Optimal. Leaf size=51 \[ \frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{\sqrt {a} (1-n) \sqrt {c x}} \]
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Rubi [A] time = 0.09, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2031, 2029, 206} \begin {gather*} \frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{\sqrt {a} (1-n) \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c x} \sqrt {a x+b x^n}} \, dx &=\frac {\sqrt {x} \int \frac {1}{\sqrt {x} \sqrt {a x+b x^n}} \, dx}{\sqrt {c x}}\\ &=\frac {\left (2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^n}}\right )}{(1-n) \sqrt {c x}}\\ &=\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{\sqrt {a} (1-n) \sqrt {c x}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 87, normalized size = 1.71 \begin {gather*} -\frac {2 \sqrt {b} x^{\frac {n+1}{2}} \sqrt {\frac {a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {1}{2}-\frac {n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} (n-1) \sqrt {c x} \sqrt {a x+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c x} \sqrt {a x+b x^n}} \, dx \end {gather*}
Verification is not applicable to the result.
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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 {1}{\sqrt {a x + b x^{n}} \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 {1}{\sqrt {c x}\, \sqrt {a x +b \,x^{n}}}\, 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 {1}{\sqrt {a x + b x^{n}} \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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {c\,x}\,\sqrt {b\,x^n+a\,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 {1}{\sqrt {c x} \sqrt {a x + b x^{n}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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