Optimal. Leaf size=104 \[ \frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3} \]
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Rubi [A] time = 0.133196, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 2028, 2008, 206} \[ \frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2028
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^n\right )^{3/2}}{c^4 x^4} \, dx &=\frac{\int \frac{\left (a x^2+b x^n\right )^{3/2}}{x^4} \, dx}{c^4}\\ &=-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3}+\frac{a \int \frac{\sqrt{a x^2+b x^n}}{x^2} \, dx}{c^4}\\ &=-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3}+\frac{a^2 \int \frac{1}{\sqrt{a x^2+b x^n}} \, dx}{c^4}\\ &=-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}\\ &=-\frac{2 a \sqrt{a x^2+b x^n}}{c^4 (2-n) x}-\frac{2 \left (a x^2+b x^n\right )^{3/2}}{3 c^4 (2-n) x^3}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^4 (2-n)}\\ \end{align*}
Mathematica [A] time = 0.157735, size = 117, normalized size = 1.12 \[ \frac{2 \left (-3 a^{3/2} \sqrt{b} x^{\frac{n}{2}+3} \sqrt{\frac{a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{1-\frac{n}{2}}}{\sqrt{b}}\right )+4 a^2 x^4+5 a b x^{n+2}+b^2 x^{2 n}\right )}{3 c^4 (n-2) x^3 \sqrt{a x^2+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.324, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{c}^{4}{x}^{4}} \left ( a{x}^{2}+b{x}^{n} \right ) ^{{\frac{3}{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*} \frac{\int \frac{{\left (a x^{2} + b x^{n}\right )}^{\frac{3}{2}}}{x^{4}}\,{d x}}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} \frac{\int \frac{a \sqrt{a x^{2} + b x^{n}}}{x^{2}}\, dx + \int \frac{b x^{n} \sqrt{a x^{2} + b x^{n}}}{x^{4}}\, dx}{c^{4}} \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 (a x^{2} + b x^{n}\right )}^{\frac{3}{2}}}{c^{4} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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