Optimal. Leaf size=50 \[ \frac{b^3 x (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 c^2 (n+1) \sqrt{c x^2}} \]
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Rubi [A] time = 0.013246, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 65} \[ \frac{b^3 x (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 c^2 (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 65
Rubi steps
\begin{align*} \int \frac{x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{(a+b x)^n}{x^4} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{b^3 x (a+b x)^{1+n} \, _2F_1\left (4,1+n;2+n;1+\frac{b x}{a}\right )}{a^4 c^2 (1+n) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0028012, size = 49, normalized size = 0.98 \[ \frac{b^3 x^5 (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 (n+1) \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{x \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, 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 \frac{{\left (b x + a\right )}^{n} x}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{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 (\frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{c^{3} x^{5}}, 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 \frac{x \left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac{5}{2}}}\, 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 \frac{{\left (b x + a\right )}^{n} x}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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