Optimal. Leaf size=299 \[ \frac {3 d e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+2 n}+\frac {e^3 x^{1+3 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p;\frac {1}{2} \left (5+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+3 n}+d^3 x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2 n},-p;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+\frac {3 d^2 e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+n}{2 n},-p;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+n} \]
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Rubi [A]
time = 0.11, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1451, 252,
251, 372, 371} \begin {gather*} d^3 x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2 n},-p;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+\frac {3 d^2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {n+1}{2 n},-p;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{n+1}+\frac {3 d e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 n+1}+\frac {e^3 x^{3 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p;\frac {1}{2} \left (5+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{3 n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 371
Rule 372
Rule 1451
Rubi steps
\begin {align*} \int \left (d+e x^n\right )^3 \left (a+c x^{2 n}\right )^p \, dx &=\int \left (d^3 \left (a+c x^{2 n}\right )^p+3 d^2 e x^n \left (a+c x^{2 n}\right )^p+3 d e^2 x^{2 n} \left (a+c x^{2 n}\right )^p+e^3 x^{3 n} \left (a+c x^{2 n}\right )^p\right ) \, dx\\ &=d^3 \int \left (a+c x^{2 n}\right )^p \, dx+\left (3 d^2 e\right ) \int x^n \left (a+c x^{2 n}\right )^p \, dx+\left (3 d e^2\right ) \int x^{2 n} \left (a+c x^{2 n}\right )^p \, dx+e^3 \int x^{3 n} \left (a+c x^{2 n}\right )^p \, dx\\ &=\left (d^3 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx+\left (3 d^2 e \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int x^n \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx+\left (3 d e^2 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int x^{2 n} \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx+\left (e^3 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int x^{3 n} \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx\\ &=\frac {3 d e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p;\frac {1}{2} \left (4+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+2 n}+\frac {e^3 x^{1+3 n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p;\frac {1}{2} \left (5+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+3 n}+d^3 x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2 n},-p;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+\frac {3 d^2 e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+n}{2 n},-p;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+n}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 213, normalized size = 0.71 \begin {gather*} x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (\frac {3 d e^2 x^{2 n} \, _2F_1\left (1+\frac {1}{2 n},-p;2+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+2 n}+\frac {e^3 x^{3 n} \, _2F_1\left (\frac {1}{2} \left (3+\frac {1}{n}\right ),-p;\frac {1}{2} \left (5+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+3 n}+d^2 \left (d \, _2F_1\left (\frac {1}{2 n},-p;1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )+\frac {3 e x^n \, _2F_1\left (\frac {1+n}{2 n},-p;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{1+n}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (d +e \,x^{n}\right )^{3} \left (a +c \,x^{2 n}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\left (a+c\,x^{2\,n}\right )}^p\,{\left (d+e\,x^n\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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