Optimal. Leaf size=162 \[ \frac {(c x)^{m+1} \left (a^2 g-a b f+b^2 e\right )}{b^3 c (m+1)}+\frac {(c x)^{m+1} \left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a b^3 c (m+1)}+\frac {x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac {g x^{2 n+1} (c x)^m}{b (m+2 n+1)} \]
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Rubi [A] time = 0.17, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1844, 20, 30, 364} \[ \frac {(c x)^{m+1} \left (a^2 b f+a^3 (-g)-a b^2 e+b^3 d\right ) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a b^3 c (m+1)}+\frac {(c x)^{m+1} \left (a^2 g-a b f+b^2 e\right )}{b^3 c (m+1)}+\frac {x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac {g x^{2 n+1} (c x)^m}{b (m+2 n+1)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 30
Rule 364
Rule 1844
Rubi steps
\begin {align*} \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx &=\int \left (\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^m}{b^3}+\frac {(b f-a g) x^n (c x)^m}{b^2}+\frac {g x^{2 n} (c x)^m}{b}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^m}{b^3 \left (a+b x^n\right )}\right ) \, dx\\ &=\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {g \int x^{2 n} (c x)^m \, dx}{b}+\frac {(b f-a g) \int x^n (c x)^m \, dx}{b^2}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) \int \frac {(c x)^m}{a+b x^n} \, dx}{b^3}\\ &=\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^3 c (1+m)}+\frac {\left (g x^{-m} (c x)^m\right ) \int x^{m+2 n} \, dx}{b}+\frac {\left ((b f-a g) x^{-m} (c x)^m\right ) \int x^{m+n} \, dx}{b^2}\\ &=\frac {(b f-a g) x^{1+n} (c x)^m}{b^2 (1+m+n)}+\frac {g x^{1+2 n} (c x)^m}{b (1+m+2 n)}+\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^3 c (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 130, normalized size = 0.80 \[ \frac {x (c x)^m \left (\frac {a^2 g-a b f+b^2 e}{m+1}+\frac {\left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a (m+1)}+\frac {b x^n (b f-a g)}{m+n+1}+\frac {b^2 g x^{2 n}}{m+2 n+1}\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{b x^{n} + a}, x\right ) \]
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 \[ \int \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}+d \right ) \left (c x \right )^{m}}{b \,x^{n}+a}\, dx \]
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 \[ {\left (b^{3} c^{m} d - a b^{2} c^{m} e + a^{2} b c^{m} f - a^{3} c^{m} g\right )} \int \frac {x^{m}}{b^{4} x^{n} + a b^{3}}\,{d x} + \frac {{\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} b^{2} c^{m} g x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )} + {\left ({\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} b^{2} c^{m} e - {\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a b c^{m} f + {\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{2} c^{m} g\right )} x x^{m} + {\left ({\left (m^{2} + 2 \, m {\left (n + 1\right )} + 2 \, n + 1\right )} b^{2} c^{m} f - {\left (m^{2} + 2 \, m {\left (n + 1\right )} + 2 \, n + 1\right )} a b c^{m} g\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{{\left (m^{3} + 3 \, m^{2} {\left (n + 1\right )} + {\left (2 \, n^{2} + 6 \, n + 3\right )} m + 2 \, n^{2} + 3 \, n + 1\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x\right )}^m\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right )}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 57.81, size = 654, normalized size = 4.04 \[ \frac {c^{m} d 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 {c^{m} d 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 {c^{m} 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 {c^{m} e 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 {c^{m} e 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 {c^{m} f 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 c^{m} f 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 {c^{m} f 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 {c^{m} g m x x^{m} x^{3 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 3 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} + \frac {3 c^{m} g x x^{m} x^{3 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 3 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} + \frac {c^{m} g x x^{m} x^{3 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 3 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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