Optimal. Leaf size=93 \[ -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n}+\frac {2 a^2 \sqrt {a+b (c x)^n}}{n}+\frac {2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b (c x)^n\right )^{5/2}}{5 n} \]
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Rubi [A] time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {367, 12, 266, 50, 63, 208} \[ \frac {2 a^2 \sqrt {a+b (c x)^n}}{n}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n}+\frac {2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b (c x)^n\right )^{5/2}}{5 n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 266
Rule 367
Rubi steps
\begin {align*} \int \frac {\left (a+b (c x)^n\right )^{5/2}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {c \left (a+b x^n\right )^{5/2}}{x} \, dx,x,c x\right )}{c}\\ &=\operatorname {Subst}\left (\int \frac {\left (a+b x^n\right )^{5/2}}{x} \, dx,x,c x\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac {a \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 a^2 \sqrt {a+b (c x)^n}}{n}+\frac {2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 a^2 \sqrt {a+b (c x)^n}}{n}+\frac {2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b (c 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 (c x)^n}\right )}{b n}\\ &=\frac {2 a^2 \sqrt {a+b (c x)^n}}{n}+\frac {2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 \left (a+b (c x)^n\right )^{5/2}}{5 n}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 0.83 \[ \frac {2 \sqrt {a+b (c x)^n} \left (23 a^2+11 a b (c x)^n+3 b^2 (c x)^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{15 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 164, normalized size = 1.76 \[ \left [\frac {15 \, a^{\frac {5}{2}} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, {\left (11 \, \left (c x\right )^{n} a b + 3 \, \left (c x\right )^{2 \, n} b^{2} + 23 \, a^{2}\right )} \sqrt {\left (c x\right )^{n} b + a}}{15 \, n}, \frac {2 \, {\left (15 \, \sqrt {-a} a^{2} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b + a} \sqrt {-a}}{a}\right ) + {\left (11 \, \left (c x\right )^{n} a b + 3 \, \left (c x\right )^{2 \, n} b^{2} + 23 \, a^{2}\right )} \sqrt {\left (c x\right )^{n} b + a}\right )}}{15 \, n}\right ] \]
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 \[ \int \frac {{\left (\left (c x\right )^{n} b + a\right )}^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.75 \[ \frac {-2 a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b \left (c x \right )^{n}+a}}{\sqrt {a}}\right )+2 \sqrt {b \left (c x \right )^{n}+a}\, a^{2}+\frac {2 \left (b \left (c x \right )^{n}+a \right )^{\frac {3}{2}} a}{3}+\frac {2 \left (b \left (c x \right )^{n}+a \right )^{\frac {5}{2}}}{5}}{n} \]
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 \[ \int \frac {{\left (\left (c x\right )^{n} b + a\right )}^{\frac {5}{2}}}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,{\left (c\,x\right )}^n\right )}^{5/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.66, size = 122, normalized size = 1.31 \[ \begin {cases} \frac {\frac {2 a^{3} \operatorname {atan}{\left (\frac {\sqrt {a + b \left (c x\right )^{n}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 a^{2} \sqrt {a + b \left (c x\right )^{n}} + \frac {2 a \left (a + b \left (c x\right )^{n}\right )^{\frac {3}{2}}}{3} + \frac {2 \left (a + b \left (c x\right )^{n}\right )^{\frac {5}{2}}}{5}}{n} & \text {for}\: n \neq 0 \\- \left (- a^{2} \sqrt {a + b} - 2 a b \sqrt {a + b} - b^{2} \sqrt {a + b}\right ) \log {\left (c x \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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