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

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

[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)*L
og[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 = 0.528035, 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 \[ \frac{\log (x)}{a^3 c^3}+\frac{b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac{d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}-\frac{b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac{b^3}{2 a (a+b x)^2 (b c-a d)^3}-\frac{d^3 (4 b c-a d)}{c^2 (c+d x) (b c-a d)^4}-\frac{d^3}{2 c (c+d x)^2 (b c-a d)^3} \]

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)*L
og[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 in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.637465, size = 218, normalized size = 0.99 \[ \frac{\log (x)}{a^3 c^3}+\frac{b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac{d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}+\frac{b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (a d-b c)^5}-\frac{b^3}{2 a (a+b x)^2 (a d-b c)^3}+\frac{d^3 (a d-4 b c)}{c^2 (c+d x) (b c-a d)^4}-\frac{d^3}{2 c (c+d x)^2 (b c-a d)^3} \]

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 = 0.025, size = 322, normalized size = 1.5 \[{\frac{{d}^{3}}{2\,c \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{d}^{4}a}{{c}^{2} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-4\,{\frac{{d}^{3}b}{c \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-{\frac{{d}^{5}\ln \left ( dx+c \right ){a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{5}}}+5\,{\frac{{d}^{4}\ln \left ( dx+c \right ) ab}{{c}^{2} \left ( ad-bc \right ) ^{5}}}-10\,{\frac{{d}^{3}\ln \left ( dx+c \right ){b}^{2}}{c \left ( ad-bc \right ) ^{5}}}+{\frac{\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}-{\frac{{b}^{3}}{2\, \left ( ad-bc \right ) ^{3}a \left ( bx+a \right ) ^{2}}}-4\,{\frac{{b}^{3}d}{ \left ( ad-bc \right ) ^{4}a \left ( bx+a \right ) }}+{\frac{{b}^{4}c}{ \left ( ad-bc \right ) ^{4}{a}^{2} \left ( bx+a \right ) }}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ){d}^{2}}{ \left ( ad-bc \right ) ^{5}a}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) cd}{ \left ( ad-bc \right ) ^{5}{a}^{2}}}+{\frac{{b}^{5}\ln \left ( bx+a \right ){c}^{2}}{ \left ( ad-bc \right ) ^{5}{a}^{3}}} \]

