Integrand size = 16, antiderivative size = 204 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=-\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )-\frac {5 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (b \left (b^2+44 a c\right )+\frac {2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x+\frac {5}{2} a^{3/2} b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )+\frac {5 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{128 c^{3/2}} \]
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Time = 0.15 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1356, 746, 828, 857, 635, 212, 738} \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\frac {5}{2} a^{3/2} b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )+\frac {5 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{128 c^{3/2}}-\frac {5 \left (\frac {2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{64 c}+x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{5/2}-\frac {5}{24} \left (7 b+\frac {6 c}{x}\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2} \]
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 828
Rule 857
Rule 1356
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x-\frac {5}{2} \text {Subst}\left (\int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x+\frac {5 \text {Subst}\left (\int \frac {\left (-8 a b c-c \left (b^2+12 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{x} \, dx,x,\frac {1}{x}\right )}{16 c} \\ & = -\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )-\frac {5 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (b \left (b^2+44 a c\right )+\frac {2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x-\frac {5 \text {Subst}\left (\int \frac {32 a^2 b c^2-\frac {1}{2} c \left (b^4-24 a b^2 c-48 a^2 c^2\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{64 c^2} \\ & = -\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )-\frac {5 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (b \left (b^2+44 a c\right )+\frac {2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x-\frac {1}{2} \left (5 a^2 b\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )+\frac {\left (5 \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{128 c} \\ & = -\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )-\frac {5 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (b \left (b^2+44 a c\right )+\frac {2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x+\left (5 a^2 b\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+\frac {b}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )+\frac {\left (5 \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+\frac {2 c}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{64 c} \\ & = -\frac {5}{24} \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \left (7 b+\frac {6 c}{x}\right )-\frac {5 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (b \left (b^2+44 a c\right )+\frac {2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} x+\frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )+\frac {5 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{128 c^{3/2}} \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.02 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=-\frac {\sqrt {a+\frac {c+b x}{x^2}} \left (15 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) x^4 \text {arctanh}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )+\sqrt {c} \left (\sqrt {c+x (b+a x)} \left (48 c^3+15 b^3 x^3+8 c^2 x (17 b+27 a x)+2 c x^2 \left (59 b^2+278 a b x-96 a^2 x^2\right )\right )+480 a^{3/2} b c x^4 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{192 c^{3/2} x^3 \sqrt {c+x (b+a x)}} \]
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Time = 0.15 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.22
method | result | size |
risch | \(-\frac {\left (556 a b c \,x^{3}+15 b^{3} x^{3}+216 a \,c^{2} x^{2}+118 b^{2} c \,x^{2}+136 b \,c^{2} x +48 c^{3}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}}{192 x^{3} c}+\frac {\left (384 c b \,a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )+128 a^{3} c \left (\frac {\sqrt {a \,x^{2}+b x +c}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2 a^{\frac {3}{2}}}\right )-\frac {\left (240 a^{2} c^{2}+120 a \,b^{2} c -5 b^{4}\right ) \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right )}{\sqrt {c}}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}{128 c \sqrt {a \,x^{2}+b x +c}}\) | \(249\) |
default | \(\frac {\left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {5}{2}} x \left (6 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b^{3} x^{3}-6 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{4} x^{4}+600 a^{\frac {7}{2}} \sqrt {a \,x^{2}+b x +c}\, b \,c^{3} x^{5}-30 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{3} c^{2} x^{5}+960 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,c^{4} x^{4}-360 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) c^{\frac {7}{2}} a^{\frac {5}{2}} b^{2} x^{4}-96 \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} c^{3} a^{\frac {3}{2}}+4 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b^{2} c \,x^{2}+240 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c^{3} x^{4}-10 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{4} c \,x^{4}+16 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b \,c^{2} x +720 a^{\frac {7}{2}} \sqrt {a \,x^{2}+b x +c}\, c^{4} x^{4}-30 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{4} c^{2} x^{4}-6 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{3} x^{5}+144 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c^{2} x^{4}-144 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} c^{2} x^{2}+660 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{2} c^{3} x^{4}+152 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b c \,x^{5}-152 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b c \,x^{3}+148 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{2} c \,x^{4}+280 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b \,c^{2} x^{5}-10 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{3} c \,x^{5}-720 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) c^{\frac {9}{2}} a^{\frac {7}{2}} x^{4}+15 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) c^{\frac {5}{2}} a^{\frac {3}{2}} b^{4} x^{4}+260 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{2} c^{2} x^{4}\right )}{384 \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c^{4} a^{\frac {3}{2}}}\) | \(701\) |
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Time = 0.43 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.70 \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\left [\frac {960 \, a^{\frac {3}{2}} b c^{2} x^{3} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) - 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {c} x^{3} \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) + 4 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{768 \, c^{2} x^{3}}, -\frac {1920 \, \sqrt {-a} a b c^{2} x^{3} \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {c} x^{3} \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) - 4 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{768 \, c^{2} x^{3}}, \frac {480 \, a^{\frac {3}{2}} b c^{2} x^{3} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) - 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) + 2 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{384 \, c^{2} x^{3}}, -\frac {960 \, \sqrt {-a} a b c^{2} x^{3} \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) - 2 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{384 \, c^{2} x^{3}}\right ] \]
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\[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\int \left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {5}{2}}\, dx \]
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\[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\int { {\left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx=\int {\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{5/2} \,d x \]
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