Optimal. Leaf size=299 \[ \frac {b^3 x (a d (1-4 n)-b c (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 n (b c-a d)^4}-\frac {d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{2 a c^2 n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac {d^2 x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^4}+\frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}+\frac {d x (a d+2 b c)}{2 a c n (b c-a d)^2 \left (c+d x^n\right )^2} \]
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Rubi [A] time = 0.55, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {414, 527, 522, 245} \[ \frac {d^2 x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^4}-\frac {d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{2 a c^2 n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac {b^3 x (a d (1-4 n)-b c (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 n (b c-a d)^4}+\frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}+\frac {d x (a d+2 b c)}{2 a c n (b c-a d)^2 \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 245
Rule 414
Rule 522
Rule 527
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^3} \, dx &=\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\int \frac {a d n+b (c-c n)+b d (1-3 n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )^3} \, dx}{a (b c-a d) n}\\ &=\frac {d (2 b c+a d) x}{2 a c (b c-a d)^2 n \left (c+d x^n\right )^2}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {\int \frac {n \left (a^2 d^2 (1-2 n)+2 b^2 c^2 (1-n)+4 a b c d n\right )+b d (2 b c+a d) (1-2 n) n x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx}{2 a c (b c-a d)^2 n^2}\\ &=\frac {d (2 b c+a d) x}{2 a c (b c-a d)^2 n \left (c+d x^n\right )^2}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {d \left (a b c d (1-6 n)-a^2 d^2 (1-2 n)-2 b^2 c^2 n\right ) x}{2 a c^2 (b c-a d)^3 n^2 \left (c+d x^n\right )}-\frac {\int \frac {n \left (2 b^3 c^3 (1-n) n+6 a b^2 c^2 d n^2+a^3 d^3 \left (1-3 n+2 n^2\right )-a^2 b c d^2 \left (1-7 n+6 n^2\right )\right )-b d (1-n) n \left (a b c d (1-6 n)-a^2 d^2 (1-2 n)-2 b^2 c^2 n\right ) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{2 a c^2 (b c-a d)^3 n^3}\\ &=\frac {d (2 b c+a d) x}{2 a c (b c-a d)^2 n \left (c+d x^n\right )^2}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {d \left (a b c d (1-6 n)-a^2 d^2 (1-2 n)-2 b^2 c^2 n\right ) x}{2 a c^2 (b c-a d)^3 n^2 \left (c+d x^n\right )}+\frac {\left (b^3 (a d (1-4 n)-b c (1-n))\right ) \int \frac {1}{a+b x^n} \, dx}{a (b c-a d)^4 n}+\frac {\left (d^2 \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-5 n+4 n^2\right )+b^2 c^2 \left (1-7 n+12 n^2\right )\right )\right ) \int \frac {1}{c+d x^n} \, dx}{2 c^2 (b c-a d)^4 n^2}\\ &=\frac {d (2 b c+a d) x}{2 a c (b c-a d)^2 n \left (c+d x^n\right )^2}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )^2}-\frac {d \left (a b c d (1-6 n)-a^2 d^2 (1-2 n)-2 b^2 c^2 n\right ) x}{2 a c^2 (b c-a d)^3 n^2 \left (c+d x^n\right )}+\frac {b^3 (a d (1-4 n)-b c (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^4 n}+\frac {d^2 \left (a^2 d^2 \left (1-3 n+2 n^2\right )-2 a b c d \left (1-5 n+4 n^2\right )+b^2 c^2 \left (1-7 n+12 n^2\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 (b c-a d)^4 n^2}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 233, normalized