Optimal. Leaf size=85 \[ -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n}+\frac {2 a^2 \sqrt {a+b x^n}}{n}+\frac {2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 n} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac {2 a^2 \sqrt {a+b x^n}}{n}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n}+\frac {2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 n} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^{5/2}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac {a \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {2 a^2 \sqrt {a+b x^n}}{n}+\frac {2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac {2 a^2 \sqrt {a+b x^n}}{n}+\frac {2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{b n}\\ &=\frac {2 a^2 \sqrt {a+b x^n}}{n}+\frac {2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 n}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 69, normalized size = 0.81 \[ \frac {2 \sqrt {a+b x^n} \left (23 a^2+11 a b x^n+3 b^2 x^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{15 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 144, normalized size = 1.69 \[ \left [\frac {15 \, a^{\frac {5}{2}} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt {b x^{n} + a}}{15 \, n}, \frac {2 \, {\left (15 \, \sqrt {-a} a^{2} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt {b x^{n} + a}\right )}}{15 \, n}\right ] \]
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 \[ \int \frac {{\left (b x^{n} + a\right )}^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 62, normalized size = 0.73 \[ \frac {-2 a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )+2 \sqrt {b \,x^{n}+a}\, a^{2}+\frac {2 \left (b \,x^{n}+a \right )^{\frac {3}{2}} a}{3}+\frac {2 \left (b \,x^{n}+a \right )^{\frac {5}{2}}}{5}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 83, normalized size = 0.98 \[ \frac {a^{\frac {5}{2}} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, {\left (3 \, {\left (b x^{n} + a\right )}^{\frac {5}{2}} + 5 \, {\left (b x^{n} + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x^{n} + a} a^{2}\right )}}{15 \, n} \]
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 \[ \int \frac {{\left (a+b\,x^n\right )}^{5/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.98, size = 117, normalized size = 1.38 \[ \frac {46 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{15 n} + \frac {a^{\frac {5}{2}} \log {\left (\frac {b x^{n}}{a} \right )}}{n} - \frac {2 a^{\frac {5}{2}} \log {\left (\sqrt {1 + \frac {b x^{n}}{a}} + 1 \right )}}{n} + \frac {22 a^{\frac {3}{2}} b x^{n} \sqrt {1 + \frac {b x^{n}}{a}}}{15 n} + \frac {2 \sqrt {a} b^{2} x^{2 n} \sqrt {1 + \frac {b x^{n}}{a}}}{5 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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