Optimal. Leaf size=122 \[ -\frac {2 a^{3/2} c^4 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(n+3) \sqrt {c x}}+\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{n+3}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)} \]
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Rubi [A] time = 0.28, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2028, 2031, 2029, 206} \begin {gather*} -\frac {2 a^{3/2} c^4 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(n+3) \sqrt {c x}}+\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{n+3}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx &=\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (3+n)}+\left (a c^3\right ) \int \sqrt {c x} \sqrt {\frac {a}{x^3}+b x^n} \, dx\\ &=\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{3+n}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (3+n)}+\left (a^2 c^6\right ) \int \frac {1}{(c x)^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx\\ &=\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{3+n}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (3+n)}+\frac {\left (a^2 c^4 \sqrt {x}\right ) \int \frac {1}{x^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx}{\sqrt {c x}}\\ &=\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{3+n}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (3+n)}-\frac {\left (2 a^2 c^4 \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 )}{(3+n) \sqrt {c x}}\\ &=\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{3+n}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (3+n)}-\frac {2 a^{3/2} c^4 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(3+n) \sqrt {c x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 100, normalized size = 0.82 \begin {gather*} \frac {2 c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n} \left (\sqrt {a+b x^{n+3}} \left (4 a+b x^{n+3}\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+3}}}{\sqrt {a}}\right )\right )}{3 (n+3) \sqrt {a+b x^{n+3}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.69, size = 123, normalized size = 1.01 \begin {gather*} \frac {(c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \left (\frac {2 c^{7/2} \left (\left (a+b x^{n+3}\right )^{3/2}+3 a \sqrt {a+b x^{n+3}}\right )}{3 (n+3)}-\frac {2 a^{3/2} c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+3}}}{\sqrt {a}}\right )}{n+3}\right )}{c^{9/2} \left (a+b x^{n+3}\right )^{3/2}} \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^{3}}\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {7}{2}}\,{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 \left (c x \right )^{\frac {7}{2}} \left (b \,x^{n}+\frac {a}{x^{3}}\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*} \int {\left (b x^{n} + \frac {a}{x^{3}}\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {7}{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 {\left (c\,x\right )}^{7/2}\,{\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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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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