### 3.283 $$\int \frac{1}{(d+e x) (b x+c x^2)^3} \, dx$$

Optimal. Leaf size=193 $\frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{c^3}{2 b^3 (b+c x)^2 (c d-b e)}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{1}{2 b^3 d x^2}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}$

[Out]

-1/(2*b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) + c^3/(2*b^3*(c*d - b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/(b^
4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) - (c^3*(6*c^2*d^2 - 15*b*c*d
*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)

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Rubi [A]  time = 0.227096, antiderivative size = 193, 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} $\frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac{c^3}{2 b^3 (b+c x)^2 (c d-b e)}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{1}{2 b^3 d x^2}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)^3),x]

[Out]

-1/(2*b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) + c^3/(2*b^3*(c*d - b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/(b^
4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) - (c^3*(6*c^2*d^2 - 15*b*c*d
*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^3) + (e^5*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) \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{1}{b^3 d x^3}+\frac{-3 c d-b e}{b^4 d^2 x^2}+\frac{6 c^2 d^2+3 b c d e+b^2 e^2}{b^5 d^3 x}+\frac{c^4}{b^3 (-c d+b e) (b+c x)^3}+\frac{c^4 (-3 c d+4 b e)}{b^4 (-c d+b e)^2 (b+c x)^2}+\frac{c^4 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 (-c d+b e)^3 (b+c x)}+\frac{e^6}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{1}{2 b^3 d x^2}+\frac{3 c d+b e}{b^4 d^2 x}+\frac{c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac{c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac{\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac{c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}\\ \end{align*}

Mathematica [A]  time = 0.196233, size = 192, normalized size = 0.99 $\frac{\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}+\frac{c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (b e-c d)^3}+\frac{c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}-\frac{c^3}{2 b^3 (b+c x)^2 (b e-c d)}+\frac{b e+3 c d}{b^4 d^2 x}-\frac{1}{2 b^3 d x^2}+\frac{e^5 \log (d+e x)}{d^3 (c d-b e)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*b^3*d*x^2) + (3*c*d + b*e)/(b^4*d^2*x) - c^3/(2*b^3*(-(c*d) + b*e)*(b + c*x)^2) + (c^3*(3*c*d - 4*b*e))/
(b^4*(c*d - b*e)^2*(b + c*x)) + ((6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[x])/(b^5*d^3) + (c^3*(6*c^2*d^2 - 15*b*
c*d*e + 10*b^2*e^2)*Log[b + c*x])/(b^5*(-(c*d) + b*e)^3) + (e^5*Log[d + e*x])/(d^3*(c*d - b*e)^3)

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Maple [A]  time = 0.061, size = 254, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,d{b}^{3}{x}^{2}}}+{\frac{e}{{d}^{2}{b}^{3}x}}+3\,{\frac{c}{d{b}^{4}x}}+{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{3}{b}^{3}}}+3\,{\frac{\ln \left ( x \right ) ce}{{d}^{2}{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}}{d{b}^{5}}}-{\frac{{c}^{3}}{ \left ( 2\,be-2\,cd \right ){b}^{3} \left ( cx+b \right ) ^{2}}}-4\,{\frac{{c}^{3}e}{ \left ( be-cd \right ) ^{2}{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{4}d}{ \left ( be-cd \right ) ^{2}{b}^{4} \left ( cx+b \right ) }}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ){e}^{2}}{ \left ( be-cd \right ) ^{3}{b}^{3}}}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ) de}{ \left ( be-cd \right ) ^{3}{b}^{4}}}+6\,{\frac{{c}^{5}\ln \left ( cx+b \right ){d}^{2}}{ \left ( be-cd \right ) ^{3}{b}^{5}}}-{\frac{{e}^{5}\ln \left ( ex+d \right ) }{{d}^{3} \left ( be-cd \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.23846, size = 593, normalized size = 3.07 \begin{align*} \frac{e^{5} \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{{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac{b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \,{\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} -{\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \,{\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} +{\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac{{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^5*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) - (6*c^5*d^2 - 15*b*c^4*d*e + 10*b^
2*c^3*e^2)*log(c*x + b)/(b^5*c^3*d^3 - 3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3) - 1/2*(b^3*c^2*d^3 - 2*b^4*c
*d^2*e + b^5*d*e^2 - 2*(6*c^5*d^3 - 9*b*c^4*d^2*e + b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^3 - (18*b*c^4*d^3 - 27*b^2*
c^3*d^2*e + 3*b^3*c^2*d*e^2 + 4*b^4*c*e^3)*x^2 - 2*(2*b^2*c^3*d^3 - 3*b^3*c^2*d^2*e + b^5*e^3)*x)/((b^4*c^4*d^
4 - 2*b^5*c^3*d^3*e + b^6*c^2*d^2*e^2)*x^4 + 2*(b^5*c^3*d^4 - 2*b^6*c^2*d^3*e + b^7*c*d^2*e^2)*x^3 + (b^6*c^2*
d^4 - 2*b^7*c*d^3*e + b^8*d^2*e^2)*x^2) + (6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^3)

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Fricas [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)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

Timed out

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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)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.3336, size = 559, normalized size = 2.9 \begin{align*} -\frac{{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac{e^{6} \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{{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac{b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 2 \,{\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \,{\left (c d - b e\right )}^{3}{\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*c^6*d^2 - 15*b*c^5*d*e + 10*b^2*c^4*e^2)*log(abs(c*x + b))/(b^5*c^4*d^3 - 3*b^6*c^3*d^2*e + 3*b^7*c^2*d*e^
2 - b^8*c*e^3) + e^6*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) + (6*c^2*
d^2 + 3*b*c*d*e + b^2*e^2)*log(abs(x))/(b^5*d^3) - 1/2*(b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*
d^2*e^3 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - b^4*c^2*d*e^4)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4
*d^4*e + 30*b^3*c^3*d^3*e^2 + b^4*c^2*d^2*e^3 - 4*b^5*c*d*e^4)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e + 3*b^
4*c^2*d^3*e^2 + b^5*c*d^2*e^3 - b^6*d*e^4)*x)/((c*d - b*e)^3*(c*x + b)^2*b^4*d^3*x^2)