Integrand size = 22, antiderivative size = 173 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {a^{3/2} \text {arctanh}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n} \]
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Time = 0.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1371, 748, 828, 857, 635, 212, 738} \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=-\frac {a^{3/2} \text {arctanh}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n}+\frac {\left (8 a c+b^2+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n} \]
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 828
Rule 857
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {\text {Subst}\left (\int \frac {(-2 a-b x) \sqrt {a+b x+c x^2}}{x} \, dx,x,x^n\right )}{2 n} \\ & = \frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}+\frac {\text {Subst}\left (\int \frac {8 a^2 c-\frac {1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{8 c n} \\ & = \frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{n}-\frac {\left (b \left (b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{16 c n} \\ & = \frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {\left (b \left (b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right )}{8 c n} \\ & = \frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\frac {2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )} \left (3 b^2+14 b c x^n+8 c \left (4 a+c x^{2 n}\right )\right )+96 a^{3/2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x^n-\sqrt {a+x^n \left (b+c x^n\right )}}{\sqrt {a}}\right )+3 \left (b^3-12 a b c\right ) \log \left (c n \left (b+2 c x^n-2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )}\right )\right )}{48 c^{3/2} n} \]
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Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {\left (8 c^{2} {\mathrm e}^{2 n \ln \left (x \right )}+14 b \,{\mathrm e}^{n \ln \left (x \right )} c +32 a c +3 b^{2}\right ) \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}}{24 c n}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c \,{\mathrm e}^{n \ln \left (x \right )}}{\sqrt {c}}+\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}\right )}{16 c^{\frac {3}{2}} n}+\frac {3 a b \ln \left (\frac {\frac {b}{2}+c \,{\mathrm e}^{n \ln \left (x \right )}}{\sqrt {c}}+\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}\right )}{4 \sqrt {c}\, n}-\frac {a^{\frac {3}{2}} \ln \left (\left (2 a +b \,{\mathrm e}^{n \ln \left (x \right )}+2 \sqrt {a}\, \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}}\right ) {\mathrm e}^{-n \ln \left (x \right )}\right )}{n}\) | \(209\) |
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Time = 0.43 (sec) , antiderivative size = 827, normalized size of antiderivative = 4.78 \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\left [\frac {48 \, a^{\frac {3}{2}} c^{2} \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{96 \, c^{2} n}, \frac {24 \, a^{\frac {3}{2}} c^{2} \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{48 \, c^{2} n}, \frac {96 \, \sqrt {-a} a c^{2} \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{96 \, c^{2} n}, \frac {48 \, \sqrt {-a} a c^{2} \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{48 \, c^{2} n}\right ] \]
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\[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\int \frac {\left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}}{x}\, dx \]
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\[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}{x} \,d x } \]
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\[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx=\int \frac {{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}}{x} \,d x \]
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