Optimal. Leaf size=33 \[ \frac{c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
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Rubi [A] time = 0.0064947, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2} (a+b x)^n}{x^5} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int (a+b x)^n \, dx}{x}\\ &=\frac{c^2 \sqrt{c x^2} (a+b x)^{1+n}}{b (1+n) x}\\ \end{align*}
Mathematica [A] time = 0.016853, size = 31, normalized size = 0.94 \[ \frac{c^3 x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ){x}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02733, size = 38, normalized size = 1.15 \begin{align*} \frac{{\left (b c^{\frac{5}{2}} x + a c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57848, size = 77, normalized size = 2.33 \begin{align*} \frac{{\left (b c^{2} x + a c^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 (c x^{2}\right )^{\frac{5}{2}}{\left (b x + a\right )}^{n}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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