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

Optimal. Leaf size=160 \[ \frac {b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac {d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}-\frac {\log (x) (3 a d+b c)}{a^2 c^4}+\frac {d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac {d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac {1}{a c^3 x} \]

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Rubi [A]  time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}+\frac {b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac {\log (x) (3 a d+b c)}{a^2 c^4}+\frac {d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac {d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac {1}{a c^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)*(c + d*x)^3),x]

[Out]

-(1/(a*c^3*x)) + d^2/(2*c^2*(b*c - a*d)*(c + d*x)^2) + (d^2*(3*b*c - 2*a*d))/(c^3*(b*c - a*d)^2*(c + d*x)) - (
(b*c + 3*a*d)*Log[x])/(a^2*c^4) + (b^4*Log[a + b*x])/(a^2*(b*c - a*d)^3) - (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2
*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^3)

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^2 (a+b x) (c+d x)^3} \, dx &=\int \left (\frac {1}{a c^3 x^2}+\frac {-b c-3 a d}{a^2 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)^3}-\frac {d^3 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)^2}-\frac {d^3 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {1}{a c^3 x}+\frac {d^2}{2 c^2 (b c-a d) (c+d x)^2}+\frac {d^2 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)}-\frac {(b c+3 a d) \log (x)}{a^2 c^4}+\frac {b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 163, normalized size = 1.02 \begin {gather*} -\frac {b^4 \log (a+b x)}{a^2 (a d-b c)^3}-\frac {\left (3 a^2 d^4-8 a b c d^3+6 b^2 c^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}+\frac {\log (x) (-3 a d-b c)}{a^2 c^4}+\frac {d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac {d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac {1}{a c^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a*c^3*x)) + d^2/(2*c^2*(b*c - a*d)*(c + d*x)^2) + (d^2*(3*b*c - 2*a*d))/(c^3*(b*c - a*d)^2*(c + d*x)) + (
(-(b*c) - 3*a*d)*Log[x])/(a^2*c^4) - (b^4*Log[a + b*x])/(a^2*(-(b*c) + a*d)^3) - ((6*b^2*c^2*d^2 - 8*a*b*c*d^3
+ 3*a^2*d^4)*Log[c + d*x])/(c^4*(b*c - a*d)^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 (a+b x) (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 24.25, size = 626, normalized size = 3.91 \begin {gather*} -\frac {2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \, {\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} + {\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x - 2 \, {\left (b^{4} c^{4} d^{2} x^{3} + 2 \, b^{4} c^{5} d x^{2} + b^{4} c^{6} x\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (6 \, a^{2} b^{2} c^{2} d^{4} - 8 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{3} + 2 \, {\left (6 \, a^{2} b^{2} c^{3} d^{3} - 8 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c d^{5}\right )} x^{2} + {\left (6 \, a^{2} b^{2} c^{4} d^{2} - 8 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4}\right )} x\right )} \log \left (d x + c\right ) + 2 \, {\left ({\left (b^{4} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{2} d^{4} + 8 \, a^{3} b c d^{5} - 3 \, a^{4} d^{6}\right )} x^{3} + 2 \, {\left (b^{4} c^{5} d - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} + {\left (b^{4} c^{6} - 6 \, a^{2} b^{2} c^{4} d^{2} + 8 \, a^{3} b c^{3} d^{3} - 3 \, a^{4} c^{2} d^{4}\right )} x\right )} \log \relax (x)}{2 \, {\left ({\left (a^{2} b^{3} c^{7} d^{2} - 3 \, a^{3} b^{2} c^{6} d^{3} + 3 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}\right )} x^{3} + 2 \, {\left (a^{2} b^{3} c^{8} d - 3 \, a^{3} b^{2} c^{7} d^{2} + 3 \, a^{4} b c^{6} d^{3} - a^{5} c^{5} d^{4}\right )} x^{2} + {\left (a^{2} b^{3} c^{9} - 3 \, a^{3} b^{2} c^{8} d + 3 \, a^{4} b c^{7} d^{2} - a^{5} c^{6} d^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*b^3*c^6 - 6*a^2*b^2*c^5*d + 6*a^3*b*c^4*d^2 - 2*a^4*c^3*d^3 + 2*(a*b^3*c^4*d^2 - 6*a^2*b^2*c^3*d^3 +
8*a^3*b*c^2*d^4 - 3*a^4*c*d^5)*x^2 + (4*a*b^3*c^5*d - 19*a^2*b^2*c^4*d^2 + 24*a^3*b*c^3*d^3 - 9*a^4*c^2*d^4)*
x - 2*(b^4*c^4*d^2*x^3 + 2*b^4*c^5*d*x^2 + b^4*c^6*x)*log(b*x + a) + 2*((6*a^2*b^2*c^2*d^4 - 8*a^3*b*c*d^5 + 3
*a^4*d^6)*x^3 + 2*(6*a^2*b^2*c^3*d^3 - 8*a^3*b*c^2*d^4 + 3*a^4*c*d^5)*x^2 + (6*a^2*b^2*c^4*d^2 - 8*a^3*b*c^3*d
^3 + 3*a^4*c^2*d^4)*x)*log(d*x + c) + 2*((b^4*c^4*d^2 - 6*a^2*b^2*c^2*d^4 + 8*a^3*b*c*d^5 - 3*a^4*d^6)*x^3 + 2
*(b^4*c^5*d - 6*a^2*b^2*c^3*d^3 + 8*a^3*b*c^2*d^4 - 3*a^4*c*d^5)*x^2 + (b^4*c^6 - 6*a^2*b^2*c^4*d^2 + 8*a^3*b*
c^3*d^3 - 3*a^4*c^2*d^4)*x)*log(x))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^6*d^3 + 3*a^4*b*c^5*d^4 - a^5*c^4*d^5)*x^3
+ 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7*d^2 + 3*a^4*b*c^6*d^3 - a^5*c^5*d^4)*x^2 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d
+ 3*a^4*b*c^7*d^2 - a^5*c^6*d^3)*x)

