Optimal. Leaf size=65 \[ \frac{c \sqrt{c x^2} (d x)^{m+4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{d^4 (m+4) x} \]
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Rubi [A] time = 0.0285395, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 16, 66, 64} \[ \frac{c \sqrt{c x^2} (d x)^{m+4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{d^4 (m+4) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 66
Rule 64
Rubi steps
\begin{align*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^n \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^3 (d x)^m (a+b x)^n \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int (d x)^{3+m} (a+b x)^n \, dx}{d^3 x}\\ &=\frac{\left (c \sqrt{c x^2} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n}\right ) \int (d x)^{3+m} \left (1+\frac{b x}{a}\right )^n \, dx}{d^3 x}\\ &=\frac{c (d x)^{4+m} \sqrt{c x^2} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} \, _2F_1\left (4+m,-n;5+m;-\frac{b x}{a}\right )}{d^4 (4+m) x}\\ \end{align*}
Mathematica [A] time = 0.0229955, size = 57, normalized size = 0.88 \[ \frac{x \left (c x^2\right )^{3/2} (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{m+4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{n}\, 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 (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n} \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 (\sqrt{c x^{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m} c x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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