Optimal. Leaf size=53 \[ \frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \]
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Rubi [A] time = 0.10, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2031, 2029, 206} \[ \frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx &=\frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {a x^3+b x^n}} \, dx}{\sqrt {x}}\\ &=\frac {\left (2 \sqrt {c x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{(3-n) \sqrt {x}}\\ &=\frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 89, normalized size = 1.68 \[ -\frac {2 \sqrt {b} \sqrt {c x} x^{\frac {n-1}{2}} \sqrt {\frac {a x^{3-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {3}{2}-\frac {n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} (n-3) \sqrt {a x^3+b x^n}} \]
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 \[ \int \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.73, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x}}{\sqrt {a \,x^{3}+b \,x^{n}}}\, dx \]
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 \[ \int \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c\,x}}{\sqrt {b\,x^n+a\,x^3}} \,d x \]
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 \[ \int \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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