Optimal. Leaf size=58 \[ -\frac {c^2}{2 a x^2}-\frac {c (b c-2 a d) \log (x)}{a^2}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 58, 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, 90}
\begin {gather*} \frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b}-\frac {c \log (x) (b c-2 a d)}{a^2}-\frac {c^2}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^2}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {c^2}{a x^2}+\frac {c (-b c+2 a d)}{a^2 x}+\frac {(-b c+a d)^2}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c^2}{2 a x^2}-\frac {c (b c-2 a d) \log (x)}{a^2}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 60, normalized size = 1.03 \begin {gather*} \frac {-a b c^2-2 b c (b c-2 a d) x^2 \log (x)+(b c-a d)^2 x^2 \log \left (a+b x^2\right )}{2 a^2 b x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 66, normalized size = 1.14
method | result | size |
default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b}-\frac {c^{2}}{2 a \,x^{2}}+\frac {c \left (2 a d -b c \right ) \ln \left (x \right )}{a^{2}}\) | \(66\) |
norman | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b}-\frac {c^{2}}{2 a \,x^{2}}+\frac {c \left (2 a d -b c \right ) \ln \left (x \right )}{a^{2}}\) | \(66\) |
risch | \(-\frac {c^{2}}{2 a \,x^{2}}+\frac {2 c \ln \left (x \right ) d}{a}-\frac {c^{2} \ln \left (x \right ) b}{a^{2}}+\frac {\ln \left (-b \,x^{2}-a \right ) d^{2}}{2 b}-\frac {\ln \left (-b \,x^{2}-a \right ) c d}{a}+\frac {b \ln \left (-b \,x^{2}-a \right ) c^{2}}{2 a^{2}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 69, normalized size = 1.19 \begin {gather*} -\frac {{\left (b c^{2} - 2 \, a c d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {c^{2}}{2 \, a x^{2}} + \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^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 73, normalized size = 1.26 \begin {gather*} -\frac {a b c^{2} - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (b x^{2} + a\right ) + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d\right )} x^{2} \log \left (x\right )}{2 \, a^{2} b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.90, size = 49, normalized size = 0.84 \begin {gather*} - \frac {c^{2}}{2 a x^{2}} + \frac {c \left (2 a d - b c\right ) \log {\left (x \right )}}{a^{2}} + \frac {\left (a d - b c\right )^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 90, normalized size = 1.55 \begin {gather*} -\frac {{\left (b c^{2} - 2 \, a c d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \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^{2} b} + \frac {b c^{2} x^{2} - 2 \, a c d x^{2} - a c^{2}}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 67, normalized size = 1.16 \begin {gather*} \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^2\,b}-\frac {c^2}{2\,a\,x^2}-\frac {\ln \left (x\right )\,\left (b\,c^2-2\,a\,c\,d\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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