Optimal. Leaf size=73 \[ \frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{2 a x^2}+\frac {d^3 x^2}{2 b} \]
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Rubi [A] time = 0.08, antiderivative size = 73, 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} \[ \frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{2 a x^2}+\frac {d^3 x^2}{2 b} \]
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 )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d^3}{b}+\frac {c^3}{a x^2}+\frac {c^2 (-b c+3 a d)}{a^2 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c^3}{2 a x^2}+\frac {d^3 x^2}{2 b}-\frac {c^2 (b c-3 a d) \log (x)}{a^2}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 1.03 \[ \frac {-2 b^2 c^2 x^2 \log (x) (b c-3 a d)+a b \left (a d^3 x^4-b c^3\right )+x^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 105, normalized size = 1.44 \[ \frac {a^{2} b d^{3} x^{4} - a b^{2} c^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x^{2} \log \relax (x)}{2 \, a^{2} b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 120, normalized size = 1.64 \[ \frac {d^{3} x^{2}}{2 \, b} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{2}} + \frac {b c^{3} x^{2} - 3 \, a c^{2} d x^{2} - a c^{3}}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 114, normalized size = 1.56 \[ \frac {d^{3} x^{2}}{2 b}-\frac {a \,d^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {3 c^{2} d \ln \relax (x )}{a}-\frac {3 c^{2} d \ln \left (b \,x^{2}+a \right )}{2 a}-\frac {b \,c^{3} \ln \relax (x )}{a^{2}}+\frac {b \,c^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{2}}+\frac {3 c \,d^{2} \ln \left (b \,x^{2}+a \right )}{2 b}-\frac {c^{3}}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 97, normalized size = 1.33 \[ \frac {d^{3} x^{2}}{2 \, b} - \frac {c^{3}}{2 \, a x^{2}} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 95, normalized size = 1.30 \[ \frac {d^3\,x^2}{2\,b}-\frac {c^3}{2\,a\,x^2}-\frac {\ln \relax (x)\,\left (b\,c^3-3\,a\,c^2\,d\right )}{a^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a^2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.21, size = 63, normalized size = 0.86 \[ \frac {d^{3} x^{2}}{2 b} - \frac {c^{3}}{2 a x^{2}} + \frac {c^{2} \left (3 a d - b c\right ) \log {\relax (x )}}{a^{2}} - \frac {\left (a d - b c\right )^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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