Optimal. Leaf size=85 \[ \frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (n+3)}-\frac {2 \sqrt {a} c \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(n+3) \sqrt {c x}} \]
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Rubi [A] time = 0.20, antiderivative size = 85, 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 = {2028, 2031, 2029, 206} \[ \frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (n+3)}-\frac {2 \sqrt {a} c \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(n+3) \sqrt {c x}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \sqrt {c x} \sqrt {\frac {a}{x^3}+b x^n} \, dx &=\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (3+n)}+\left (a c^3\right ) \int \frac {1}{(c x)^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx\\ &=\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (3+n)}+\frac {\left (a c \sqrt {x}\right ) \int \frac {1}{x^{5/2} \sqrt {\frac {a}{x^3}+b x^n}} \, dx}{\sqrt {c x}}\\ &=\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (3+n)}-\frac {\left (2 a c \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 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (3+n)}-\frac {2 \sqrt {a} c \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 = 0.07, size = 85, normalized size = 1.00 \[ \frac {x \sqrt {c x} \sqrt {\frac {a}{x^3}+b x^n} \left (2 \sqrt {a+b x^{n+3}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+3}}}{\sqrt {a}}\right )\right )}{(n+3) \sqrt {a+b x^{n+3}}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \sqrt {b x^{n} + \frac {a}{x^{3}}} \sqrt {c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \sqrt {c x}\, \sqrt {b \,x^{n}+\frac {a}{x^{3}}}\, dx \]
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 \[ \int \sqrt {b x^{n} + \frac {a}{x^{3}}} \sqrt {c x}\,{d x} \]
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 \[ \int \sqrt {c\,x}\,\sqrt {b\,x^n+\frac {a}{x^3}} \,d x \]
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 \[ \int \sqrt {c x} \sqrt {\frac {a}{x^{3}} + b x^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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