Optimal. Leaf size=98 \[ -\frac {2 a^{3/2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{n+2}+\frac {2 a c^2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2}+\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)} \]
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Rubi [A] time = 0.17, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {12, 2028, 2007, 2029, 206} \begin {gather*} -\frac {2 a^{3/2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{n+2}+\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)}+\frac {2 a c^2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2007
Rule 2028
Rule 2029
Rubi steps
\begin {align*} \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx &=c^2 \int x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx\\ &=\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (2+n)}+\left (a c^2\right ) \int \sqrt {\frac {a}{x^2}+b x^n} \, dx\\ &=\frac {2 a c^2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}+\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (2+n)}+\left (a^2 c^2\right ) \int \frac {1}{x^2 \sqrt {\frac {a}{x^2}+b x^n}} \, dx\\ &=\frac {2 a c^2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}+\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (2+n)}-\frac {\left (2 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{2+n}\\ &=\frac {2 a c^2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}+\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (2+n)}-\frac {2 a^{3/2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{2+n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 94, normalized size = 0.96 \begin {gather*} \frac {2 c^2 x \sqrt {\frac {a}{x^2}+b x^n} \left (\sqrt {a+b x^{n+2}} \left (4 a+b x^{n+2}\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+2}}}{\sqrt {a}}\right )\right )}{3 (n+2) \sqrt {a+b x^{n+2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 108, normalized size = 1.10 \begin {gather*} \frac {x \sqrt {\frac {a}{x^2}+b x^n} \left (\frac {2 c^2 \left (\left (a+b x^{n+2}\right )^{3/2}+3 a \sqrt {a+b x^{n+2}}\right )}{3 (n+2)}-\frac {2 a^{3/2} c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+2}}}{\sqrt {a}}\right )}{n+2}\right )}{\sqrt {a+b x^{n+2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \,x^{n}+\frac {a}{x^{2}}\right )^{\frac {3}{2}} c^{2} x^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{2} \int {\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} x^{2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int c^2\,x^2\,{\left (b\,x^n+\frac {a}{x^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{2} \left (\int a \sqrt {\frac {a}{x^{2}} + b x^{n}}\, dx + \int b x^{2} x^{n} \sqrt {\frac {a}{x^{2}} + b x^{n}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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