Integrand size = 23, antiderivative size = 178 \[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{8 b^3 c n}-\frac {5 a x^{-2 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{12 b^2 c n}+\frac {x^{-n} (c x)^{7 n/2} \sqrt {a+b x^n}}{3 b c n}-\frac {5 a^3 x^{-7 n/2} (c x)^{7 n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{8 b^{7/2} c n} \]
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Time = 0.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {364, 362, 294, 212} \[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {5 a^3 x^{-7 n/2} (c x)^{7 n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{8 b^{7/2} c n}+\frac {5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{8 b^3 c n}-\frac {5 a x^{-2 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{12 b^2 c n}+\frac {x^{-n} (c x)^{7 n/2} \sqrt {a+b x^n}}{3 b c n} \]
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Rule 212
Rule 294
Rule 362
Rule 364
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-7 n/2} (c x)^{7 n/2}\right ) \int \frac {x^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx}{c} \\ & = \frac {\left (2 a^3 x^{-7 n/2} (c x)^{7 n/2}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1-b x^2\right )^4} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{c n} \\ & = \frac {x^{-n} (c x)^{7 n/2} \sqrt {a+b x^n}}{3 b c n}-\frac {\left (5 a^3 x^{-7 n/2} (c x)^{7 n/2}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1-b x^2\right )^3} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{3 b c n} \\ & = -\frac {5 a x^{-2 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{12 b^2 c n}+\frac {x^{-n} (c x)^{7 n/2} \sqrt {a+b x^n}}{3 b c n}+\frac {\left (5 a^3 x^{-7 n/2} (c x)^{7 n/2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-b x^2\right )^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{4 b^2 c n} \\ & = \frac {5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{8 b^3 c n}-\frac {5 a x^{-2 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{12 b^2 c n}+\frac {x^{-n} (c x)^{7 n/2} \sqrt {a+b x^n}}{3 b c n}-\frac {\left (5 a^3 x^{-7 n/2} (c x)^{7 n/2}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{8 b^3 c n} \\ & = \frac {5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{8 b^3 c n}-\frac {5 a x^{-2 n} (c x)^{7 n/2} \sqrt {a+b x^n}}{12 b^2 c n}+\frac {x^{-n} (c x)^{7 n/2} \sqrt {a+b x^n}}{3 b c n}-\frac {5 a^3 x^{-7 n/2} (c x)^{7 n/2} \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{8 b^{7/2} c n} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.75 \[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {x^{-7 n/2} (c x)^{7 n/2} \sqrt {a+b x^n} \left (\sqrt {b} x^{n/2} \sqrt {1+\frac {b x^n}{a}} \left (15 a^2-10 a b x^n+8 b^2 x^{2 n}\right )-15 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )\right )}{24 b^{7/2} c n \sqrt {1+\frac {b x^n}{a}}} \]
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\[\int \frac {\left (c x \right )^{-1+\frac {7 n}{2}}}{\sqrt {a +b \,x^{n}}}d x\]
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Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.30 \[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\left [\frac {15 \, a^{3} \sqrt {b} c^{\frac {7}{2} \, n - 1} \log \left (2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right ) + 2 \, {\left (8 \, b^{3} c^{\frac {7}{2} \, n - 1} x^{\frac {5}{2} \, n} - 10 \, a b^{2} c^{\frac {7}{2} \, n - 1} x^{\frac {3}{2} \, n} + 15 \, a^{2} b c^{\frac {7}{2} \, n - 1} x^{\frac {1}{2} \, n}\right )} \sqrt {b x^{n} + a}}{48 \, b^{4} n}, \frac {15 \, a^{3} \sqrt {-b} c^{\frac {7}{2} \, n - 1} \arctan \left (\frac {\sqrt {-b} x^{\frac {1}{2} \, n}}{\sqrt {b x^{n} + a}}\right ) + {\left (8 \, b^{3} c^{\frac {7}{2} \, n - 1} x^{\frac {5}{2} \, n} - 10 \, a b^{2} c^{\frac {7}{2} \, n - 1} x^{\frac {3}{2} \, n} + 15 \, a^{2} b c^{\frac {7}{2} \, n - 1} x^{\frac {1}{2} \, n}\right )} \sqrt {b x^{n} + a}}{24 \, b^{4} n}\right ] \]
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Time = 7.73 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.07 \[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {5 a^{\frac {5}{2}} c^{\frac {7 n}{2} - 1} x^{\frac {n}{2}}}{8 b^{3} n \sqrt {1 + \frac {b x^{n}}{a}}} + \frac {5 a^{\frac {3}{2}} c^{\frac {7 n}{2} - 1} x^{\frac {3 n}{2}}}{24 b^{2} n \sqrt {1 + \frac {b x^{n}}{a}}} - \frac {\sqrt {a} c^{\frac {7 n}{2} - 1} x^{\frac {5 n}{2}}}{12 b n \sqrt {1 + \frac {b x^{n}}{a}}} - \frac {5 a^{3} c^{\frac {7 n}{2} - 1} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}} n} + \frac {c^{\frac {7 n}{2} - 1} x^{\frac {7 n}{2}}}{3 \sqrt {a} n \sqrt {1 + \frac {b x^{n}}{a}}} \]
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\[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]
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\[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]
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Timed out. \[ \int \frac {(c x)^{-1+\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {{\left (c\,x\right )}^{\frac {7\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \]
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