size = 0.78 \[ \frac {x \left (\frac {2 b^3 n (a d (1-4 n)+b c (n-1)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2}+\frac {d^2 \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^3}+\frac {2 b^3 n (b c-a d)}{a \left (a+b x^n\right )}+\frac {d^2 (a d-b c) (a d (2 n-1)+b (c-6 c n))}{c^2 \left (c+d x^n\right )}+\frac {d^2 n (b c-a d)^2}{c \left (c+d x^n\right )^2}\right )}{2 n^2 (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} d^{3} x^{5 \, n} + a^{2} c^{3} + {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{4 \, n} + {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3 \, n} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2 \, n} + {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{n}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{\left (b \,x^{n}+a \right )^{2} \left (d \,x^{n}+c \right )^{3}}\, dx \]
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 \[ {\left ({\left (12 \, n^{2} - 7 \, n + 1\right )} b^{2} c^{2} d^{2} - 2 \, {\left (4 \, n^{2} - 5 \, n + 1\right )} a b c d^{3} + {\left (2 \, n^{2} - 3 \, n + 1\right )} a^{2} d^{4}\right )} \int \frac {1}{2 \, {\left (b^{4} c^{7} n^{2} - 4 \, a b^{3} c^{6} d n^{2} + 6 \, a^{2} b^{2} c^{5} d^{2} n^{2} - 4 \, a^{3} b c^{4} d^{3} n^{2} + a^{4} c^{3} d^{4} n^{2} + {\left (b^{4} c^{6} d n^{2} - 4 \, a b^{3} c^{5} d^{2} n^{2} + 6 \, a^{2} b^{2} c^{4} d^{3} n^{2} - 4 \, a^{3} b c^{3} d^{4} n^{2} + a^{4} c^{2} d^{5} n^{2}\right )} x^{n}\right )}}\,{d x} - {\left (a b^{3} d {\left (4 \, n - 1\right )} - b^{4} c {\left (n - 1\right )}\right )} \int \frac {1}{a^{2} b^{4} c^{4} n - 4 \, a^{3} b^{3} c^{3} d n + 6 \, a^{4} b^{2} c^{2} d^{2} n - 4 \, a^{5} b c d^{3} n + a^{6} d^{4} n + {\left (a b^{5} c^{4} n - 4 \, a^{2} b^{4} c^{3} d n + 6 \, a^{3} b^{3} c^{2} d^{2} n - 4 \, a^{4} b^{2} c d^{3} n + a^{5} b d^{4} n\right )} x^{n}}\,{d x} + \frac {{\left (a b^{2} c d^{3} {\left (6 \, n - 1\right )} - a^{2} b d^{4} {\left (2 \, n - 1\right )} + 2 \, b^{3} c^{2} d^{2} n\right )} x x^{2 \, n} + {\left (a b^{2} c^{2} d^{2} {\left (7 \, n - 1\right )} - a^{3} d^{4} {\left (2 \, n - 1\right )} + 4 \, b^{3} c^{3} d n + 3 \, a^{2} b c d^{3} n\right )} x x^{n} + {\left (a^{2} b c^{2} d^{2} {\left (7 \, n - 1\right )} - a^{3} c d^{3} {\left (3 \, n - 1\right )} + 2 \, b^{3} c^{4} n\right )} x}{2 \, {\left (a^{2} b^{3} c^{7} n^{2} - 3 \, a^{3} b^{2} c^{6} d n^{2} + 3 \, a^{4} b c^{5} d^{2} n^{2} - a^{5} c^{4} d^{3} n^{2} + {\left (a b^{4} c^{5} d^{2} n^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} n^{2} + 3 \, a^{3} b^{2} c^{3} d^{4} n^{2} - a^{4} b c^{2} d^{5} n^{2}\right )} x^{3 \, n} + {\left (2 \, a b^{4} c^{6} d n^{2} - 5 \, a^{2} b^{3} c^{5} d^{2} n^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} n^{2} + a^{4} b c^{3} d^{4} n^{2} - a^{5} c^{2} d^{5} n^{2}\right )} x^{2 \, n} + {\left (a b^{4} c^{7} n^{2} - a^{2} b^{3} c^{6} d n^{2} - 3 \, a^{3} b^{2} c^{5} d^{2} n^{2} + 5 \, a^{4} b c^{4} d^{3} n^{2} - 2 \, a^{5} c^{3} d^{4} n^{2}\right )} x^{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,x^n\right )}^2\,{\left (c+d\,x^n\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x^{n}\right )^{2} \left (c + d x^{n}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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