Optimal. Leaf size=113 \[ -\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n} \]
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Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {272, 43, 44, 65,
214} \begin {gather*} -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int x^{-1-3 n} \sqrt {a+b x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}+\frac {b \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^n\right )}{6 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^n\right )}{8 a n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{16 a^2 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{8 a^2 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 81, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a} x^{-3 n} \sqrt {a+b x^n} \left (-8 a^2-2 a b x^n+3 b^2 x^{2 n}\right )-3 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{24 a^{5/2} n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{-1-3 n} \sqrt {a +b \,x^{n}}\, dx\]
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 [A]
time = 0.41, size = 183, normalized size = 1.62 \begin {gather*} \left [\frac {3 \, \sqrt {a} b^{3} x^{3 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{48 \, a^{3} n x^{3 \, n}}, \frac {3 \, \sqrt {-a} b^{3} x^{3 \, n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{24 \, a^{3} n x^{3 \, n}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,x^n}}{x^{3\,n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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