Optimal. Leaf size=86 \[ \frac {(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac {\frac {a^2}{c^2}-\frac {b^2}{d^2}}{2 \left (c+d x^2\right )}+\frac {a^2 \log (x)}{c^3}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 86, 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 {\frac {a^2}{c^2}-\frac {b^2}{d^2}}{2 \left (c+d x^2\right )}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac {a^2 \log (x)}{c^3}+\frac {(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{c^3 x}-\frac {(b c-a d)^2}{c d (c+d x)^3}+\frac {b^2 c^2-a^2 d^2}{c^2 d (c+d x)^2}-\frac {a^2 d}{c^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac {\frac {a^2}{c^2}-\frac {b^2}{d^2}}{2 \left (c+d x^2\right )}+\frac {a^2 \log (x)}{c^3}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 103, normalized size = 1.20 \begin {gather*} \frac {b^2 c^2-2 a b c d+a^2 d^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac {-b^2 c^2+a^2 d^2}{2 c^2 d^2 \left (c+d x^2\right )}+\frac {a^2 \log (x)}{c^3}-\frac {a^2 \log \left (c+d x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 98, normalized size = 1.14
method | result | size |
norman | \(\frac {-\frac {\left (a^{2} d -a b c \right ) x^{2}}{c^{2}}-\frac {\left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x^{4}}{4 c^{3}}}{\left (d \,x^{2}+c \right )^{2}}+\frac {a^{2} \ln \left (x \right )}{c^{3}}-\frac {a^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{3}}\) | \(88\) |
risch | \(\frac {\frac {\left (a^{2} d^{2}-b^{2} c^{2}\right ) x^{2}}{2 c^{2} d}+\frac {3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}}{4 d^{2} c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {a^{2} \ln \left (x \right )}{c^{3}}-\frac {a^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{3}}\) | \(96\) |
default | \(-\frac {-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{2} \left (d \,x^{2}+c \right )^{2}}-\frac {c \left (a^{2} d^{2}-b^{2} c^{2}\right )}{d^{2} \left (d \,x^{2}+c \right )}+a^{2} \ln \left (d \,x^{2}+c \right )}{2 c^{3}}+\frac {a^{2} \ln \left (x \right )}{c^{3}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 109, normalized size = 1.27 \begin {gather*} -\frac {b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2} + 2 \, {\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}}{4 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )}} - \frac {a^{2} \log \left (d x^{2} + c\right )}{2 \, c^{3}} + \frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs.
\(2 (80) = 160\).
time = 1.03, size = 163, normalized size = 1.90 \begin {gather*} -\frac {b^{2} c^{4} + 2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{3} d - a^{2} c d^{3}\right )} x^{2} + 2 \, {\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \left (x\right )}{4 \, {\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.70, size = 107, normalized size = 1.24 \begin {gather*} \frac {a^{2} \log {\left (x \right )}}{c^{3}} - \frac {a^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{3}} + \frac {3 a^{2} c d^{2} - 2 a b c^{2} d - b^{2} c^{3} + x^{2} \cdot \left (2 a^{2} d^{3} - 2 b^{2} c^{2} d\right )}{4 c^{4} d^{2} + 8 c^{3} d^{3} x^{2} + 4 c^{2} d^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 110, normalized size = 1.28 \begin {gather*} \frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac {a^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{3}} + \frac {3 \, a^{2} d^{4} x^{4} - 2 \, b^{2} c^{3} d x^{2} + 8 \, a^{2} c d^{3} x^{2} - b^{2} c^{4} - 2 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}}{4 \, {\left (d x^{2} + c\right )}^{2} c^{3} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 106, normalized size = 1.23 \begin {gather*} \frac {a^2\,\ln \left (x\right )}{c^3}-\frac {a^2\,\ln \left (d\,x^2+c\right )}{2\,c^3}-\frac {\frac {-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2}{4\,c\,d^2}-\frac {x^2\,\left (a^2\,d^2-b^2\,c^2\right )}{2\,c^2\,d}}{c^2+2\,c\,d\,x^2+d^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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