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 [A]  time = 1.4323, size = 1085, normalized size = 4.91 \[ -\frac{{\left (b^{5} c^{2} - 5 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} \log \left (b x + a\right )}{a^{3} b^{5} c^{5} - 5 \, a^{4} b^{4} c^{4} d + 10 \, a^{5} b^{3} c^{3} d^{2} - 10 \, a^{6} b^{2} c^{2} d^{3} + 5 \, a^{7} b c d^{4} - a^{8} d^{5}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 5 \, a b c d^{4} + a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{5} c^{8} - 5 \, a b^{4} c^{7} d + 10 \, a^{2} b^{3} c^{6} d^{2} - 10 \, a^{3} b^{2} c^{5} d^{3} + 5 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}} + \frac{3 \, a b^{4} c^{5} - 9 \, a^{2} b^{3} c^{4} d - 9 \, a^{4} b c^{2} d^{3} + 3 \, a^{5} c d^{4} + 2 \,{\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} - 4 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{3} +{\left (4 \, b^{5} c^{4} d - 13 \, a b^{4} c^{3} d^{2} - 18 \, a^{2} b^{3} c^{2} d^{3} - 13 \, a^{3} b^{2} c d^{4} + 4 \, a^{4} b d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{5} - a b^{4} c^{4} d - 9 \, a^{2} b^{3} c^{3} d^{2} - 9 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} + a^{5} d^{5}\right )} x}{2 \,{\left (a^{4} b^{4} c^{8} - 4 \, a^{5} b^{3} c^{7} d + 6 \, a^{6} b^{2} c^{6} d^{2} - 4 \, a^{7} b c^{5} d^{3} + a^{8} c^{4} d^{4} +{\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6}\right )} x^{4} + 2 \,{\left (a^{2} b^{6} c^{7} d - 3 \, a^{3} b^{5} c^{6} d^{2} + 2 \, a^{4} b^{4} c^{5} d^{3} + 2 \, a^{5} b^{3} c^{4} d^{4} - 3 \, a^{6} b^{2} c^{3} d^{5} + a^{7} b c^{2} d^{6}\right )} x^{3} +{\left (a^{2} b^{6} c^{8} - 9 \, a^{4} b^{4} c^{6} d^{2} + 16 \, a^{5} b^{3} c^{5} d^{3} - 9 \, a^{6} b^{2} c^{4} d^{4} + a^{8} c^{2} d^{6}\right )} x^{2} + 2 \,{\left (a^{3} b^{5} c^{8} - 3 \, a^{4} b^{4} c^{7} d + 2 \, a^{5} b^{3} c^{6} d^{2} + 2 \, a^{6} b^{2} c^{5} d^{3} - 3 \, a^{7} b c^{4} d^{4} + a^{8} c^{3} d^{5}\right )} x\right )}} + \frac{\log \left (x\right )}{a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*(d*x + c)^3*x),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 - 10*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 [A]  time = 48.9203, size = 2201, normalized size = 9.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 9*a^4*b^3*c^5*d^2 - 9*a^5*b^2*c^4*d^3 +
12*a^6*b*c^3*d^4 - 3*a^7*c^2*d^5 + 2*(a*b^6*c^5*d^2 - 5*a^2*b^5*c^4*d^3 + 5*a^4*
b^3*c^2*d^5 - a^5*b^2*c*d^6)*x^3 + (4*a*b^6*c^6*d - 17*a^2*b^5*c^5*d^2 - 5*a^3*b
^4*c^4*d^3 + 5*a^4*b^3*c^3*d^4 + 17*a^5*b^2*c^2*d^5 - 4*a^6*b*c*d^6)*x^2 + 2*(a*
b^6*c^7 - 2*a^2*b^5*c^6*d - 8*a^3*b^4*c^5*d^2 + 8*a^5*b^2*c^3*d^4 + 2*a^6*b*c^2*
d^5 - a^7*c*d^6)*x - 2*(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 + (b^
7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c
^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4)*x^3 + (b^7*c^7 - a*b^6*c^6*d -
9*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4)*x^2 + 2*(a*b^6*c^7
- 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 + 10*a^4*b^3*c^4*d^3)*x)*log(b*x + a) + 2*
(10*a^5*b^2*c^4*d^3 - 5*a^6*b*c^3*d^4 + a^7*c^2*d^5 + (10*a^3*b^4*c^2*d^5 - 5*a^
4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(10*a^3*b^4*c^3*d^4 + 5*a^4*b^3*c^2*d^5 - 4*a
^5*b^2*c*d^6 + a^6*b*d^7)*x^3 + (10*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 9*a^5
*b^2*c^2*d^5 - a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(10*a^4*b^3*c^4*d^3 + 5*a^5*b^2*c^
3*d^4 - 4*a^6*b*c^2*d^5 + a^7*c*d^6)*x)*log(d*x + c) + 2*(a^2*b^5*c^7 - 5*a^3*b^
4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^
5 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5
*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c
^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6
*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2
*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b
^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)*log(x))/(a^5*b^
5*c^10 - 5*a^6*b^4*c^9*d + 10*a^7*b^3*c^8*d^2 - 10*a^8*b^2*c^7*d^3 + 5*a^9*b*c^6
*d^4 - a^10*c^5*d^5 + (a^3*b^7*c^8*d^2 - 5*a^4*b^6*c^7*d^3 + 10*a^5*b^5*c^6*d^4
- 10*a^6*b^4*c^5*d^5 + 5*a^7*b^3*c^4*d^6 - a^8*b^2*c^3*d^7)*x^4 + 2*(a^3*b^7*c^9
*d - 4*a^4*b^6*c^8*d^2 + 5*a^5*b^5*c^7*d^3 - 5*a^7*b^3*c^5*d^5 + 4*a^8*b^2*c^4*d
^6 - a^9*b*c^3*d^7)*x^3 + (a^3*b^7*c^10 - a^4*b^6*c^9*d - 9*a^5*b^5*c^8*d^2 + 25
*a^6*b^4*c^7*d^3 - 25*a^7*b^3*c^6*d^4 + 9*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6 - a^10
*c^3*d^7)*x^2 + 2*(a^4*b^6*c^10 - 4*a^5*b^5*c^9*d + 5*a^6*b^4*c^8*d^2 - 5*a^8*b^
2*c^6*d^4 + 4*a^9*b*c^5*d^5 - a^10*c^4*d^6)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError