Optimal. Leaf size=84 \[ \frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (n+1)}-\frac {2 \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(n+1) \sqrt {c x}} \]
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Rubi [A] time = 0.17, antiderivative size = 84, 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} \begin {gather*} \frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (n+1)}-\frac {2 \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{(n+1) \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x}+b x^n}}{\sqrt {c x}} \, dx &=\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (1+n)}+(a c) \int \frac {1}{(c x)^{3/2} \sqrt {\frac {a}{x}+b x^n}} \, dx\\ &=\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (1+n)}+\frac {\left (a \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {\frac {a}{x}+b x^n}} \, dx}{\sqrt {c x}}\\ &=\frac {2 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (1+n)}-\frac {\left (2 a \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 \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}{c (1+n)}-\frac {2 \sqrt {a} \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 = 0.07, size = 85, normalized size = 1.01 \begin {gather*} \frac {x \sqrt {\frac {a}{x}+b x^n} \left (2 \sqrt {a+b x^{n+1}}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+1}}}{\sqrt {a}}\right )\right )}{(n+1) \sqrt {c x} \sqrt {a+b x^{n+1}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.27, size = 104, normalized size = 1.24 \begin {gather*} \frac {\sqrt {c x} \sqrt {\frac {a}{x}+b x^n} \left (\frac {2 \sqrt {a+b x^{n+1}}}{\sqrt {c} (n+1)}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+1}}}{\sqrt {a}}\right )}{\sqrt {c} (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 \frac {\sqrt {b x^{n} + \frac {a}{x}}}{\sqrt {c x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b \,x^{n}+\frac {a}{x}}}{\sqrt {c x}}\, 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 {\sqrt {b x^{n} + \frac {a}{x}}}{\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 \frac {\sqrt {b\,x^n+\frac {a}{x}}}{\sqrt {c\,x}} \,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 \frac {\sqrt {\frac {a}{x} + b x^{n}}}{\sqrt {c x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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