Optimal. Leaf size=73 \[ -\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a b^3}+\frac {d^2 x^2 (3 b c-a d)}{2 b^2}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^4}{4 b} \]
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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, 72} \begin {gather*} \frac {d^2 x^2 (3 b c-a d)}{2 b^2}-\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a b^3}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^4}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x \left (a+b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^3}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d^2 (3 b c-a d)}{b^2}+\frac {c^3}{a x}+\frac {d^3 x}{b}+\frac {(-b c+a d)^3}{a b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {d^2 (3 b c-a d) x^2}{2 b^2}+\frac {d^3 x^4}{4 b}+\frac {c^3 \log (x)}{a}-\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.89 \begin {gather*} \frac {a b d^2 x^2 \left (-2 a d+6 b c+b d x^2\right )-2 (b c-a d)^3 \log \left (a+b x^2\right )+4 b^3 c^3 \log (x)}{4 a b^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 \left (a+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.51, size = 101, normalized size = 1.38 \begin {gather*} \frac {a b^{2} d^{3} x^{4} + 4 \, b^{3} c^{3} \log \relax (x) + 2 \, {\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{2} - 2 \, {\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 )}{4 \, a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 99, normalized size = 1.36 \begin {gather*} \frac {c^{3} \log \left (x^{2}\right )}{2 \, a} + \frac {b d^{3} x^{4} + 6 \, b c d^{2} x^{2} - 2 \, a d^{3} x^{2}}{4 \, b^{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 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 1.59 \begin {gather*} \frac {d^{3} x^{4}}{4 b}-\frac {a \,d^{3} x^{2}}{2 b^{2}}+\frac {3 c \,d^{2} x^{2}}{2 b}+\frac {a^{2} d^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}-\frac {3 a c \,d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {c^{3} \ln \relax (x )}{a}-\frac {c^{3} \ln \left (b \,x^{2}+a \right )}{2 a}+\frac {3 c^{2} d \ln \left (b \,x^{2}+a \right )}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 98, normalized size = 1.34 \begin {gather*} \frac {c^{3} \log \left (x^{2}\right )}{2 \, a} + \frac {b d^{3} x^{4} + 2 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x^{2}}{4 \, b^{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 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 97, normalized size = 1.33 \begin {gather*} \frac {d^3\,x^4}{4\,b}-x^2\,\left (\frac {a\,d^3}{2\,b^2}-\frac {3\,c\,d^2}{2\,b}\right )+\frac {c^3\,\ln \relax (x)}{a}+\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\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.77, size = 65, normalized size = 0.89 \begin {gather*} x^{2} \left (- \frac {a d^{3}}{2 b^{2}} + \frac {3 c d^{2}}{2 b}\right ) + \frac {d^{3} x^{4}}{4 b} + \frac {c^{3} \log {\relax (x )}}{a} + \frac {\left (a d - b c\right )^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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