Optimal. Leaf size=322 \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac {d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \]
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Rubi [A] time = 0.54, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {594, 459, 364} \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 459
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 )^3} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}-\frac {\int \frac {(e x)^m \left (c+d x^n\right ) \left (-c (a B (1+m)-A b (1+m-2 n))+d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{\left (a+b x^n\right )^2} \, dx}{2 a b n}\\ &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\int \frac {(e x)^m \left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))+d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) x^n\right )}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-\frac {a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))}{b}\right ) \int \frac {(e x)^m}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-\frac {a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))}{b}\right ) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{2 a^3 b^2 e (1+m) n^2}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 168, normalized size = 0.52 \[ \frac {x (e x)^m \left (\frac {(A b-a B) (b c-a d)^2 \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3}+\frac {(b c-a d) (-3 a B d+2 A b d+b B c) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^2}+\frac {d (-3 a B d+A b d+2 b B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a}+B d^2\right )}{b^3 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d^{2} x^{3 \, n} + A c^{2} + {\left (2 \, B c d + A d^{2}\right )} x^{2 \, n} + {\left (B c^{2} + 2 \, A c d\right )} x^{n}\right )} \left (e x\right )^{m}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (d \,x^{n}+c \right )^{2} \left (e x \right )^{m}}{\left (b \,x^{n}+a \right )^{3}}\, dx \]
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 ({\left (m^{2} - m {\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, 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} + m {\left (n + 2\right )} + 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} + m {\left (n + 2\right )} + 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 )} \int \frac {x^{m}}{2 \, {\left (a^{2} b^{4} n^{2} x^{n} + a^{3} b^{3} n^{2}\right )}}\,{d x} + \frac {2 \, B a^{2} b^{2} d^{2} e^{m} n^{2} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )} - {\left ({\left ({\left (m^{2} - m {\left (3 \, n - 2\right )} - 3 \, n + 1\right )} a b^{3} c^{2} e^{m} - 2 \, {\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} a^{2} b^{2} c d e^{m} + {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a^{3} b d^{2} e^{m}\right )} A - {\left ({\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} a^{2} b^{2} c^{2} e^{m} - 2 \, {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a^{3} b c d e^{m} + {\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{4} d^{2} e^{m}\right )} B\right )} x x^{m} - {\left ({\left ({\left (m^{2} - 2 \, m {\left (n - 1\right )} - 2 \, n + 1\right )} b^{4} c^{2} e^{m} - 2 \, {\left (m^{2} + 2 \, m + 1\right )} a b^{3} c d e^{m} + {\left (m^{2} + 2 \, m {\left (n + 1\right )} + 2 \, n + 1\right )} a^{2} b^{2} d^{2} e^{m}\right )} A - {\left ({\left (m^{2} + 2 \, m + 1\right )} a b^{3} c^{2} e^{m} - 2 \, {\left (m^{2} + 2 \, m {\left (n + 1\right )} + 2 \, n + 1\right )} a^{2} b^{2} c d e^{m} + {\left (m^{2} + 2 \, m {\left (2 \, n + 1\right )} + 4 \, n^{2} + 4 \, n + 1\right )} a^{3} b d^{2} e^{m}\right )} B\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{2 \, {\left ({\left (m n^{2} + n^{2}\right )} a^{2} b^{5} x^{2 \, n} + 2 \, {\left (m n^{2} + n^{2}\right )} a^{3} b^{4} x^{n} + {\left (m n^{2} + n^{2}\right )} a^{4} b^{3}\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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (c+d\,x^n\right )}^2}{{\left (a+b\,x^n\right )}^3} \,d x \]
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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