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 = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 364
Rule 365
Rule 370
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*}
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Mathematica [A] time = 0.23, 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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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.61, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (b \left (c \,x^{q}\right )^{n}+a \right )^{p}\, dx \]
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 \[ \int {\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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