Optimal. Leaf size=75 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{5/2} n}+\frac {2}{a^2 n \sqrt {a+b (c x)^n}}+\frac {2}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {367, 12, 266, 51, 63, 208} \[ \frac {2}{a^2 n \sqrt {a+b (c x)^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{5/2} n}+\frac {2}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 208
Rule 266
Rule 367
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b (c x)^n\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {c}{x \left (a+b x^n\right )^{5/2}} \, dx,x,c x\right )}{c}\\ &=\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )^{5/2}} \, dx,x,c x\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2}{3 a n \left (a+b (c x)^n\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,(c x)^n\right )}{a n}\\ &=\frac {2}{3 a n \left (a+b (c x)^n\right )^{3/2}}+\frac {2}{a^2 n \sqrt {a+b (c x)^n}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^n\right )}{a^2 n}\\ &=\frac {2}{3 a n \left (a+b (c x)^n\right )^{3/2}}+\frac {2}{a^2 n \sqrt {a+b (c x)^n}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b (c x)^n}\right )}{a^2 b n}\\ &=\frac {2}{3 a n \left (a+b (c x)^n\right )^{3/2}}+\frac {2}{a^2 n \sqrt {a+b (c x)^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{a^{5/2} n}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 43, normalized size = 0.57 \[ \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b (c x)^n}{a}+1\right )}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 262, normalized size = 3.49 \[ \left [\frac {3 \, {\left (2 \, \left (c x\right )^{n} a^{\frac {3}{2}} b + \left (c x\right )^{2 \, n} \sqrt {a} b^{2} + a^{\frac {5}{2}}\right )} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, {\left (3 \, \left (c x\right )^{n} a b + 4 \, a^{2}\right )} \sqrt {\left (c x\right )^{n} b + a}}{3 \, {\left (2 \, \left (c x\right )^{n} a^{4} b n + \left (c x\right )^{2 \, n} a^{3} b^{2} n + a^{5} n\right )}}, \frac {2 \, {\left (3 \, {\left (2 \, \left (c x\right )^{n} \sqrt {-a} a b + \left (c x\right )^{2 \, n} \sqrt {-a} b^{2} + \sqrt {-a} a^{2}\right )} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b + a} \sqrt {-a}}{a}\right ) + {\left (3 \, \left (c x\right )^{n} a b + 4 \, a^{2}\right )} \sqrt {\left (c x\right )^{n} b + a}\right )}}{3 \, {\left (2 \, \left (c x\right )^{n} a^{4} b n + \left (c x\right )^{2 \, n} a^{3} b^{2} n + a^{5} n\right )}}\right ] \]
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 \[ \int \frac {1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac {5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.79 \[ \frac {\frac {2}{3 \left (b \left (c x \right )^{n}+a \right )^{\frac {3}{2}} a}-\frac {2 \arctanh \left (\frac {\sqrt {b \left (c x \right )^{n}+a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{\sqrt {b \left (c x \right )^{n}+a}\, a^{2}}}{n} \]
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 \[ \int \frac {1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac {5}{2}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (a+b\,{\left (c\,x\right )}^n\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.79, size = 70, normalized size = 0.93 \[ \frac {2}{3 a n \left (a + b \left (c x\right )^{n}\right )^{\frac {3}{2}}} + \frac {2}{a^{2} n \sqrt {a + b \left (c x\right )^{n}}} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {a + b \left (c x\right )^{n}}}{\sqrt {- a}} \right )}}{a^{2} n \sqrt {- a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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