Optimal. Leaf size=194 \[ \frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+1}{2 n};-p,1;\frac {m+1}{2 n}+1;-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d (m+1)}-\frac {e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+n+1}{2 n};-p,1;\frac {m+3 n+1}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (m+n+1)} \]
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Rubi [A] time = 0.22, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1562, 511, 510} \[ \frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+1}{2 n};-p,1;\frac {m+1}{2 n}+1;-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d (m+1)}-\frac {e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+n+1}{2 n};-p,1;\frac {m+3 n+1}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 1562
Rubi steps
\begin {align*} \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx &=\left (x^{-m} (f x)^m\right ) \int \left (\frac {d x^m \left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}}+\frac {e x^{m+n} \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}}\right ) \, dx\\ &=\left (d x^{-m} (f x)^m\right ) \int \frac {x^m \left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}} \, dx+\left (e x^{-m} (f x)^m\right ) \int \frac {x^{m+n} \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}} \, dx\\ &=\left (d x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^m \left (1+\frac {c x^{2 n}}{a}\right )^p}{d^2-e^2 x^{2 n}} \, dx+\left (e x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^{m+n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{-d^2+e^2 x^{2 n}} \, dx\\ &=\frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+m}{2 n};-p,1;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d (1+m)}-\frac {e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+m+n}{2 n};-p,1;\frac {1+m+3 n}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+m+n)}\\ \end {align*}
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Mathematica [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (c \,x^{2 n}+a \right )^{p}}{e \,x^{n}+d}\, dx \]
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 \[ \int \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{e x^{n} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+c\,x^{2\,n}\right )}^p\,{\left (f\,x\right )}^m}{d+e\,x^n} \,d x \]
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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