Optimal. Leaf size=123 \[ \frac {b^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a (b c-a d)^2}+\frac {d x (b c (1-2 n)-a d (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 n (b c-a d)^2}-\frac {d x}{c n (b c-a d) \left (c+d x^n\right )} \]
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Rubi [A] time = 0.15, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {414, 522, 245} \[ \frac {b^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a (b c-a d)^2}+\frac {d x (b c (1-2 n)-a d (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 n (b c-a d)^2}-\frac {d x}{c n (b c-a d) \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
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Rule 245
Rule 414
Rule 522
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx &=-\frac {d x}{c (b c-a d) n \left (c+d x^n\right )}+\frac {\int \frac {b c n+a (d-d n)+b d (1-n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{c (b c-a d) n}\\ &=-\frac {d x}{c (b c-a d) n \left (c+d x^n\right )}+\frac {b^2 \int \frac {1}{a+b x^n} \, dx}{(b c-a d)^2}-\frac {(d (a d (1-n)-b (c-2 c n))) \int \frac {1}{c+d x^n} \, dx}{c (b c-a d)^2 n}\\ &=-\frac {d x}{c (b c-a d) n \left (c+d x^n\right )}+\frac {b^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a (b c-a d)^2}+\frac {d (b c (1-2 n)-a d (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^2 n}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 121, normalized size = 0.98 \[ \frac {x \left (b^2 c^2 n \left (c+d x^n\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )+a d \left (\left (c+d x^n\right ) (a d (n-1)+b (c-2 c n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )+c (a d-b c)\right )\right )}{a c^2 n (b c-a d)^2 \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b d^{2} x^{3 \, n} + a c^{2} + {\left (2 \, b c d + a d^{2}\right )} x^{2 \, n} + {\left (b c^{2} + 2 \, a c d\right )} x^{n}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.93, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right ) \left (d \,x^{n}+c \right )^{2}}\, dx \]
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 \[ b^{2} \int \frac {1}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{n}}\,{d x} - {\left (b c d {\left (2 \, n - 1\right )} - a d^{2} {\left (n - 1\right )}\right )} \int \frac {1}{b^{2} c^{4} n - 2 \, a b c^{3} d n + a^{2} c^{2} d^{2} n + {\left (b^{2} c^{3} d n - 2 \, a b c^{2} d^{2} n + a^{2} c d^{3} n\right )} x^{n}}\,{d x} - \frac {d x}{b c^{3} n - a c^{2} d n + {\left (b c^{2} d n - a c d^{2} n\right )} x^{n}} \]
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 \[ \int \frac {1}{\left (a+b\,x^n\right )\,{\left (c+d\,x^n\right )}^2} \,d x \]
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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