Optimal. Leaf size=91 \[ \frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n} \]
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Rubi [A] time = 0.04, antiderivative size = 91, 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 = {357, 355, 288, 206} \[ \frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 355
Rule 357
Rubi steps
\begin {align*} \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx &=\frac {\left (x^{-3 n/2} (c x)^{3 n/2}\right ) \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx}{c}\\ &=\frac {\left (2 a x^{-3 n/2} (c x)^{3 n/2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-b x^2\right )^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{c n}\\ &=\frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {\left (a x^{-3 n/2} (c x)^{3 n/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{b c n}\\ &=\frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 110, normalized size = 1.21 \[ \frac {a x^{1-\frac {3 n}{2}} (c x)^{\frac {3 n}{2}-1} \sqrt {\frac {b x^n}{a}+1} \left (\sqrt {b} x^{n/2} \sqrt {\frac {a+b x^n}{a}}-\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )\right )}{b^{3/2} n \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 142, normalized size = 1.56 \[ \left [\frac {2 \, \sqrt {b x^{n} + a} b c^{\frac {3}{2} \, n - 1} x^{\frac {1}{2} \, n} + a \sqrt {b} c^{\frac {3}{2} \, n - 1} \log \left (2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right )}{2 \, b^{2} n}, \frac {\sqrt {b x^{n} + a} b c^{\frac {3}{2} \, n - 1} x^{\frac {1}{2} \, n} + a \sqrt {-b} c^{\frac {3}{2} \, n - 1} \arctan \left (\frac {\sqrt {-b} x^{\frac {1}{2} \, n}}{\sqrt {b x^{n} + a}}\right )}{b^{2} n}\right ] \]
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 \[ \int \frac {\left (c x\right )^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {3 n}{2}-1}}{\sqrt {b \,x^{n}+a}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{\frac {3\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.79, size = 66, normalized size = 0.73 \[ \frac {\sqrt {a} c^{\frac {3 n}{2}} x^{\frac {n}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{b c n} - \frac {a c^{\frac {3 n}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}} c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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