Optimal. Leaf size=268 \[ -\frac {(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac {d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac {d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.67, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {594, 570, 20, 30, 364} \[ -\frac {(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac {d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac {d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 20
Rule 30
Rule 364
Rule 570
Rule 594
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^2} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )}-\frac {\int \frac {(e x)^m \left (c+d x^n\right ) \left (-c (a B (1+m)-A b (1+m-n))+d (A b (1+m+n)-a B (1+m+2 n)) x^n\right )}{a+b x^n} \, dx}{a b n}\\ &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )}-\frac {\int \left (\frac {d (A b (2 b c (1+m)-a d (1+m+n))-a B (2 b c (1+m+n)-a d (1+m+2 n))) (e x)^m}{b^2}+\frac {d^2 (A b (1+m+n)-a B (1+m+2 n)) x^n (e x)^m}{b}+\frac {(b c-a d) (A b (b c (1+m-n)-a d (1+m+n))-a B (b c (1+m)-a d (1+m+2 n))) (e x)^m}{b^2 \left (a+b x^n\right )}\right ) \, dx}{a b n}\\ &=-\frac {d (A b (2 b c (1+m)-a d (1+m+n))-a B (2 b c (1+m+n)-a d (1+m+2 n))) (e x)^{1+m}}{a b^3 e (1+m) n}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )}-\frac {\left (d^2 (A b (1+m+n)-a B (1+m+2 n))\right ) \int x^n (e x)^m \, dx}{a b^2 n}-\frac {((b c-a d) (A b (b c (1+m-n)-a d (1+m+n))-a B (b c (1+m)-a d (1+m+2 n)))) \int \frac {(e x)^m}{a+b x^n} \, dx}{a b^3 n}\\ &=-\frac {d (A b (2 b c (1+m)-a d (1+m+n))-a B (2 b c (1+m+n)-a d (1+m+2 n))) (e x)^{1+m}}{a b^3 e (1+m) n}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )}-\frac {(b c-a d) (A b (b c (1+m-n)-a d (1+m+n))-a B (b c (1+m)-a d (1+m+2 n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^2 b^3 e (1+m) n}-\frac {\left (d^2 (A b (1+m+n)-a B (1+m+2 n)) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx}{a b^2 n}\\ &=-\frac {d^2 (A b (1+m+n)-a B (1+m+2 n)) x^{1+n} (e x)^m}{a b^2 n (1+m+n)}-\frac {d (A b (2 b c (1+m)-a d (1+m+n))-a B (2 b c (1+m+n)-a d (1+m+2 n))) (e x)^{1+m}}{a b^3 e (1+m) n}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )}-\frac {(b c-a d) (A b (b c (1+m-n)-a d (1+m+n))-a B (b c (1+m)-a d (1+m+2 n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^2 b^3 e (1+m) n}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 159, normalized size = 0.59 \[ \frac {x (e x)^m \left (\frac {(A b-a B) (b c-a d)^2 \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^2 (m+1)}+\frac {(b c-a d) (-3 a B d+2 A b d+b B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a (m+1)}+\frac {d (-2 a B d+A b d+2 b B c)}{m+1}+\frac {b B d^2 x^n}{m+n+1}\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d^{2} x^{3 \, n} + A c^{2} + {\left (2 \, B c d + A d^{2}\right )} x^{2 \, n} + {\left (B c^{2} + 2 \, A c d\right )} x^{n}\right )} \left (e x\right )^{m}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (d \,x^{n}+c \right )^{2} \left (e x \right )^{m}}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
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 \[ -{\left ({\left (a^{2} b d^{2} e^{m} {\left (m + n + 1\right )} + b^{3} c^{2} e^{m} {\left (m - n + 1\right )} - 2 \, a b^{2} c d e^{m} {\left (m + 1\right )}\right )} A - {\left (a^{3} d^{2} e^{m} {\left (m + 2 \, n + 1\right )} - 2 \, a^{2} b c d e^{m} {\left (m + n + 1\right )} + a b^{2} c^{2} e^{m} {\left (m + 1\right )}\right )} B\right )} \int \frac {x^{m}}{a b^{4} n x^{n} + a^{2} b^{3} n}\,{d x} + \frac {{\left (m n + n\right )} B a b^{2} d^{2} e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )} + {\left ({\left ({\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} b^{3} c^{2} e^{m} - 2 \, {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a b^{2} c d e^{m} + {\left (m^{2} + 2 \, m {\left (n + 1\right )} + n^{2} + 2 \, n + 1\right )} a^{2} b d^{2} e^{m}\right )} A - {\left ({\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a b^{2} c^{2} e^{m} - 2 \, {\left (m^{2} + 2 \, m {\left (n + 1\right )} + n^{2} + 2 \, n + 1\right )} a^{2} b c d e^{m} + {\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{3} d^{2} e^{m}\right )} B\right )} x x^{m} + {\left ({\left (m n + n^{2} + n\right )} A a b^{2} d^{2} e^{m} + {\left (2 \, {\left (m n + n^{2} + n\right )} a b^{2} c d e^{m} - {\left (m n + 2 \, n^{2} + n\right )} a^{2} b d^{2} e^{m}\right )} B\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{{\left (m^{2} n + {\left (n^{2} + 2 \, n\right )} m + n^{2} + n\right )} a b^{4} x^{n} + {\left (m^{2} n + {\left (n^{2} + 2 \, n\right )} m + n^{2} + n\right )} a^{2} b^{3}} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (c+d\,x^n\right )}^2}{{\left (a+b\,x^n\right )}^2} \,d x \]
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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