### 3.284 $$\int \frac{1}{(d+e x)^2 (b x+c x^2)^3} \, dx$$

Optimal. Leaf size=230 $\frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 b e+3 c d}{b^4 d^3 x}-\frac{1}{2 b^3 d^2 x^2}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}$

[Out]

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

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Rubi [A]  time = 0.33923, antiderivative size = 230, 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} $\frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 b e+3 c d}{b^4 d^3 x}-\frac{1}{2 b^3 d^2 x^2}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.29037, size = 230, normalized size = 1. $\frac{3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac{3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac{c^4 (5 b e-3 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac{c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac{2 b e+3 c d}{b^4 d^3 x}-\frac{1}{2 b^3 d^2 x^2}-\frac{e^5}{d^3 (d+e x) (c d-b e)^3}+\frac{3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.069, size = 306, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,{d}^{2}{b}^{3}{x}^{2}}}+2\,{\frac{e}{{d}^{3}{b}^{3}x}}+3\,{\frac{c}{{d}^{2}{b}^{4}x}}+3\,{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{4}{b}^{3}}}+6\,{\frac{\ln \left ( x \right ) ce}{{d}^{3}{b}^{4}}}+6\,{\frac{\ln \left ( x \right ){c}^{2}}{{d}^{2}{b}^{5}}}+{\frac{{c}^{4}}{2\, \left ( be-cd \right ) ^{2}{b}^{3} \left ( cx+b \right ) ^{2}}}+5\,{\frac{{c}^{4}e}{ \left ( be-cd \right ) ^{3}{b}^{3} \left ( cx+b \right ) }}-3\,{\frac{{c}^{5}d}{ \left ( be-cd \right ) ^{3}{b}^{4} \left ( cx+b \right ) }}-15\,{\frac{{c}^{4}\ln \left ( cx+b \right ){e}^{2}}{ \left ( be-cd \right ) ^{4}{b}^{3}}}+18\,{\frac{{c}^{5}\ln \left ( cx+b \right ) de}{ \left ( be-cd \right ) ^{4}{b}^{4}}}-6\,{\frac{{c}^{6}\ln \left ( cx+b \right ){d}^{2}}{ \left ( be-cd \right ) ^{4}{b}^{5}}}+{\frac{{e}^{5}}{{d}^{3} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{{e}^{6}\ln \left ( ex+d \right ) b}{{d}^{4} \left ( be-cd \right ) ^{4}}}+6\,{\frac{{e}^{5}\ln \left ( ex+d \right ) c}{{d}^{3} \left ( be-cd \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.29149, size = 1015, normalized size = 4.41 \begin{align*} -\frac{3 \,{\left (2 \, c^{6} d^{2} - 6 \, b c^{5} d e + 5 \, b^{2} c^{4} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}} + \frac{3 \,{\left (2 \, c d e^{5} - b e^{6}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \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} - 6 \,{\left (2 \, c^{6} d^{4} e - 4 \, b c^{5} d^{3} e^{2} + b^{2} c^{4} d^{2} e^{3} + b^{3} c^{3} d e^{4} - b^{4} c^{2} e^{5}\right )} x^{4} - 3 \,{\left (4 \, c^{6} d^{5} - 2 \, b c^{5} d^{4} e - 10 \, b^{2} c^{4} d^{3} e^{2} + 5 \, b^{3} c^{3} d^{2} e^{3} + 3 \, b^{4} c^{2} d e^{4} - 4 \, b^{5} c e^{5}\right )} x^{3} -{\left (18 \, b c^{5} d^{5} - 32 \, b^{2} c^{4} d^{4} e + b^{3} c^{3} d^{3} e^{2} + 13 \, b^{4} c^{2} d^{2} e^{3} - 6 \, b^{6} e^{5}\right )} x^{2} -{\left (4 \, b^{2} c^{4} d^{5} - 9 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + 5 \, b^{5} c d^{2} e^{3} - 3 \, b^{6} d e^{4}\right )} x}{2 \,{\left ({\left (b^{4} c^{5} d^{6} e - 3 \, b^{5} c^{4} d^{5} e^{2} + 3 \, b^{6} c^{3} d^{4} e^{3} - b^{7} c^{2} d^{3} e^{4}\right )} x^{5} +{\left (b^{4} c^{5} d^{7} - b^{5} c^{4} d^{6} e - 3 \, b^{6} c^{3} d^{5} e^{2} + 5 \, b^{7} c^{2} d^{4} e^{3} - 2 \, b^{8} c d^{3} e^{4}\right )} x^{4} +{\left (2 \, b^{5} c^{4} d^{7} - 5 \, b^{6} c^{3} d^{6} e + 3 \, b^{7} c^{2} d^{5} e^{2} + b^{8} c d^{4} e^{3} - b^{9} d^{3} e^{4}\right )} x^{3} +{\left (b^{6} c^{3} d^{7} - 3 \, b^{7} c^{2} d^{6} e + 3 \, b^{8} c d^{5} e^{2} - b^{9} d^{4} e^{3}\right )} x^{2}\right )}} + \frac{3 \,{\left (2 \, c^{2} d^{2} + 2 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(2*c^6*d^2 - 6*b*c^5*d*e + 5*b^2*c^4*e^2)*log(c*x + b)/(b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 -
4*b^8*c*d*e^3 + b^9*e^4) + 3*(2*c*d*e^5 - b*e^6)*log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 -
4*b^3*c*d^5*e^3 + b^4*d^4*e^4) - 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 - 6*(2*c^6
*d^4*e - 4*b*c^5*d^3*e^2 + b^2*c^4*d^2*e^3 + b^3*c^3*d*e^4 - b^4*c^2*e^5)*x^4 - 3*(4*c^6*d^5 - 2*b*c^5*d^4*e -
10*b^2*c^4*d^3*e^2 + 5*b^3*c^3*d^2*e^3 + 3*b^4*c^2*d*e^4 - 4*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 32*b^2*c^4*d^4*
e + b^3*c^3*d^3*e^2 + 13*b^4*c^2*d^2*e^3 - 6*b^6*e^5)*x^2 - (4*b^2*c^4*d^5 - 9*b^3*c^3*d^4*e + 3*b^4*c^2*d^3*e
^2 + 5*b^5*c*d^2*e^3 - 3*b^6*d*e^4)*x)/((b^4*c^5*d^6*e - 3*b^5*c^4*d^5*e^2 + 3*b^6*c^3*d^4*e^3 - b^7*c^2*d^3*e
^4)*x^5 + (b^4*c^5*d^7 - b^5*c^4*d^6*e - 3*b^6*c^3*d^5*e^2 + 5*b^7*c^2*d^4*e^3 - 2*b^8*c*d^3*e^4)*x^4 + (2*b^5
*c^4*d^7 - 5*b^6*c^3*d^6*e + 3*b^7*c^2*d^5*e^2 + b^8*c*d^4*e^3 - b^9*d^3*e^4)*x^3 + (b^6*c^3*d^7 - 3*b^7*c^2*d
^6*e + 3*b^8*c*d^5*e^2 - b^9*d^4*e^3)*x^2) + 3*(2*c^2*d^2 + 2*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^4)

