Optimal. Leaf size=81 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )}{a^{5/2} n}+\frac {2}{a^2 n \sqrt {b (c x)^n-a}}-\frac {2}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {367, 12, 266, 51, 63, 205} \[ \frac {2}{a^2 n \sqrt {b (c x)^n-a}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )}{a^{5/2} n}-\frac {2}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 205
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 \tan ^{-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 = 0.04, size = 46, normalized size = 0.57 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {b (c x)^n}{a}\right )}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 277, normalized size = 3.42 \[ \left [-\frac {3 \, {\left (2 \, \left (c x\right )^{n} \sqrt {-a} a b - \left (c x\right )^{2 \, n} \sqrt {-a} b^{2} - \sqrt {-a} a^{2}\right )} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b - a} \sqrt {-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, {\left (3 \, \left (c x\right )^{n} a b - 4 \, a^{2}\right )} \sqrt {\left (c x\right )^{n} b - a}}{3 \, {\left (2 \, \left (c x\right )^{n} a^{4} b n - \left (c x\right )^{2 \, n} a^{3} b^{2} n - a^{5} n\right )}}, \frac {2 \, {\left (3 \, {\left (2 \, \left (c x\right )^{n} a^{\frac {3}{2}} b - \left (c x\right )^{2 \, n} \sqrt {a} b^{2} - a^{\frac {5}{2}}\right )} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b - a}}{\sqrt {a}}\right ) - {\left (3 \, \left (c x\right )^{n} a b - 4 \, a^{2}\right )} \sqrt {\left (c x\right )^{n} b - a}\right )}}{3 \, {\left (2 \, \left (c x\right )^{n} a^{4} b n - \left (c x\right )^{2 \, n} a^{3} b^{2} n - a^{5} n\right )}}\right ] \]
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 = 70, normalized size = 0.86 \[ -\frac {2}{3 \left (b \left (c x \right )^{n}-a \right )^{\frac {3}{2}} a n}+\frac {2 \arctan \left (\frac {\sqrt {b \left (c x \right )^{n}-a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}} n}+\frac {2}{\sqrt {b \left (c x \right )^{n}-a}\, a^{2} n} \]
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 \[ \int \frac {1}{x\,{\left (b\,{\left (c\,x\right )}^n-a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.63, size = 63, normalized size = 0.78 \[ - \frac {2}{3 a n \left (- a + b \left (c x\right )^{n}\right )^{\frac {3}{2}}} + \frac {2}{a^{2} n \sqrt {- a + b \left (c x\right )^{n}}} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {- a + b \left (c x\right )^{n}}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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