Optimal. Leaf size=72 \[ \frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{a^{3/2} (2-n)}-\frac {2 c^2 x}{a (2-n) \sqrt {a x^2+b x^n}} \]
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Rubi [A] time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2030, 2008, 206} \begin {gather*} \frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{a^{3/2} (2-n)}-\frac {2 c^2 x}{a (2-n) \sqrt {a x^2+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2008
Rule 2030
Rubi steps
\begin {align*} \int \frac {c^2 x^2}{\left (a x^2+b x^n\right )^{3/2}} \, dx &=c^2 \int \frac {x^2}{\left (a x^2+b x^n\right )^{3/2}} \, dx\\ &=-\frac {2 c^2 x}{a (2-n) \sqrt {a x^2+b x^n}}+\frac {c^2 \int \frac {1}{\sqrt {a x^2+b x^n}} \, dx}{a}\\ &=-\frac {2 c^2 x}{a (2-n) \sqrt {a x^2+b x^n}}+\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^n}}\right )}{a (2-n)}\\ &=-\frac {2 c^2 x}{a (2-n) \sqrt {a x^2+b x^n}}+\frac {2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{a^{3/2} (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 91, normalized size = 1.26 \begin {gather*} \frac {2 c^2 \left (\sqrt {a} x-\sqrt {b} x^{n/2} \sqrt {\frac {a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{1-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{a^{3/2} (n-2) \sqrt {a x^2+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c^2 x^2}{\left (a x^2+b x^n\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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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 \frac {c^{2} x^{2}}{{\left (a x^{2} + b x^{n}\right )}^{\frac {3}{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 \frac {c^{2} x^{2}}{\left (a \,x^{2}+b \,x^{n}\right )^{\frac {3}{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 \frac {x^{2}}{{\left (a x^{2} + b x^{n}\right )}^{\frac {3}{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 \frac {c^2\,x^2}{{\left (b\,x^n+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} \int \frac {x^{2}}{a x^{2} \sqrt {a x^{2} + b x^{n}} + b x^{n} \sqrt {a x^{2} + b x^{n}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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