Optimal. Leaf size=90 \[ \frac {2}{a c^4 (3+n) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}-\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{a^{3/2} c^5 (3+n) \sqrt {c x}} \]
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Rubi [A]
time = 0.14, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 = {2055, 2056,
2054, 212} \begin {gather*} \frac {2}{a c^4 (n+3) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}-\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{a^{3/2} c^5 (n+3) \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2054
Rule 2055
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{11/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}} \, dx &=\frac {2}{a c^4 (3+n) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}+\frac {\int \frac {1}{(c x)^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx}{a c^3}\\ &=\frac {2}{a c^4 (3+n) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}+\frac {\sqrt {x} \int \frac {1}{x^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx}{a c^5 \sqrt {c x}}\\ &=\frac {2}{a c^4 (3+n) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}-\frac {\left (2 \sqrt {x}\right ) \text {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 )}{a c^5 (3+n) \sqrt {c x}}\\ &=\frac {2}{a c^4 (3+n) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}-\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{a^{3/2} c^5 (3+n) \sqrt {c x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 78, normalized size = 0.87 \begin {gather*} \frac {2 \left (\sqrt {a}-\sqrt {a+b x^{3+n}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{3+n}}}{\sqrt {a}}\right )\right )}{a^{3/2} c^4 (3+n) (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c x \right )^{\frac {11}{2}} \left (\frac {a}{x^{3}}+b \,x^{n}\right )^{\frac {3}{2}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{{\left (c\,x\right )}^{11/2}\,{\left (b\,x^n+\frac {a}{x^3}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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