Optimal. Leaf size=95 \[ \frac {d \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac {b \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a x^2 (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 95, 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 = {508, 364} \[ \frac {d \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac {b \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a x^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 508
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\frac {b \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx}{b c-a d}-\frac {d \int \frac {1}{x^3 \left (c+d x^n\right )} \, dx}{b c-a d}\\ &=-\frac {b \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a (b c-a d) x^2}+\frac {d \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 c (b c-a d) x^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 77, normalized size = 0.81 \[ \frac {b c \, _2F_1\left (1,-\frac {2}{n};\frac {n-2}{n};-\frac {b x^n}{a}\right )-a d \, _2F_1\left (1,-\frac {2}{n};\frac {n-2}{n};-\frac {d x^n}{c}\right )}{2 a c x^2 (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b d x^{3} x^{2 \, n} + {\left (b c + a d\right )} x^{3} x^{n} + a c x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right ) \left (d \,x^{n}+c \right ) x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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