Optimal. Leaf size=120 \[ \frac {(e x)^{m+1} (A b-a B) (b c-a d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a b^2 e (m+1)}+\frac {(e x)^{m+1} (-a B d+A b d+b B c)}{b^2 e (m+1)}+\frac {B d x^{n+1} (e x)^m}{b (m+n+1)} \]
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Rubi [A] time = 0.12, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {570, 20, 30, 364} \[ \frac {(e x)^{m+1} (A b-a B) (b c-a d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a b^2 e (m+1)}+\frac {(e x)^{m+1} (-a B d+A b d+b B c)}{b^2 e (m+1)}+\frac {B d x^{n+1} (e x)^m}{b (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 30
Rule 364
Rule 570
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx &=\int \left (\frac {(b B c+A b d-a B d) (e x)^m}{b^2}+\frac {B d x^n (e x)^m}{b}+\frac {(A b-a B) (b c-a d) (e x)^m}{b^2 \left (a+b x^n\right )}\right ) \, dx\\ &=\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {(B d) \int x^n (e x)^m \, dx}{b}+\frac {((A b-a B) (b c-a d)) \int \frac {(e x)^m}{a+b x^n} \, dx}{b^2}\\ &=\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {(A b-a B) (b c-a d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^2 e (1+m)}+\frac {\left (B d x^{-m} (e x)^m\right ) \int x^{m+n} \, dx}{b}\\ &=\frac {B d x^{1+n} (e x)^m}{b (1+m+n)}+\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {(A b-a B) (b c-a d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^2 e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 95, normalized size = 0.79 \[ \frac {x (e x)^m \left (\frac {(a B-A b) (a d-b c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a (m+1)}+\frac {-a B d+A b d+b B c}{m+1}+\frac {b B d x^n}{m+n+1}\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d x^{2 \, n} + A c + {\left (B c + A d\right )} x^{n}\right )} \left (e x\right )^{m}}{b x^{n} + a}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (d \,x^{n}+c \right ) \left (e x \right )^{m}}{b \,x^{n}+a}\, dx \]
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 \[ {\left ({\left (b^{2} c e^{m} - a b d e^{m}\right )} A - {\left (a b c e^{m} - a^{2} d e^{m}\right )} B\right )} \int \frac {x^{m}}{b^{3} x^{n} + a b^{2}}\,{d x} + \frac {B b d e^{m} {\left (m + 1\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )} + {\left (A b d e^{m} {\left (m + n + 1\right )} + {\left (b c e^{m} {\left (m + n + 1\right )} - a d e^{m} {\left (m + n + 1\right )}\right )} B\right )} x x^{m}}{{\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} b^{2}} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,\left (c+d\,x^n\right )}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.14, size = 666, normalized size = 5.55 \[ \frac {A c e^{m} m x x^{m} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A c e^{m} x x^{m} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A d e^{m} m x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {A d e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {A d e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c e^{m} m x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B d e^{m} m x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {2 B d e^{m} x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {B d e^{m} x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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