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 = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2008
Rule 2028
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*}
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Mathematica [A] time = 0.17, size = 117, normalized size = 1.12 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^2+b x^n\right )^{3/2}}{c^4 x^4} \, 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 {{\left (a x^{2} + b x^{n}\right )}^{\frac {3}{2}}}{c^{4} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{2}+b \,x^{n}\right )^{\frac {3}{2}}}{c^{4} x^{4}}\, 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*} \frac {\int \frac {{\left (a x^{2} + b x^{n}\right )}^{\frac {3}{2}}}{x^{4}}\,{d x}}{c^{4}} \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 {{\left (b\,x^n+a\,x^2\right )}^{3/2}}{c^4\,x^4} \,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*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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