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

Optimal. Leaf size=221 \[ -\frac{b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac{d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}+\frac{b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac{\log (x)}{a^3 c^3}+\frac{b^3}{2 a (a+b x)^2 (b c-a d)^3}-\frac{d^3 (4 b c-a d)}{c^2 (c+d x) (b c-a d)^4}-\frac{d^3}{2 c (c+d x)^2 (b c-a d)^3} \]

[Out]

b^3/(2*a*(b*c - a*d)^3*(a + b*x)^2) + (b^3*(b*c - 4*a*d))/(a^2*(b*c - a*d)^4*(a + b*x)) - d^3/(2*c*(b*c - a*d)
^3*(c + d*x)^2) - (d^3*(4*b*c - a*d))/(c^2*(b*c - a*d)^4*(c + d*x)) + Log[x]/(a^3*c^3) - (b^3*(b^2*c^2 - 5*a*b
*c*d + 10*a^2*d^2)*Log[a + b*x])/(a^3*(b*c - a*d)^5) + (d^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[c + d*x])/(
c^3*(b*c - a*d)^5)

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Rubi [A]  time = 0.243741, antiderivative size = 221, 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} \[ -\frac{b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac{d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}+\frac{b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac{\log (x)}{a^3 c^3}+\frac{b^3}{2 a (a+b x)^2 (b c-a d)^3}-\frac{d^3 (4 b c-a d)}{c^2 (c+d x) (b c-a d)^4}-\frac{d^3}{2 c (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^3*(c + d*x)^3),x]

[Out]

b^3/(2*a*(b*c - a*d)^3*(a + b*x)^2) + (b^3*(b*c - 4*a*d))/(a^2*(b*c - a*d)^4*(a + b*x)) - d^3/(2*c*(b*c - a*d)
^3*(c + d*x)^2) - (d^3*(4*b*c - a*d))/(c^2*(b*c - a*d)^4*(c + d*x)) + Log[x]/(a^3*c^3) - (b^3*(b^2*c^2 - 5*a*b
*c*d + 10*a^2*d^2)*Log[a + b*x])/(a^3*(b*c - a*d)^5) + (d^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[c + d*x])/(
c^3*(b*c - a*d)^5)

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)^3 (c+d x)^3} \, dx &=\int \left (\frac{1}{a^3 c^3 x}+\frac{b^4}{a (-b c+a d)^3 (a+b x)^3}+\frac{b^4 (-b c+4 a d)}{a^2 (-b c+a d)^4 (a+b x)^2}+\frac{b^4 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right )}{a^3 (-b c+a d)^5 (a+b x)}+\frac{d^4}{c (b c-a d)^3 (c+d x)^3}+\frac{d^4 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)^2}+\frac{d^4 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^5 (c+d x)}\right ) \, dx\\ &=\frac{b^3}{2 a (b c-a d)^3 (a+b x)^2}+\frac{b^3 (b c-4 a d)}{a^2 (b c-a d)^4 (a+b x)}-\frac{d^3}{2 c (b c-a d)^3 (c+d x)^2}-\frac{d^3 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)}+\frac{\log (x)}{a^3 c^3}-\frac{b^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac{d^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}\\ \end{align*}

Mathematica [A]  time = 0.384677, size = 218, normalized size = 0.99 \[ \frac{b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (a d-b c)^5}+\frac{d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}+\frac{b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac{\log (x)}{a^3 c^3}-\frac{b^3}{2 a (a+b x)^2 (a d-b c)^3}+\frac{d^3 (a d-4 b c)}{c^2 (c+d x) (b c-a d)^4}-\frac{d^3}{2 c (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^3/(2*a*(-(b*c) + a*d)^3*(a + b*x)^2) + (b^3*(b*c - 4*a*d))/(a^2*(b*c - a*d)^4*(a + b*x)) - d^3/(2*c*(b*c -
a*d)^3*(c + d*x)^2) + (d^3*(-4*b*c + a*d))/(c^2*(b*c - a*d)^4*(c + d*x)) + Log[x]/(a^3*c^3) + (b^3*(b^2*c^2 -
5*a*b*c*d + 10*a^2*d^2)*Log[a + b*x])/(a^3*(-(b*c) + a*d)^5) + (d^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[c +
d*x])/(c^3*(b*c - a*d)^5)

