Optimal. Leaf size=145 \[ \frac {b}{a (b c-a d) n x^2 \left (a+b x^n\right )}+\frac {b (2 a d (1+n)-b c (2+n)) \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a^2 (b c-a d)^2 n x^2}-\frac {d^2 \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 c (b c-a d)^2 x^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {518, 611, 371}
\begin {gather*} \frac {b (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a^2 n x^2 (b c-a d)^2}-\frac {d^2 \, _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)^2}+\frac {b}{a n x^2 (b c-a d) \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 518
Rule 611
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac {b}{a (b c-a d) n x^2 \left (a+b x^n\right )}-\frac {\int \frac {a d n-b c (2+n)-b d (2+n) x^n}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{a (b c-a d) n}\\ &=\frac {b}{a (b c-a d) n x^2 \left (a+b x^n\right )}-\frac {\int \left (\frac {b (2 a d (1+n)-b c (2+n))}{(b c-a d) x^3 \left (a+b x^n\right )}+\frac {a d^2 n}{(-b c+a d) x^3 \left (c+d x^n\right )}\right ) \, dx}{a (b c-a d) n}\\ &=\frac {b}{a (b c-a d) n x^2 \left (a+b x^n\right )}+\frac {d^2 \int \frac {1}{x^3 \left (c+d x^n\right )} \, dx}{(b c-a d)^2}-\frac {(b (2 a d (1+n)-b c (2+n))) \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx}{a (b c-a d)^2 n}\\ &=\frac {b}{a (b c-a d) n x^2 \left (a+b x^n\right )}+\frac {b (2 a d (1+n)-b c (2+n)) \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a^2 (b c-a d)^2 n x^2}-\frac {d^2 \, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 c (b c-a d)^2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 136, normalized size = 0.94 \begin {gather*} \frac {b c (2 a d (1+n)-b c (2+n)) \left (a+b x^n\right ) \, _2F_1\left (1,-\frac {2}{n};\frac {-2+n}{n};-\frac {b x^n}{a}\right )-a \left (2 b c (-b c+a d)+a d^2 n \left (a+b x^n\right ) \, _2F_1\left (1,-\frac {2}{n};\frac {-2+n}{n};-\frac {d x^n}{c}\right )\right )}{2 a^2 c (b c-a d)^2 n x^2 \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^3\,{\left (a+b\,x^n\right )}^2\,\left (c+d\,x^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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