Optimal. Leaf size=94 \[ \frac{2 c^3 \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{a^{3/2} (3-n) \sqrt{x}}-\frac{2 c^2 (c x)^{3/2}}{a (3-n) \sqrt{a x^3+b x^n}} \]
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Rubi [A] time = 0.159572, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2030, 2031, 2029, 206} \[ \frac{2 c^3 \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{a^{3/2} (3-n) \sqrt{x}}-\frac{2 c^2 (c x)^{3/2}}{a (3-n) \sqrt{a x^3+b x^n}} \]
Antiderivative was successfully verified.
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Rule 2030
Rule 2031
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{(c x)^{7/2}}{\left (a x^3+b x^n\right )^{3/2}} \, dx &=-\frac{2 c^2 (c x)^{3/2}}{a (3-n) \sqrt{a x^3+b x^n}}+\frac{c^3 \int \frac{\sqrt{c x}}{\sqrt{a x^3+b x^n}} \, dx}{a}\\ &=-\frac{2 c^2 (c x)^{3/2}}{a (3-n) \sqrt{a x^3+b x^n}}+\frac{\left (c^3 \sqrt{c x}\right ) \int \frac{\sqrt{x}}{\sqrt{a x^3+b x^n}} \, dx}{a \sqrt{x}}\\ &=-\frac{2 c^2 (c x)^{3/2}}{a (3-n) \sqrt{a x^3+b x^n}}+\frac{\left (2 c^3 \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{a (3-n) \sqrt{x}}\\ &=-\frac{2 c^2 (c x)^{3/2}}{a (3-n) \sqrt{a x^3+b x^n}}+\frac{2 c^3 \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{a^{3/2} (3-n) \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.187117, size = 109, normalized size = 1.16 \[ \frac{2 c^3 \sqrt{c x} \left (\sqrt{a} x^{3/2}-\sqrt{b} x^{n/2} \sqrt{\frac{a x^{3-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{3}{2}-\frac{n}{2}}}{\sqrt{b}}\right )\right )}{a^{3/2} (n-3) \sqrt{x} \sqrt{a x^3+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{{\frac{7}{2}}} \left ( a{x}^{3}+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*} \int \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (a x^{3} + b x^{n}\right )}^{\frac{3}{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 (c x\right )^{\frac{7}{2}}}{{\left (a x^{3} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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