^6)*e^(-2)*log(abs(-2*c*d*e + 2*c*d^2*e/(x*e + d) + b*
e^2 - 2*b*d*e^2/(x*e + d) - abs(b)*e^2)/abs(-2*c*d*e + 2*c*d^2*e/(x*e + d) + b*e^2 - 2*b*d*e^2/(x*e + d) + abs
(b)*e^2))/((b^2*c^3*d^6 - 3*b^3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)*abs(b)) - (2*c*d*e^3 - b*e^4)*log(a
bs(-c + 2*c*d/(x*e + d) - c*d^2/(x*e + d)^2 - b*e/(x*e + d) + b*d*e/(x*e + d)^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/((c^2*d^4*e^4 - 2*b*c*d^3*e^5 + b^2*d^2*e^6)*(x*e + d)) - ((2*c^4*d^3*e -
3*b*c^3*d^2*e^2 + 3*b^2*c^2*d*e^3 - b^3*c*e^4)/(c*d^2 - b*d*e) - (2*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*
d^2*e^4 - 4*b^3*c*d*e^5 + b^4*e^6)*e^(-1)/((c*d^2 - b*d*e)*(x*e + d)))/((c*d - b*e)^2*b^2*(c - 2*c*d/(x*e + d)
+ c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2)*d^2)