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giac [B]  time = 0.98, size = 335, normalized size = 2.09 \begin {gather*} \frac {b^{5} \log \left ({\left | b x + a \right |}\right )}{a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}} - \frac {{\left (6 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}} - \frac {{\left (b c + 3 \, a d\right )} \log \left ({\left | x \right |}\right )}{a^{2} c^{4}} - \frac {2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \, {\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} + {\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} a^{2} c^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^5*log(abs(b*x + a))/(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3) - (6*b^2*c^2*d^3 - 8*a*b*c
*d^4 + 3*a^2*d^5)*log(abs(d*x + c))/(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c^4*d^4) - (b*c + 3*a
*d)*log(abs(x))/(a^2*c^4) - 1/2*(2*a*b^3*c^6 - 6*a^2*b^2*c^5*d + 6*a^3*b*c^4*d^2 - 2*a^4*c^3*d^3 + 2*(a*b^3*c^
4*d^2 - 6*a^2*b^2*c^3*d^3 + 8*a^3*b*c^2*d^4 - 3*a^4*c*d^5)*x^2 + (4*a*b^3*c^5*d - 19*a^2*b^2*c^4*d^2 + 24*a^3*
b*c^3*d^3 - 9*a^4*c^2*d^4)*x)/((b*c - a*d)^3*(d*x + c)^2*a^2*c^4*x)

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maple [A]  time = 0.02, size = 216, normalized size = 1.35 \begin {gather*} \frac {3 a^{2} d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{4}}-\frac {8 a b \,d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{3}}-\frac {b^{4} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{2}}+\frac {6 b^{2} d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{2}}-\frac {2 a \,d^{3}}{\left (a d -b c \right )^{2} \left (d x +c \right ) c^{3}}+\frac {3 b \,d^{2}}{\left (a d -b c \right )^{2} \left (d x +c \right ) c^{2}}-\frac {d^{2}}{2 \left (a d -b c \right ) \left (d x +c \right )^{2} c^{2}}-\frac {3 d \ln \relax (x )}{a \,c^{4}}-\frac {b \ln \relax (x )}{a^{2} c^{3}}-\frac {1}{a \,c^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/c^3/x-3/a/c^4*ln(x)*d-1/a^2/c^3*ln(x)*b-1/2*d^2/c^2/(a*d-b*c)/(d*x+c)^2-2*d^3/c^3/(a*d-b*c)^2/(d*x+c)*a+3
*d^2/c^2/(a*d-b*c)^2/(d*x+c)*b+3*d^4/c^4/(a*d-b*c)^3*ln(d*x+c)*a^2-8*d^3/c^3/(a*d-b*c)^3*ln(d*x+c)*a*b+6*d^2/c
^2/(a*d-b*c)^3*ln(d*x+c)*b^2-b^4/a^2/(a*d-b*c)^3*ln(b*x+a)

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maxima [B]  time = 1.27, size = 353, normalized size = 2.21 \begin {gather*} \frac {b^{4} \log \left (b x + a\right )}{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}} - \frac {{\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}} - \frac {2 \, b^{2} c^{4} - 4 \, a b c^{3} d + 2 \, a^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{2} d^{2} - 5 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + {\left (4 \, b^{2} c^{3} d - 15 \, a b c^{2} d^{2} + 9 \, a^{2} c d^{3}\right )} x}{2 \, {\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{3} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x\right )}} - \frac {{\left (b c + 3 \, a d\right )} \log \relax (x)}{a^{2} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.93, size = 302, normalized size = 1.89 \begin {gather*} -\frac {\frac {1}{a\,c}+\frac {x^2\,\left (3\,a^2\,d^4-5\,a\,b\,c\,d^3+b^2\,c^2\,d^2\right )}{a\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (9\,a^2\,d^3-15\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )}{2\,a\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2\,x+2\,c\,d\,x^2+d^2\,x^3}-\frac {b^4\,\ln \left (a+b\,x\right )}{a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,d^4-8\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{-a^3\,c^4\,d^3+3\,a^2\,b\,c^5\,d^2-3\,a\,b^2\,c^6\,d+b^3\,c^7}-\frac {\ln \relax (x)\,\left (3\,a\,d+b\,c\right )}{a^2\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x)*(c + d*x)^3),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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