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 = 0.06, 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 = {2056, 2054,
212} \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 212
Rule 2054
Rule 2056
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 ) \text {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 = 0.19, size = 87, normalized size = 1.71 \begin {gather*} -\frac {2 \sqrt {b} x^{\frac {1+n}{2}} \sqrt {1+\frac {a x^{1-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {1}{2}-\frac {n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} (-1+n) \sqrt {c x} \sqrt {a x+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {c x}\, \sqrt {a x +b \,x^{n}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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