Optimal. Leaf size=145 \[ -\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{9/2} n}+\frac {35 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^4 n}-\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n} \]
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Rubi [A] time = 0.07, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {266, 51, 63, 208} \[ -\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}+\frac {35 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^4 n}-\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{9/2} n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x}} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}-\frac {(7 b) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,x^n\right )}{8 a n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}+\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^n\right )}{48 a^2 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}-\frac {\left (35 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^n\right )}{64 a^3 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}+\frac {35 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^4 n}+\frac {\left (35 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{128 a^4 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}+\frac {35 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^4 n}+\frac {\left (35 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^4 n}\\ &=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}+\frac {35 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^4 n}-\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{9/2} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.28 \[ -\frac {2 b^4 \sqrt {a+b x^n} \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};\frac {b x^n}{a}+1\right )}{a^5 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 209, normalized size = 1.44 \[ \left [\frac {105 \, \sqrt {a} b^{4} x^{4 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (105 \, a b^{3} x^{3 \, n} - 70 \, a^{2} b^{2} x^{2 \, n} + 56 \, a^{3} b x^{n} - 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{384 \, a^{5} n x^{4 \, n}}, \frac {105 \, \sqrt {-a} b^{4} x^{4 \, n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (105 \, a b^{3} x^{3 \, n} - 70 \, a^{2} b^{2} x^{2 \, n} + 56 \, a^{3} b x^{n} - 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{192 \, a^{5} n x^{4 \, n}}\right ] \]
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 \[ \int \frac {x^{-4 \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^{-4 n -1}}{\sqrt {b \,x^{n}+a}}\, dx \]
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 \[ \int \frac {x^{-4 \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
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 \[ \int \frac {1}{x^{4\,n+1}\,\sqrt {a+b\,x^n}} \,d x \]
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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