Optimal. Leaf size=90 \[ -\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}-\frac {x^4 (a d+b c)}{4 b^2 d^2}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}+\frac {x^8}{8 b d} \]
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Rubi [A] time = 0.10, antiderivative size = 90, 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 = {446, 72} \[ -\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}-\frac {x^4 (a d+b c)}{4 b^2 d^2}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}+\frac {x^8}{8 b d} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x) (c+d x)} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {-b c-a d}{b^2 d^2}+\frac {x}{b d}-\frac {a^3}{b^2 (b c-a d) (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx,x,x^4\right )\\ &=-\frac {(b c+a d) x^4}{4 b^2 d^2}+\frac {x^8}{8 b d}-\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 92, normalized size = 1.02 \[ -\frac {a^3 \log \left (a+b x^4\right )}{4 b^3 (b c-a d)}+\frac {x^4 (-a d-b c)}{4 b^2 d^2}+\frac {c^3 \log \left (c+d x^4\right )}{4 d^3 (b c-a d)}+\frac {x^8}{8 b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 7.13, size = 100, normalized size = 1.11 \[ \frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{8} - 2 \, a^{3} d^{3} \log \left (b x^{4} + a\right ) + 2 \, b^{3} c^{3} \log \left (d x^{4} + c\right ) - 2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{4}}{8 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 88, normalized size = 0.98 \[ -\frac {a^{3} \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )}} + \frac {c^{3} \log \left ({\left | d x^{4} + c \right |}\right )}{4 \, {\left (b c d^{3} - a d^{4}\right )}} + \frac {b d x^{8} - 2 \, b c x^{4} - 2 \, a d x^{4}}{8 \, b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 89, normalized size = 0.99 \[ \frac {x^{8}}{8 b d}-\frac {a \,x^{4}}{4 b^{2} d}-\frac {c \,x^{4}}{4 b \,d^{2}}+\frac {a^{3} \ln \left (b \,x^{4}+a \right )}{4 \left (a d -b c \right ) b^{3}}-\frac {c^{3} \ln \left (d \,x^{4}+c \right )}{4 \left (a d -b c \right ) d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 84, normalized size = 0.93 \[ -\frac {a^{3} \log \left (b x^{4} + a\right )}{4 \, {\left (b^{4} c - a b^{3} d\right )}} + \frac {c^{3} \log \left (d x^{4} + c\right )}{4 \, {\left (b c d^{3} - a d^{4}\right )}} + \frac {b d x^{8} - 2 \, {\left (b c + a d\right )} x^{4}}{8 \, b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.88, size = 88, normalized size = 0.98 \[ \frac {x^8}{8\,b\,d}-\frac {c^3\,\ln \left (d\,x^4+c\right )}{4\,\left (a\,d^4-b\,c\,d^3\right )}-\frac {a^3\,\ln \left (b\,x^4+a\right )}{4\,\left (b^4\,c-a\,b^3\,d\right )}-\frac {x^4\,\left (a\,d+b\,c\right )}{4\,b^2\,d^2} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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