Optimal. Leaf size=212 \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (A b (a d (m-2 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-n+1)))}{a^2 e (m+1) n (b c-a d)^2}-\frac {d (e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c e (m+1) (b c-a d)^2}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right )} \]
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Rubi [A] time = 0.53, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {595, 597, 364} \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (A b (a d (m-2 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-n+1)))}{a^2 e (m+1) n (b c-a d)^2}-\frac {d (e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c e (m+1) (b c-a d)^2}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 595
Rule 597
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}-\frac {\int \frac {(e x)^m \left (-a B c (1+m)+A b c (1+m-n)+a A d n+(A b-a B) d (1+m-n) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{a (b c-a d) n}\\ &=\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}-\frac {\int \left (\frac {(-A b (a d (1+m-2 n)-b c (1+m-n))-a B (b c (1+m)-a d (1+m-n))) (e x)^m}{(b c-a d) \left (a+b x^n\right )}+\frac {a d (-B c+A d) n (e x)^m}{(-b c+a d) \left (c+d x^n\right )}\right ) \, dx}{a (b c-a d) n}\\ &=\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}-\frac {(d (B c-A d)) \int \frac {(e x)^m}{c+d x^n} \, dx}{(b c-a d)^2}+\frac {(A b (a d (1+m-2 n)-b c (1+m-n))+a B (b c (1+m)-a d (1+m-n))) \int \frac {(e x)^m}{a+b x^n} \, dx}{a (b c-a d)^2 n}\\ &=\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}+\frac {(A b (a d (1+m-2 n)-b c (1+m-n))+a B (b c (1+m)-a d (1+m-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 c-a d)^2 e (1+m) n}-\frac {d (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c (b c-a d)^2 e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 152, normalized size = 0.72 \[ -\frac {x (e x)^m \left (a^2 d (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )+a b c (A d-B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )-c (A b-a B) (b c-a d) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )\right )}{a^2 c (m+1) (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b^{2} d x^{3 \, n} + a^{2} c + {\left (b^{2} c + 2 \, a b d\right )} x^{2 \, n} + {\left (2 \, a b c + a^{2} d\right )} x^{n}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m}}{\left (b \,x^{n}+a \right )^{2} \left (d \,x^{n}+c \right )}\, dx \]
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 \[ -\frac {{\left (B a e^{m} - A b e^{m}\right )} x x^{m}}{a^{2} b c n - a^{3} d n + {\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} - {\left ({\left (b^{2} c e^{m} {\left (m - n + 1\right )} - a b d e^{m} {\left (m - 2 \, n + 1\right )}\right )} A + {\left (a^{2} d e^{m} {\left (m - n + 1\right )} - a b c e^{m} {\left (m + 1\right )}\right )} B\right )} \int \frac {x^{m}}{a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n + {\left (a b^{3} c^{2} n - 2 \, a^{2} b^{2} c d n + a^{3} b d^{2} n\right )} x^{n}}\,{d x} - {\left (B c d e^{m} - A d^{2} e^{m}\right )} \int \frac {x^{m}}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^2\,\left (c+d\,x^n\right )} \,d x \]
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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