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

[Out]

Timed out

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Giac [B]  time = 1.35597, size = 1133, normalized size = 4.93 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(4*c^6*d^6*e^2 - 12*b*c^5*d^5*e^3 + 10*b^2*c^4*d^4*e^4 - 2*b^5*c*d*e^7 + b^6*e^8)*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^4*c^4*d^8 - 4*b^5*c^3*d^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3 + b
^8*d^4*e^4)*abs(b)) - 3/2*(2*c*d*e^5 - b*e^6)*log(abs(-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^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4) - e^11/((
c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*c*d^4*e^8 - b^3*d^3*e^9)*(x*e + d)) + 1/2*(12*c^7*d^5*e - 30*b*c^6*d^4*e
^2 + 16*b^2*c^5*d^3*e^3 + 6*b^3*c^4*d^2*e^4 - 14*b^4*c^3*d*e^5 + 5*b^5*c^2*e^6 - 2*(18*c^7*d^6*e^2 - 54*b*c^6*
d^5*e^3 + 47*b^2*c^5*d^4*e^4 - 4*b^3*c^4*d^3*e^5 - 29*b^4*c^3*d^2*e^6 + 22*b^5*c^2*d*e^7 - 5*b^6*c*e^8)*e^(-1)
/(x*e + d) + (36*c^7*d^7*e^3 - 126*b*c^6*d^6*e^4 + 144*b^2*c^5*d^5*e^5 - 45*b^3*c^4*d^4*e^6 - 70*b^4*c^3*d^3*e
^7 + 87*b^5*c^2*d^2*e^8 - 36*b^6*c*d*e^9 + 5*b^7*e^10)*e^(-2)/(x*e + d)^2 - 6*(2*c^7*d^8*e^4 - 8*b*c^6*d^7*e^5
+ 11*b^2*c^5*d^6*e^6 - 5*b^3*c^4*d^5*e^7 - 5*b^4*c^3*d^4*e^8 + 9*b^5*c^2*d^3*e^9 - 5*b^6*c*d^2*e^10 + b^7*d*e
^11)*e^(-3)/(x*e + d)^3)/((c*d - b*e)^4*b^4*(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)^2*d^4)