Optimal. Leaf size=80 \[ -\frac {c^2}{2 a^2 x^2}-\frac {(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {2 c (b c-a d) \log (x)}{a^3}+\frac {c (b c-a d) \log \left (a+b x^2\right )}{a^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 80, 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 {c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac {2 c \log (x) (b c-a d)}{a^3}-\frac {(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {c^2}{2 a^2 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 )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^2}{x^2 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {c^2}{a^2 x^2}+\frac {2 c (-b c+a d)}{a^3 x}+\frac {(-b c+a d)^2}{a^2 (a+b x)^2}-\frac {2 b c (-b c+a d)}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c^2}{2 a^2 x^2}-\frac {(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {2 c (b c-a d) \log (x)}{a^3}+\frac {c (b c-a d) \log \left (a+b x^2\right )}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 72, normalized size = 0.90 \begin {gather*} -\frac {\frac {a c^2}{x^2}+\frac {a (b c-a d)^2}{b \left (a+b x^2\right )}+4 c (b c-a d) \log (x)-2 c (b c-a d) \log \left (a+b x^2\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 77, normalized size = 0.96
method | result | size |
default | \(\frac {\left (a d -b c \right ) \left (-2 c \ln \left (b \,x^{2}+a \right )-\frac {a \left (a d -b c \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{3}}-\frac {c^{2}}{2 a^{2} x^{2}}+\frac {2 c \left (a d -b c \right ) \ln \left (x \right )}{a^{3}}\) | \(77\) |
norman | \(\frac {-\frac {c^{2}}{2 a}+\frac {\left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x^{4}}{2 a^{3}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {2 c \left (a d -b c \right ) \ln \left (x \right )}{a^{3}}-\frac {c \left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{a^{3}}\) | \(92\) |
risch | \(\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x^{2}}{2 a^{2} b}-\frac {c^{2}}{2 a}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {2 c \ln \left (x \right ) d}{a^{2}}-\frac {2 c^{2} \ln \left (x \right ) b}{a^{3}}-\frac {c \ln \left (b \,x^{2}+a \right ) d}{a^{2}}+\frac {c^{2} \ln \left (b \,x^{2}+a \right ) b}{a^{3}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 100, normalized size = 1.25 \begin {gather*} -\frac {a b c^{2} + {\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{2 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )}} + \frac {{\left (b c^{2} - a c d\right )} \log \left (b x^{2} + a\right )}{a^{3}} - \frac {{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{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. 159 vs.
\(2 (76) = 152\).
time = 0.61, size = 159, normalized size = 1.99 \begin {gather*} -\frac {a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 2 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.76, size = 92, normalized size = 1.15 \begin {gather*} \frac {- a b c^{2} + x^{2} \left (- a^{2} d^{2} + 2 a b c d - 2 b^{2} c^{2}\right )}{2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{4}} + \frac {2 c \left (a d - b c\right ) \log {\left (x \right )}}{a^{3}} - \frac {c \left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 109, normalized size = 1.36 \begin {gather*} -\frac {{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} + \frac {{\left (b^{2} c^{2} - a b c d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b} - \frac {2 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + a b c^{2}}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 100, normalized size = 1.25 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (b\,c^2-a\,c\,d\right )}{a^3}-\frac {\frac {c^2}{2\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,b}}{b\,x^4+a\,x^2}-\frac {\ln \left (x\right )\,\left (2\,b\,c^2-2\,a\,c\,d\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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