Optimal. Leaf size=297 \[ \frac {d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{c (m+1)}+\frac {e x^{n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+n+1}{n},-p;\frac {m+2 n+1}{n};-\frac {b x^n}{a}\right )}{m+n+1}+\frac {f x^{2 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2 n+1}{n},-p;\frac {m+3 n+1}{n};-\frac {b x^n}{a}\right )}{m+2 n+1}+\frac {g x^{3 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+3 n+1}{n},-p;\frac {m+4 n+1}{n};-\frac {b x^n}{a}\right )}{m+3 n+1} \]
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Rubi [A] time = 0.21, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1844, 365, 364, 20} \[ \frac {d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{c (m+1)}+\frac {e x^{n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+n+1}{n},-p;\frac {m+2 n+1}{n};-\frac {b x^n}{a}\right )}{m+n+1}+\frac {f x^{2 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2 n+1}{n},-p;\frac {m+3 n+1}{n};-\frac {b x^n}{a}\right )}{m+2 n+1}+\frac {g x^{3 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+3 n+1}{n},-p;\frac {m+4 n+1}{n};-\frac {b x^n}{a}\right )}{m+3 n+1} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 365
Rule 1844
Rubi steps
\begin {align*} \int (c x)^m \left (a+b x^n\right )^p \left (d+e x^n+f x^{2 n}+g x^{3 n}\right ) \, dx &=\int \left (d (c x)^m \left (a+b x^n\right )^p+e x^n (c x)^m \left (a+b x^n\right )^p+f x^{2 n} (c x)^m \left (a+b x^n\right )^p+g x^{3 n} (c x)^m \left (a+b x^n\right )^p\right ) \, dx\\ &=d \int (c x)^m \left (a+b x^n\right )^p \, dx+e \int x^n (c x)^m \left (a+b x^n\right )^p \, dx+f \int x^{2 n} (c x)^m \left (a+b x^n\right )^p \, dx+g \int x^{3 n} (c x)^m \left (a+b x^n\right )^p \, dx\\ &=\left (e x^{-m} (c x)^m\right ) \int x^{m+n} \left (a+b x^n\right )^p \, dx+\left (f x^{-m} (c x)^m\right ) \int x^{m+2 n} \left (a+b x^n\right )^p \, dx+\left (g x^{-m} (c x)^m\right ) \int x^{m+3 n} \left (a+b x^n\right )^p \, dx+\left (d \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int (c x)^m \left (1+\frac {b x^n}{a}\right )^p \, dx\\ &=\frac {d (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{n},-p;\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{c (1+m)}+\left (e x^{-m} (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int x^{m+n} \left (1+\frac {b x^n}{a}\right )^p \, dx+\left (f x^{-m} (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int x^{m+2 n} \left (1+\frac {b x^n}{a}\right )^p \, dx+\left (g x^{-m} (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int x^{m+3 n} \left (1+\frac {b x^n}{a}\right )^p \, dx\\ &=\frac {d (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{n},-p;\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{c (1+m)}+\frac {e x^{1+n} (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+n}{n},-p;\frac {1+m+2 n}{n};-\frac {b x^n}{a}\right )}{1+m+n}+\frac {f x^{1+2 n} (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+2 n}{n},-p;\frac {1+m+3 n}{n};-\frac {b x^n}{a}\right )}{1+m+2 n}+\frac {g x^{1+3 n} (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+3 n}{n},-p;\frac {1+m+4 n}{n};-\frac {b x^n}{a}\right )}{1+m+3 n}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 204, normalized size = 0.69 \[ x (c x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (\frac {d \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{m+1}+x^n \left (\frac {e \, _2F_1\left (\frac {m+n+1}{n},-p;\frac {m+2 n+1}{n};-\frac {b x^n}{a}\right )}{m+n+1}+x^n \left (\frac {f \, _2F_1\left (\frac {m+2 n+1}{n},-p;\frac {m+3 n+1}{n};-\frac {b x^n}{a}\right )}{m+2 n+1}+\frac {g x^n \, _2F_1\left (\frac {m+3 n+1}{n},-p;\frac {m+4 n+1}{n};-\frac {b x^n}{a}\right )}{m+3 n+1}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}+d \right ) \left (c x \right )^{m} \left (b \,x^{n}+a \right )^{p}\, dx \]
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 \[ \int {\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m}\,{d x} \]
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 \[ \int {\left (c\,x\right )}^m\,{\left (a+b\,x^n\right )}^p\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right ) \,d x \]
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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