Optimal. Leaf size=73 \[ -\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n}+\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n} \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {12, 266, 50, 63, 208} \begin {gather*} -\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n}+\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx &=\frac {\int \frac {\left (a+b x^n\right )^{3/2}}{x} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{b c n}\\ &=\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {a+b x^n} \left (4 a+b x^n\right )-6 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{3 c n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 68, normalized size = 0.93 \begin {gather*} \frac {2 \left (\left (a+b x^n\right )^{3/2}+3 a \sqrt {a+b x^n}\right )}{3 c n}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 120, normalized size = 1.64 \begin {gather*} \left [\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (b x^{n} + 4 \, a\right )} \sqrt {b x^{n} + a}}{3 \, c n}, \frac {2 \, {\left (3 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (b x^{n} + 4 \, a\right )} \sqrt {b x^{n} + a}\right )}}{3 \, c n}\right ] \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*} \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}}}{c x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 51, normalized size = 0.70 \begin {gather*} \frac {-2 a^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )+2 \sqrt {b \,x^{n}+a}\, a +\frac {2 \left (b \,x^{n}+a \right )^{\frac {3}{2}}}{3}}{c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 73, normalized size = 1.00 \begin {gather*} \frac {\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, {\left ({\left (b x^{n} + a\right )}^{\frac {3}{2}} + 3 \, \sqrt {b x^{n} + a} a\right )}}{n}}{3 \, c} \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 {{\left (a+b\,x^n\right )}^{3/2}}{c\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.08, size = 88, normalized size = 1.21 \begin {gather*} \frac {\frac {8 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{3 n} + \frac {a^{\frac {3}{2}} \log {\left (\frac {b x^{n}}{a} \right )}}{n} - \frac {2 a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b x^{n}}{a}} + 1 \right )}}{n} + \frac {2 \sqrt {a} b x^{n} \sqrt {1 + \frac {b x^{n}}{a}}}{3 n}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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