Optimal. Leaf size=126 \[ \frac {b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^2}-\frac {\log (x) (a d+2 b c)}{a^3 c^2}-\frac {b^2}{2 a^2 \left (a+b x^2\right ) (b c-a d)}-\frac {1}{2 a^2 c x^2}+\frac {d^3 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 88} \begin {gather*} -\frac {b^2}{2 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac {b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^2}-\frac {\log (x) (a d+2 b c)}{a^3 c^2}-\frac {1}{2 a^2 c x^2}+\frac {d^3 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 c x^2}+\frac {-2 b c-a d}{a^3 c^2 x}-\frac {b^3}{a^2 (-b c+a d) (a+b x)^2}-\frac {b^3 (-2 b c+3 a d)}{a^3 (-b c+a d)^2 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^2 c x^2}-\frac {b^2}{2 a^2 (b c-a d) \left (a+b x^2\right )}-\frac {(2 b c+a d) \log (x)}{a^3 c^2}+\frac {b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^2}+\frac {d^3 \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 119, normalized size = 0.94 \begin {gather*} \frac {1}{2} \left (\frac {b^2 (2 b c-3 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^2}-\frac {2 \log (x) (a d+2 b c)}{a^3 c^2}+\frac {b^2}{a^2 \left (a+b x^2\right ) (a d-b c)}-\frac {1}{a^2 c x^2}+\frac {d^3 \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 5.58, size = 303, normalized size = 2.40 \begin {gather*} -\frac {a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d + a^{4} c d^{2} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2} - {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - {\left (a^{3} b d^{3} x^{4} + a^{4} d^{3} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left ({\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{3} d + a^{5} b c^{2} d^{2}\right )} x^{4} + {\left (a^{4} b^{2} c^{4} - 2 \, a^{5} b c^{3} d + a^{6} c^{2} d^{2}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 257, normalized size = 2.04 \begin {gather*} \frac {d^{4} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} + \frac {{\left (2 \, b^{4} c - 3 \, a b^{3} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} + \frac {a^{2} b d^{3} x^{4} - 4 \, b^{3} c^{3} x^{2} + 6 \, a b^{2} c^{2} d x^{2} - 2 \, a^{2} b c d^{2} x^{2} + a^{3} d^{3} x^{2} - 2 \, a b^{2} c^{3} + 4 \, a^{2} b c^{2} d - 2 \, a^{3} c d^{2}}{4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left (b x^{4} + a x^{2}\right )}} - \frac {{\left (2 \, b c + a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 170, normalized size = 1.35 \begin {gather*} \frac {b^{2} d}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a}-\frac {b^{3} c}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b^{2} d \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2} a^{2}}+\frac {b^{3} c \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{2} a^{3}}+\frac {d^{3} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2} c^{2}}-\frac {d \ln \relax (x )}{a^{2} c^{2}}-\frac {2 b \ln \relax (x )}{a^{3} c}-\frac {1}{2 a^{2} c \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 189, normalized size = 1.50 \begin {gather*} \frac {d^{3} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} + \frac {{\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}} - \frac {a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{4} + {\left (a^{3} b c^{2} - a^{4} c d\right )} x^{2}\right )}} - \frac {{\left (2 \, b c + a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 171, normalized size = 1.36 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (2\,b^3\,c-3\,a\,b^2\,d\right )}{2\,a^5\,d^2-4\,a^4\,b\,c\,d+2\,a^3\,b^2\,c^2}-\frac {\frac {1}{2\,a\,c}-\frac {x^2\,\left (2\,b^2\,c-a\,b\,d\right )}{2\,a^2\,c\,\left (a\,d-b\,c\right )}}{b\,x^4+a\,x^2}+\frac {d^3\,\ln \left (d\,x^2+c\right )}{2\,\left (a^2\,c^2\,d^2-2\,a\,b\,c^3\,d+b^2\,c^4\right )}-\frac {\ln \relax (x)\,\left (a\,d+2\,b\,c\right )}{a^3\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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