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Maple [A]  time = 0.014, size = 322, normalized size = 1.5 \begin{align*}{\frac{{d}^{3}}{2\,c \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{d}^{4}a}{{c}^{2} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-4\,{\frac{{d}^{3}b}{c \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-{\frac{{d}^{5}\ln \left ( dx+c \right ){a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{5}}}+5\,{\frac{{d}^{4}\ln \left ( dx+c \right ) ab}{{c}^{2} \left ( ad-bc \right ) ^{5}}}-10\,{\frac{{d}^{3}\ln \left ( dx+c \right ){b}^{2}}{c \left ( ad-bc \right ) ^{5}}}+{\frac{\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}-{\frac{{b}^{3}}{2\, \left ( ad-bc \right ) ^{3}a \left ( bx+a \right ) ^{2}}}-4\,{\frac{{b}^{3}d}{ \left ( ad-bc \right ) ^{4}a \left ( bx+a \right ) }}+{\frac{{b}^{4}c}{ \left ( ad-bc \right ) ^{4}{a}^{2} \left ( bx+a \right ) }}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ){d}^{2}}{ \left ( ad-bc \right ) ^{5}a}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) cd}{ \left ( ad-bc \right ) ^{5}{a}^{2}}}+{\frac{{b}^{5}\ln \left ( bx+a \right ){c}^{2}}{ \left ( ad-bc \right ) ^{5}{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.25224, size = 1085, normalized size = 4.91 \begin{align*} -\frac{{\left (b^{5} c^{2} - 5 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} \log \left (b x + a\right )}{a^{3} b^{5} c^{5} - 5 \, a^{4} b^{4} c^{4} d + 10 \, a^{5} b^{3} c^{3} d^{2} - 10 \, a^{6} b^{2} c^{2} d^{3} + 5 \, a^{7} b c d^{4} - a^{8} d^{5}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 5 \, a b c d^{4} + a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{5} c^{8} - 5 \, a b^{4} c^{7} d + 10 \, a^{2} b^{3} c^{6} d^{2} - 10 \, a^{3} b^{2} c^{5} d^{3} + 5 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}} + \frac{3 \, a b^{4} c^{5} - 9 \, a^{2} b^{3} c^{4} d - 9 \, a^{4} b c^{2} d^{3} + 3 \, a^{5} c d^{4} + 2 \,{\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} - 4 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{3} +{\left (4 \, b^{5} c^{4} d - 13 \, a b^{4} c^{3} d^{2} - 18 \, a^{2} b^{3} c^{2} d^{3} - 13 \, a^{3} b^{2} c d^{4} + 4 \, a^{4} b d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{5} - a b^{4} c^{4} d - 9 \, a^{2} b^{3} c^{3} d^{2} - 9 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} + a^{5} d^{5}\right )} x}{2 \,{\left (a^{4} b^{4} c^{8} - 4 \, a^{5} b^{3} c^{7} d + 6 \, a^{6} b^{2} c^{6} d^{2} - 4 \, a^{7} b c^{5} d^{3} + a^{8} c^{4} d^{4} +{\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6}\right )} x^{4} + 2 \,{\left (a^{2} b^{6} c^{7} d - 3 \, a^{3} b^{5} c^{6} d^{2} + 2 \, a^{4} b^{4} c^{5} d^{3} + 2 \, a^{5} b^{3} c^{4} d^{4} - 3 \, a^{6} b^{2} c^{3} d^{5} + a^{7} b c^{2} d^{6}\right )} x^{3} +{\left (a^{2} b^{6} c^{8} - 9 \, a^{4} b^{4} c^{6} d^{2} + 16 \, a^{5} b^{3} c^{5} d^{3} - 9 \, a^{6} b^{2} c^{4} d^{4} + a^{8} c^{2} d^{6}\right )} x^{2} + 2 \,{\left (a^{3} b^{5} c^{8} - 3 \, a^{4} b^{4} c^{7} d + 2 \, a^{5} b^{3} c^{6} d^{2} + 2 \, a^{6} b^{2} c^{5} d^{3} - 3 \, a^{7} b c^{4} d^{4} + a^{8} c^{3} d^{5}\right )} x\right )}} + \frac{\log \left (x\right )}{a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^5*c^2 - 5*a*b^4*c*d + 10*a^2*b^3*d^2)*log(b*x + a)/(a^3*b^5*c^5 - 5*a^4*b^4*c^4*d + 10*a^5*b^3*c^3*d^2 - 1
0*a^6*b^2*c^2*d^3 + 5*a^7*b*c*d^4 - a^8*d^5) + (10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(d*x + c)/(b^5*c^8
- 5*a*b^4*c^7*d + 10*a^2*b^3*c^6*d^2 - 10*a^3*b^2*c^5*d^3 + 5*a^4*b*c^4*d^4 - a^5*c^3*d^5) + 1/2*(3*a*b^4*c^5
- 9*a^2*b^3*c^4*d - 9*a^4*b*c^2*d^3 + 3*a^5*c*d^4 + 2*(b^5*c^3*d^2 - 4*a*b^4*c^2*d^3 - 4*a^2*b^3*c*d^4 + a^3*b
^2*d^5)*x^3 + (4*b^5*c^4*d - 13*a*b^4*c^3*d^2 - 18*a^2*b^3*c^2*d^3 - 13*a^3*b^2*c*d^4 + 4*a^4*b*d^5)*x^2 + 2*(
b^5*c^5 - a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2 - 9*a^3*b^2*c^2*d^3 - a^4*b*c*d^4 + a^5*d^5)*x)/(a^4*b^4*c^8 - 4*a^5
*b^3*c^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4 + (a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + 6*a^4*
b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6)*x^4 + 2*(a^2*b^6*c^7*d - 3*a^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*
d^3 + 2*a^5*b^3*c^4*d^4 - 3*a^6*b^2*c^3*d^5 + a^7*b*c^2*d^6)*x^3 + (a^2*b^6*c^8 - 9*a^4*b^4*c^6*d^2 + 16*a^5*b
^3*c^5*d^3 - 9*a^6*b^2*c^4*d^4 + a^8*c^2*d^6)*x^2 + 2*(a^3*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a
^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^5)*x) + log(x)/(a^3*c^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/x/(b*x+a)^3/(d*x+c)^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/x/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError