Optimal. Leaf size=51 \[ \frac {d^2 x^2}{2 b}+\frac {c^2 \log (x)}{a}-\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 51, 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 = {457, 84}
\begin {gather*} -\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a b^2}+\frac {c^2 \log (x)}{a}+\frac {d^2 x^2}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^2}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {d^2}{b}+\frac {c^2}{a x}-\frac {(-b c+a d)^2}{a b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {d^2 x^2}{2 b}+\frac {c^2 \log (x)}{a}-\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.98 \begin {gather*} \frac {a b d^2 x^2+2 b^2 c^2 \log (x)-(b c-a d)^2 \log \left (a+b x^2\right )}{2 a b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 59, normalized size = 1.16
method | result | size |
default | \(\frac {d^{2} x^{2}}{2 b}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 a \,b^{2}}+\frac {c^{2} \ln \left (x \right )}{a}\) | \(59\) |
norman | \(\frac {d^{2} x^{2}}{2 b}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 a \,b^{2}}+\frac {c^{2} \ln \left (x \right )}{a}\) | \(59\) |
risch | \(\frac {d^{2} x^{2}}{2 b}-\frac {a \ln \left (b \,x^{2}+a \right ) d^{2}}{2 b^{2}}+\frac {\ln \left (b \,x^{2}+a \right ) c d}{b}-\frac {\ln \left (b \,x^{2}+a \right ) c^{2}}{2 a}+\frac {c^{2} \ln \left (x \right )}{a}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 61, normalized size = 1.20 \begin {gather*} \frac {d^{2} x^{2}}{2 \, b} + \frac {c^{2} \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 59, normalized size = 1.16 \begin {gather*} \frac {a b d^{2} x^{2} + 2 \, b^{2} c^{2} \log \left (x\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.77, size = 41, normalized size = 0.80 \begin {gather*} \frac {d^{2} x^{2}}{2 b} + \frac {c^{2} \log {\left (x \right )}}{a} - \frac {\left (a d - b c\right )^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 62, normalized size = 1.22 \begin {gather*} \frac {d^{2} x^{2}}{2 \, b} + \frac {c^{2} \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 58, normalized size = 1.14 \begin {gather*} \frac {d^2\,x^2}{2\,b}+\frac {c^2\,\ln \left (x\right )}{a}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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