Optimal. Leaf size=86 \[ \frac{(d x)^{m+1} \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{m+1}{n q};\frac{m+1}{n q}+1;-\frac{b \left (c x^q\right )^n}{a}\right )}{d (m+1)} \]
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Rubi [A] time = 0.0437766, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {370, 365, 364} \[ \frac{(d x)^{m+1} \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{m+1}{n q};\frac{m+1}{n q}+1;-\frac{b \left (c x^q\right )^n}{a}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 370
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \, dx &=\operatorname{Subst}\left (\int (d x)^m \left (a+b c^n x^{n q}\right )^p \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right )\\ &=\operatorname{Subst}\left (\left (\left (a+b c^n x^{n q}\right )^p \left (1+\frac{b c^n x^{n q}}{a}\right )^{-p}\right ) \int (d x)^m \left (1+\frac{b c^n x^{n q}}{a}\right )^p \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right )\\ &=\frac{(d x)^{1+m} \left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac{b \left (c x^q\right )^n}{a}\right )^{-p} \, _2F_1\left (-p,\frac{1+m}{n q};1+\frac{1+m}{n q};-\frac{b \left (c x^q\right )^n}{a}\right )}{d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.13734, size = 82, normalized size = 0.95 \[ \frac{x (d x)^m \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{m+1}{n q};\frac{m+1}{n q}+1;-\frac{b \left (c x^q\right )^n}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.085, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \left (d x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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