Optimal. Leaf size=54 \[ -\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{\sqrt {a} c^2 (n+3) \sqrt {c x}} \]
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Rubi [A] time = 0.14, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2031, 2029, 206} \begin {gather*} -\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{\sqrt {a} c^2 (n+3) \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx &=\frac {\sqrt {x} \int \frac {1}{x^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx}{c^2 \sqrt {c x}}\\ &=-\frac {\left (2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{c^2 (3+n) \sqrt {c x}}\\ &=-\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{\sqrt {a} c^2 (3+n) \sqrt {c x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 1.26 \begin {gather*} -\frac {2 x \sqrt {a+b x^{n+3}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+3}}}{\sqrt {a}}\right )}{\sqrt {a} (n+3) (c x)^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.20, size = 70, normalized size = 1.30 \begin {gather*} -\frac {2 \sqrt {a+b x^{n+3}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+3}}}{\sqrt {a}}\right )}{\sqrt {a} c (n+3) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}} \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 \frac {1}{\sqrt {b x^{n} + \frac {a}{x^{3}}} \left (c x\right )^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c x \right )^{\frac {5}{2}} \sqrt {b \,x^{n}+\frac {a}{x^{3}}}}\, 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*} \int \frac {1}{\sqrt {b x^{n} + \frac {a}{x^{3}}} \left (c x\right )^{\frac {5}{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 = -0.02 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{5/2}\,\sqrt {b\,x^n+\frac {a}{x^3}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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