Optimal. Leaf size=79 \[ \frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)} \]
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Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {465, 481, 205} \begin {gather*} \frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 465
Rule 481
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 (b c-a d)}+\frac {c \operatorname {Subst}\left (\int \frac {1}{c+d x^2} \, dx,x,x^2\right )}{2 (b c-a d)}\\ &=-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} (b c-a d)}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 \sqrt {d} (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.84 \begin {gather*} \frac {\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{\sqrt {d}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b}}}{2 b c-2 a d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.49, size = 325, normalized size = 4.11 \begin {gather*} \left [-\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x^{4} + 2 \, b x^{2} \sqrt {-\frac {a}{b}} - a}{b x^{4} + a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{4} - 2 \, d x^{2} \sqrt {-\frac {c}{d}} - c}{d x^{4} + c}\right )}{4 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{2} \sqrt {\frac {a}{b}}}{a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{4} - 2 \, d x^{2} \sqrt {-\frac {c}{d}} - c}{d x^{4} + c}\right )}{4 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {c}{d}} \arctan \left (\frac {d x^{2} \sqrt {\frac {c}{d}}}{c}\right ) - \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{4} + 2 \, b x^{2} \sqrt {-\frac {a}{b}} - a}{b x^{4} + a}\right )}{4 \, {\left (b c - a d\right )}}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{2} \sqrt {\frac {a}{b}}}{a}\right ) - \sqrt {\frac {c}{d}} \arctan \left (\frac {d x^{2} \sqrt {\frac {c}{d}}}{c}\right )}{2 \, {\left (b c - a d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 59, normalized size = 0.75 \begin {gather*} -\frac {a \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 60, normalized size = 0.76 \begin {gather*} \frac {a \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -b c \right ) \sqrt {a b}}-\frac {c \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -b c \right ) \sqrt {c d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 59, normalized size = 0.75 \begin {gather*} -\frac {a \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 379, normalized size = 4.80 \begin {gather*} \frac {\ln \left (d^2\,{\left (-a\,b\right )}^{5/2}+b^4\,c^2\,\sqrt {-a\,b}-b^5\,c^2\,x^2+2\,b^2\,c\,d\,{\left (-a\,b\right )}^{3/2}-a^2\,b^3\,d^2\,x^2+2\,a\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,b^2\,c-4\,a\,b\,d}-\frac {\ln \left (d^2\,{\left (-a\,b\right )}^{5/2}+b^4\,c^2\,\sqrt {-a\,b}+b^5\,c^2\,x^2+2\,b^2\,c\,d\,{\left (-a\,b\right )}^{3/2}+a^2\,b^3\,d^2\,x^2-2\,a\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,\left (b^2\,c-a\,b\,d\right )}-\frac {\ln \left (b^2\,{\left (-c\,d\right )}^{5/2}+a^2\,d^4\,\sqrt {-c\,d}+a^2\,d^5\,x^2+2\,a\,b\,d^2\,{\left (-c\,d\right )}^{3/2}+b^2\,c^2\,d^3\,x^2-2\,a\,b\,c\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,\left (a\,d^2-b\,c\,d\right )}+\frac {\ln \left (b^2\,{\left (-c\,d\right )}^{5/2}+a^2\,d^4\,\sqrt {-c\,d}-a^2\,d^5\,x^2+2\,a\,b\,d^2\,{\left (-c\,d\right )}^{3/2}-b^2\,c^2\,d^3\,x^2+2\,a\,b\,c\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,a\,d^2-4\,b\,c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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