Optimal. Leaf size=142 \[ \frac {5 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{7/2} n}-\frac {5 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^3 n}+\frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n} \]
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Rubi [A] time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {266, 47, 51, 63, 208} \[ \frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}-\frac {5 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^3 n}+\frac {5 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{7/2} n}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int x^{-1-4 n} \sqrt {a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^5} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n}-\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^n\right )}{48 a n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n}+\frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^n\right )}{64 a^2 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n}+\frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}-\frac {5 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^3 n}-\frac {\left (5 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{128 a^3 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n}+\frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}-\frac {5 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^3 n}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{64 a^3 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n}+\frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}-\frac {5 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^3 n}+\frac {5 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{7/2} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.30 \[ -\frac {2 b^4 \left (a+b x^n\right )^{3/2} \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {b x^n}{a}+1\right )}{3 a^5 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 209, normalized size = 1.47 \[ \left [\frac {15 \, \sqrt {a} b^{4} x^{4 \, n} \log \left (\frac {b x^{n} + 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, {\left (15 \, a b^{3} x^{3 \, n} - 10 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} + 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{384 \, a^{4} n x^{4 \, n}}, -\frac {15 \, \sqrt {-a} b^{4} x^{4 \, n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{3} x^{3 \, n} - 10 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} + 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{192 \, a^{4} n x^{4 \, n}}\right ] \]
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 \[ \int \sqrt {b x^{n} + a} x^{-4 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,x^{n}+a}\, x^{-4 n -1}\, dx \]
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 \[ \int \sqrt {b x^{n} + a} x^{-4 \, n - 1}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,x^n}}{x^{4\,n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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