Optimal. Leaf size=74 \[ -\frac {c}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 74, 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, 78}
\begin {gather*} -\frac {c}{2 d \left (c+d x^2\right ) (b c-a d)}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x) (c+d x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a b}{(b c-a d)^2 (a+b x)}+\frac {c}{(b c-a d) (c+d x)^2}+\frac {a d}{(-b c+a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 74, normalized size = 1.00 \begin {gather*} \frac {c}{2 d (-b c+a d) \left (c+d x^2\right )}-\frac {a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac {a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 68, normalized size = 0.92
method | result | size |
default | \(-\frac {a \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2}}+\frac {\frac {c \left (a d -b c \right )}{d \left (d \,x^{2}+c \right )}+a \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2}}\) | \(68\) |
norman | \(-\frac {x^{2}}{2 \left (a d -b c \right ) \left (d \,x^{2}+c \right )}-\frac {a \ln \left (b \,x^{2}+a \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a \ln \left (d \,x^{2}+c \right )}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(94\) |
risch | \(\frac {c}{2 d \left (a d -b c \right ) \left (d \,x^{2}+c \right )}-\frac {a \ln \left (-b \,x^{2}-a \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a \ln \left (d \,x^{2}+c \right )}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 105, normalized size = 1.42 \begin {gather*} -\frac {a \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {a \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {c}{2 \, {\left (b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 117, normalized size = 1.58 \begin {gather*} -\frac {b c^{2} - a c d + {\left (a d^{2} x^{2} + a c d\right )} \log \left (b x^{2} + a\right ) - {\left (a d^{2} x^{2} + a c d\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs.
\(2 (58) = 116\).
time = 1.34, size = 253, normalized size = 3.42 \begin {gather*} \frac {a \log {\left (x^{2} + \frac {- \frac {a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac {a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac {a \log {\left (x^{2} + \frac {\frac {a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac {a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac {c}{2 a c d^{2} - 2 b c^{2} d + x^{2} \cdot \left (2 a d^{3} - 2 b c d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.57, size = 91, normalized size = 1.23 \begin {gather*} -\frac {\frac {a d^{2} \log \left ({\left | b - \frac {b c}{d x^{2} + c} + \frac {a d}{d x^{2} + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac {c d}{{\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 173, normalized size = 2.34 \begin {gather*} -\frac {b\,c^2-c\,\left (a\,d-a\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}\right )+a\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}}{2\,a^2\,c\,d^3+2\,a^2\,d^4\,x^2-4\,a\,b\,c^2\,d^2-4\,a\,b\,c\,d^3\,x^2+2\,b^2\,c^3\,d+2\,b^2\,c^2\,d^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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