### 3.266 $$\int \frac{1}{(d+e x)^3 (b x+c x^2)} \, dx$$

Optimal. Leaf size=134 $\frac{e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{c^3 \log (b+c x)}{b (c d-b e)^3}-\frac{e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac{e}{2 d (d+e x)^2 (c d-b e)}+\frac{\log (x)}{b d^3}$

[Out]

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

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Rubi [A]  time = 0.119824, antiderivative size = 134, 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} $\frac{e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{c^3 \log (b+c x)}{b (c d-b e)^3}-\frac{e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}-\frac{e}{2 d (d+e x)^2 (c d-b e)}+\frac{\log (x)}{b d^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^3*(b*x + c*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.257857, size = 116, normalized size = 0.87 $\frac{\frac{e \left (\frac{d (c d-b e) (c d (5 d+4 e x)-b e (3 d+2 e x))}{(d+e x)^2}-2 \left (b^2 e^2-3 b c d e+3 c^2 d^2\right ) \log (d+e x)\right )}{d^3}+\frac{2 c^3 \log (b+c x)}{b}}{2 (b e-c d)^3}+\frac{\log (x)}{b d^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.056, size = 184, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( x \right ) }{{d}^{3}b}}+{\frac{{c}^{3}\ln \left ( cx+b \right ) }{ \left ( be-cd \right ) ^{3}b}}+{\frac{e}{2\,d \left ( be-cd \right ) \left ( ex+d \right ) ^{2}}}+{\frac{{e}^{2}b}{{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{ce}{d \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}-{\frac{{e}^{3}\ln \left ( ex+d \right ){b}^{2}}{{d}^{3} \left ( be-cd \right ) ^{3}}}+3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) bc}{{d}^{2} \left ( be-cd \right ) ^{3}}}-3\,{\frac{e\ln \left ( ex+d \right ){c}^{2}}{d \left ( be-cd \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x),x)

[Out]

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

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Maxima [B]  time = 1.15808, size = 359, normalized size = 2.68 \begin{align*} -\frac{c^{3} \log \left (c x + b\right )}{b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}} + \frac{{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\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{5 \, c d^{2} e - 3 \, b d e^{2} + 2 \,{\left (2 \, c d e^{2} - b e^{3}\right )} x}{2 \,{\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} +{\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}} + \frac{\log \left (x\right )}{b d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 45.3123, size = 1008, normalized size = 7.52 \begin{align*} -\frac{5 \, b c^{2} d^{4} e - 8 \, b^{2} c d^{3} e^{2} + 3 \, b^{3} d^{2} e^{3} + 2 \,{\left (2 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x + 2 \,{\left (c^{3} d^{3} e^{2} x^{2} + 2 \, c^{3} d^{4} e x + c^{3} d^{5}\right )} \log \left (c x + b\right ) - 2 \,{\left (3 \, b c^{2} d^{4} e - 3 \, b^{2} c d^{3} e^{2} + b^{3} d^{2} e^{3} +{\left (3 \, b c^{2} d^{2} e^{3} - 3 \, b^{2} c d e^{4} + b^{3} e^{5}\right )} x^{2} + 2 \,{\left (3 \, b c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{2} e^{3} + b^{3} d e^{4}\right )} x\right )} \log \left (e x + d\right ) - 2 \,{\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3} +{\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{2} + 2 \,{\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (b c^{3} d^{8} - 3 \, b^{2} c^{2} d^{7} e + 3 \, b^{3} c d^{6} e^{2} - b^{4} d^{5} e^{3} +{\left (b c^{3} d^{6} e^{2} - 3 \, b^{2} c^{2} d^{5} e^{3} + 3 \, b^{3} c d^{4} e^{4} - b^{4} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (b c^{3} d^{7} e - 3 \, b^{2} c^{2} d^{6} e^{2} + 3 \, b^{3} c d^{5} e^{3} - b^{4} d^{4} e^{4}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x),x, algorithm="fricas")

[Out]

-1/2*(5*b*c^2*d^4*e - 8*b^2*c*d^3*e^2 + 3*b^3*d^2*e^3 + 2*(2*b*c^2*d^3*e^2 - 3*b^2*c*d^2*e^3 + b^3*d*e^4)*x +
2*(c^3*d^3*e^2*x^2 + 2*c^3*d^4*e*x + c^3*d^5)*log(c*x + b) - 2*(3*b*c^2*d^4*e - 3*b^2*c*d^3*e^2 + b^3*d^2*e^3
+ (3*b*c^2*d^2*e^3 - 3*b^2*c*d*e^4 + b^3*e^5)*x^2 + 2*(3*b*c^2*d^3*e^2 - 3*b^2*c*d^2*e^3 + b^3*d*e^4)*x)*log(e
*x + d) - 2*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3*d^2*e^3 + (c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*
c*d*e^4 - b^3*e^5)*x^2 + 2*(c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*log(x))/(b*c^3*d^8 -
3*b^2*c^2*d^7*e + 3*b^3*c*d^6*e^2 - b^4*d^5*e^3 + (b*c^3*d^6*e^2 - 3*b^2*c^2*d^5*e^3 + 3*b^3*c*d^4*e^4 - b^4*
d^3*e^5)*x^2 + 2*(b*c^3*d^7*e - 3*b^2*c^2*d^6*e^2 + 3*b^3*c*d^5*e^3 - b^4*d^4*e^4)*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)**3/(c*x**2+b*x),x)

[Out]

Timed out

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Giac [A]  time = 1.27047, size = 306, normalized size = 2.28 \begin{align*} -\frac{c^{4} \log \left ({\left | c x + b \right |}\right )}{b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} - b^{4} c e^{3}} + \frac{{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac{\log \left ({\left | x \right |}\right )}{b d^{3}} - \frac{5 \, c^{2} d^{4} e - 8 \, b c d^{3} e^{2} + 3 \, b^{2} d^{2} e^{3} + 2 \,{\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (x e + d\right )}^{2} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x),x, algorithm="giac")

[Out]

-c^4*log(abs(c*x + b))/(b*c^4*d^3 - 3*b^2*c^3*d^2*e + 3*b^3*c^2*d*e^2 - b^4*c*e^3) + (3*c^2*d^2*e^2 - 3*b*c*d*
e^3 + b^2*e^4)*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*e^4) + log(abs(x))/(
b*d^3) - 1/2*(5*c^2*d^4*e - 8*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 2*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)*x)/(
(c*d - b*e)^3*(x*e + d)^2*d^3)