Optimal. Leaf size=117 \[ -\frac {2 a^{3/2} c \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(n+1) \sqrt {c x}}+\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{n+1}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)} \]
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Rubi [A] time = 0.23, antiderivative size = 117, 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 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(n+1) \sqrt {c x}}+\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{n+1}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \sqrt {c x} \left (\frac {a}{x}+b x^n\right )^{3/2} \, dx &=\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (1+n)}+(a c) \int \frac {\sqrt {\frac {a}{x}+b x^n}}{\sqrt {c x}} \, dx\\ &=\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{1+n}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (1+n)}+\left (a^2 c^2\right ) \int \frac {1}{(c x)^{3/2} \sqrt {\frac {a}{x}+b x^n}} \, dx\\ &=\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{1+n}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (1+n)}+\frac {\left (a^2 c \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {\frac {a}{x}+b x^n}} \, dx}{\sqrt {c x}}\\ &=\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{1+n}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (1+n)}-\frac {\left (2 a^2 c \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(1+n) \sqrt {c x}}\\ &=\frac {2 a \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{1+n}+\frac {2 (c x)^{3/2} \left (\frac {a}{x}+b x^n\right )^{3/2}}{3 c (1+n)}-\frac {2 a^{3/2} c \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(1+n) \sqrt {c x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 97, normalized size = 0.83 \begin {gather*} \frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n} \left (\sqrt {a+b x^{n+1}} \left (4 a+b x^{n+1}\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+1}}}{\sqrt {a}}\right )\right )}{3 (n+1) \sqrt {a+b x^{n+1}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.50, size = 123, normalized size = 1.05 \begin {gather*} \frac {\sqrt {c x} \sqrt {\frac {a}{x}+b x^n} \left (\frac {2 \sqrt {c} \left (\left (a+b x^{n+1}\right )^{3/2}+3 a \sqrt {a+b x^{n+1}}\right )}{3 (n+1)}-\frac {2 a^{3/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+1}}}{\sqrt {a}}\right )}{n+1}\right )}{\sqrt {c} \sqrt {a+b x^{n+1}}} \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}\right )}^{\frac {3}{2}} \sqrt {c x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c x}\, \left (b \,x^{n}+\frac {a}{x}\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}\right )}^{\frac {3}{2}} \sqrt {c x}\,{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 \sqrt {c\,x}\,{\left (b\,x^n+\frac {a}{x}\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*} \int \sqrt {c x} \left (\frac {a}{x} + b x^{n}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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