Optimal. Leaf size=98 \[ \frac {(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{2 a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^3}{2 a^2 x^2} \]
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Rubi [A] time = 0.11, antiderivative size = 98, 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 c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}+\frac {(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{2 a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {c^3}{2 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c^3}{a^2 x^2}+\frac {c^2 (-2 b c+3 a d)}{a^3 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)^2}+\frac {(-b c+a d)^2 (2 b c+a d)}{a^3 b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c^3}{2 a^2 x^2}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{2 a^3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 87, normalized size = 0.89 \begin {gather*} \frac {\frac {a (a d-b c)^3}{b^2 \left (a+b x^2\right )}+\frac {(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{b^2}+2 c^2 \log (x) (3 a d-2 b c)-\frac {a c^3}{x^2}}{2 a^3} \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 {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.89, size = 209, normalized size = 2.13 \begin {gather*} -\frac {a^{2} b^{2} c^{3} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{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 \left (b x^{2} + a\right ) + 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 \relax (x)}{2 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 157, normalized size = 1.60 \begin {gather*} -\frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b^{2}} - \frac {a^{2} b d^{3} x^{4} + 4 \, b^{3} c^{3} x^{2} - 6 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 156, normalized size = 1.59 \begin {gather*} \frac {a \,d^{3}}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {3 c^{2} d}{2 \left (b \,x^{2}+a \right ) a}-\frac {b \,c^{3}}{2 \left (b \,x^{2}+a \right ) a^{2}}+\frac {3 c^{2} d \ln \relax (x )}{a^{2}}-\frac {3 c^{2} d \ln \left (b \,x^{2}+a \right )}{2 a^{2}}-\frac {2 b \,c^{3} \ln \relax (x )}{a^{3}}+\frac {b \,c^{3} \ln \left (b \,x^{2}+a \right )}{a^{3}}-\frac {3 c \,d^{2}}{2 \left (b \,x^{2}+a \right ) b}+\frac {d^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{2}}-\frac {c^{3}}{2 a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 141, normalized size = 1.44 \begin {gather*} -\frac {a b^{2} c^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x^{2}\right )}} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 135, normalized size = 1.38 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{2\,a^3\,b^2}-\frac {\ln \relax (x)\,\left (2\,b\,c^3-3\,a\,c^2\,d\right )}{a^3}-\frac {\frac {c^3}{2\,a}-\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,b^2}}{b\,x^4+a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.18, size = 128, normalized size = 1.31 \begin {gather*} \frac {- a b^{2} c^{3} + x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{4}} + \frac {c^{2} \left (3 a d - 2 b c\right ) \log {\relax (x )}}{a^{3}} + \frac {\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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