Optimal. Leaf size=68 \[ \frac {\left (c x^2\right )^p (d x)^{m+1} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac {b x}{a}\right )}{d (m+2 p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 64, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 66, 64} \[ \frac {x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac {b x}{a}\right )}{m+2 p+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 64
Rule 66
Rubi steps
\begin {align*} \int (d x)^m \left (c x^2\right )^p (a+b x)^n \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{2 p} (d x)^m (a+b x)^n \, dx\\ &=\left (x^{-m-2 p} (d x)^m \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^n \, dx\\ &=\left (x^{-m-2 p} (d x)^m \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int x^{m+2 p} \left (1+\frac {b x}{a}\right )^n \, dx\\ &=\frac {x (d x)^m \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-n,1+m+2 p;2+m+2 p;-\frac {b x}{a}\right )}{1+m+2 p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 64, normalized size = 0.94 \[ \frac {x \left (c x^2\right )^p (d x)^m (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac {b x}{a}\right )}{m+2 p+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} \left (d x\right )^{m}, x\right ) \]
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 \[ \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}\right )^{p} \left (d x \right )^{m} \left (b x +a \right )^{n}\, dx \]
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 \[ \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (c\,x^2\right )}^p\,{\left (a+b\,x\right )}^n \,d x \]
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 \[ \int \left (c x^{2}\right )^{p} \left (d x\right )^{m} \left (a + b x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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