Optimal. Leaf size=66 \[ \frac{2 a^2 \sqrt{a+b x^n}}{b^3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 b^3 n}-\frac{4 a \left (a+b x^n\right )^{3/2}}{3 b^3 n} \]
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Rubi [A] time = 0.0298117, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac{2 a^2 \sqrt{a+b x^n}}{b^3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 b^3 n}-\frac{4 a \left (a+b x^n\right )^{3/2}}{3 b^3 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+3 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a+b x}}-\frac{2 a \sqrt{a+b x}}{b^2}+\frac{(a+b x)^{3/2}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{2 a^2 \sqrt{a+b x^n}}{b^3 n}-\frac{4 a \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 b^3 n}\\ \end{align*}
Mathematica [A] time = 0.0195867, size = 44, normalized size = 0.67 \[ \frac{2 \sqrt{a+b x^n} \left (8 a^2-4 a b x^n+3 b^2 x^{2 n}\right )}{15 b^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 41, normalized size = 0.6 \begin{align*}{\frac{6\,{b}^{2} \left ({x}^{n} \right ) ^{2}-8\,a{x}^{n}b+16\,{a}^{2}}{15\,{b}^{3}n}\sqrt{a+b{x}^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979454, size = 72, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (3 \, b^{3} x^{3 \, n} - a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )}}{15 \, \sqrt{b x^{n} + a} b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.98488, size = 89, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2 \, n} - 4 \, a b x^{n} + 8 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 44.0415, size = 916, normalized size = 13.88 \begin{align*} \text{result too large to display} \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{x^{3 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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