Optimal. Leaf size=302 \[ \frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+1}{2 n};-p,2;\frac {m+1}{2 n}+1;-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}+\frac {e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+2 n+1}{2 n};-p,2;\frac {m+4 n+1}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)}-\frac {2 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+n+1}{2 n};-p,2;\frac {m+3 n+1}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)} \]
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Rubi [A] time = 0.33, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1562, 511, 510} \[ \frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+1}{2 n};-p,2;\frac {m+1}{2 n}+1;-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}-\frac {2 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+n+1}{2 n};-p,2;\frac {m+3 n+1}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)}+\frac {e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {m+2 n+1}{2 n};-p,2;\frac {m+4 n+1}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)} \]
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}{\left (d+e x^n\right )^2} \, dx &=\left (x^{-m} (f x)^m\right ) \int \left (\frac {d^2 x^m \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2}-\frac {2 d e x^{m+n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}+\frac {e^2 x^{m+2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2}\right ) \, dx\\ &=\left (d^2 x^{-m} (f x)^m\right ) \int \frac {x^m \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-\left (2 d e x^{-m} (f x)^m\right ) \int \frac {x^{m+n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+\left (e^2 x^{-m} (f x)^m\right ) \int \frac {x^{m+2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx\\ &=\left (d^2 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}{\left (d^2-e^2 x^{2 n}\right )^2} \, dx-\left (2 d 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}{\left (-d^2+e^2 x^{2 n}\right )^2} \, dx+\left (e^2 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+2 n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^2} \, 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,2;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+m)}-\frac {2 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,2;\frac {1+m+3 n}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (1+m+n)}+\frac {e^2 x^{1+2 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+2 n}{2 n};-p,2;\frac {1+m+4 n}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+m+2 n)}\\ \end {align*}
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Mathematica [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\left (f x \right )^{m} \left (c \,x^{2 n}+a \right )^{p}}{\left (e \,x^{n}+d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+c\,x^{2\,n}\right )}^p\,{\left (f\,x\right )}^m}{{\left (d+e\,x^n\right )}^2} \,d x \]
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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