Optimal. Leaf size=122 \[ \frac{2 a^{3/2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c^2 (1-n) \sqrt{c x}}-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \]
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Rubi [A] time = 0.188308, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2028, 2031, 2029, 206} \[ \frac{2 a^{3/2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c^2 (1-n) \sqrt{c x}}-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2028
Rule 2031
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx &=-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac{a \int \frac{\sqrt{a x+b x^n}}{(c x)^{3/2}} \, dx}{c}\\ &=-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac{a^2 \int \frac{1}{\sqrt{c x} \sqrt{a x+b x^n}} \, dx}{c^2}\\ &=-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac{\left (a^2 \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{a x+b x^n}} \, dx}{c^2 \sqrt{c x}}\\ &=-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac{\left (2 a^2 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c^2 (1-n) \sqrt{c x}}\\ &=-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac{2 a^{3/2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c^2 (1-n) \sqrt{c x}}\\ \end{align*}
Mathematica [A] time = 0.250859, size = 120, normalized size = 0.98 \[ \frac{x \left (-6 a^{3/2} \sqrt{b} x^{\frac{n+3}{2}} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )+8 a^2 x^2+10 a b x^{n+1}+2 b^2 x^{2 n}\right )}{3 (n-1) (c x)^{5/2} \sqrt{a x+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.323, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+b{x}^{n} \right ) ^{{\frac{3}{2}}} \left ( cx \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 (a x + b x^{n}\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \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(-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 \frac{{\left (a x + b x^{n}\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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