3.298 \(\int \frac{1}{x (a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=172 \[ -\frac{d^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^4}-\frac{b^3 (b c-4 a d) \log (a+b x)}{a^2 (b c-a d)^4}+\frac{\log (x)}{a^2 c^3}+\frac{b^3}{a (a+b x) (b c-a d)^3}+\frac{d^2 (3 b c-a d)}{c^2 (c+d x) (b c-a d)^3}+\frac{d^2}{2 c (c+d x)^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.174053, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ -\frac{d^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^4}-\frac{b^3 (b c-4 a d) \log (a+b x)}{a^2 (b c-a d)^4}+\frac{\log (x)}{a^2 c^3}+\frac{b^3}{a (a+b x) (b c-a d)^3}+\frac{d^2 (3 b c-a d)}{c^2 (c+d x) (b c-a d)^3}+\frac{d^2}{2 c (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^2*(c + d*x)^3),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.248134, size = 173, normalized size = 1.01 \[ -\frac{d^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^4}+\frac{b^3 (4 a d-b c) \log (a+b x)}{a^2 (b c-a d)^4}+\frac{\log (x)}{a^2 c^3}-\frac{b^3}{a (a+b x) (a d-b c)^3}+\frac{d^2 (3 b c-a d)}{c^2 (c+d x) (b c-a d)^3}+\frac{d^2}{2 c (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 242, normalized size = 1.4 \begin{align*}{\frac{{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+{\frac{{d}^{3}a}{{c}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-3\,{\frac{{d}^{2}b}{c \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-{\frac{{d}^{4}\ln \left ( dx+c \right ){a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) ab}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-6\,{\frac{{d}^{2}\ln \left ( dx+c \right ){b}^{2}}{c \left ( ad-bc \right ) ^{4}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}{c}^{3}}}-{\frac{{b}^{3}}{ \left ( ad-bc \right ) ^{3}a \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}a}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.28059, size = 697, normalized size = 4.05 \begin{align*} -\frac{{\left (b^{4} c - 4 \, a b^{3} d\right )} \log \left (b x + a\right )}{a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}} - \frac{{\left (6 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}} + \frac{2 \, b^{3} c^{4} + 7 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} + 2 \,{\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} c d^{3} - a^{2} b d^{4}\right )} x^{2} +{\left (4 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x}{2 \,{\left (a^{2} b^{3} c^{7} - 3 \, a^{3} b^{2} c^{6} d + 3 \, a^{4} b c^{5} d^{2} - a^{5} c^{4} d^{3} +{\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5}\right )} x^{3} +{\left (2 \, a b^{4} c^{6} d - 5 \, a^{2} b^{3} c^{5} d^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} x^{2} +{\left (a b^{4} c^{7} - a^{2} b^{3} c^{6} d - 3 \, a^{3} b^{2} c^{5} d^{2} + 5 \, a^{4} b c^{4} d^{3} - 2 \, a^{5} c^{3} d^{4}\right )} x\right )}} + \frac{\log \left (x\right )}{a^{2} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 134.312, size = 2060, normalized size = 11.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b^4*c^6 - 2*a^2*b^3*c^5*d + 7*a^3*b^2*c^4*d^2 - 10*a^4*b*c^3*d^3 + 3*a^5*c^2*d^4 + 2*(a*b^4*c^4*d^2 +
2*a^2*b^3*c^3*d^3 - 4*a^3*b^2*c^2*d^4 + a^4*b*c*d^5)*x^2 + (4*a*b^4*c^5*d + 3*a^2*b^3*c^4*d^2 - 4*a^3*b^2*c^3
*d^3 - 5*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x - 2*(a*b^4*c^6 - 4*a^2*b^3*c^5*d + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3)*x^3
+ (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 - 4*a^2*b^3*c^3*d^3)*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 8*a^2*b^3*c^4*d^2)*x)*
log(b*x + a) - 2*(6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a
^4*b*d^6)*x^3 + (12*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*x^2 + (6*a^2*b^3*c^4*d^2 +
8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*x)*log(d*x + c) + 2*(a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^
2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4 + (b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d
^5 + a^4*b*d^6)*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 +
a^5*d^6)*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d
^5)*x)*log(x))/(a^3*b^4*c^9 - 4*a^4*b^3*c^8*d + 6*a^5*b^2*c^7*d^2 - 4*a^6*b*c^6*d^3 + a^7*c^5*d^4 + (a^2*b^5*c
^7*d^2 - 4*a^3*b^4*c^6*d^3 + 6*a^4*b^3*c^5*d^4 - 4*a^5*b^2*c^4*d^5 + a^6*b*c^3*d^6)*x^3 + (2*a^2*b^5*c^8*d - 7
*a^3*b^4*c^7*d^2 + 8*a^4*b^3*c^6*d^3 - 2*a^5*b^2*c^5*d^4 - 2*a^6*b*c^4*d^5 + a^7*c^3*d^6)*x^2 + (a^2*b^5*c^9 -
2*a^3*b^4*c^8*d - 2*a^4*b^3*c^7*d^2 + 8*a^5*b^2*c^6*d^3 - 7*a^6*b*c^5*d^4 + 2*a^7*c^4*d^5)*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/x/(b*x+a)**2/(d*x+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.19735, size = 419, normalized size = 2.44 \begin{align*} \frac{1}{2} \,{\left (\frac{2 \, b^{6}}{{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )}{\left (b x + a\right )}} - \frac{2 \,{\left (6 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{7} - 4 \, a b^{4} c^{6} d + 6 \, a^{2} b^{3} c^{5} d^{2} - 4 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4}} + \frac{2 \, \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} b c^{3}} - \frac{7 \, b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + \frac{2 \,{\left (4 \, b^{4} c^{3} d^{3} - 5 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5}\right )}}{{\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{4} b{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} c^{3}}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^6/((a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*(b*x + a)) - 2*(6*b^2*c^2*d^2 - 4*a*
b*c*d^3 + a^2*d^4)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^7 - 4*a*b^4*c^6*d + 6*a^2*b^3*c^5*d^2 -
4*a^3*b^2*c^4*d^3 + a^4*b*c^3*d^4) + 2*log(abs(-a/(b*x + a) + 1))/(a^2*b*c^3) - (7*b^2*c^2*d^4 - 2*a*b*c*d^5 +
2*(4*b^4*c^3*d^3 - 5*a*b^3*c^2*d^4 + a^2*b^2*c*d^5)/((b*x + a)*b))/((b*c - a*d)^4*b*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)^2*c^3))*b