Integrand size = 16, antiderivative size = 220 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {5 b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{2 a^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1356, 754, 836, 820, 738, 212} \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {5 b \text {arctanh}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{2 a^{7/2}}-\frac {2 x \left (32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}-32 a b^2 c+5 b^4\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}+\frac {x \left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 x \left (-2 a c+b^2+\frac {b c}{x}\right )}{3 a \left (b^2-4 a c\right ) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}} \]
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Rule 212
Rule 738
Rule 754
Rule 820
Rule 836
Rule 1356
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-5 b^2+16 a c\right )-3 b c x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a \left (b^2-4 a c\right )} \\ & = -\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {4 \text {Subst}\left (\int \frac {\frac {1}{4} \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )+\frac {1}{2} b c \left (5 b^2-28 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{3 a^2 \left (b^2-4 a c\right )^2} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{2 a^3} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {(5 b) \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 )}{a^3} \\ & = \frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac {2 \left (b^2-2 a c+\frac {b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2}}-\frac {2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac {b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}-\frac {5 b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{2 a^{7/2}} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\frac {2 \sqrt {a} (c+x (b+a x)) \left (15 b^6 x^2+8 a^2 b c^2 x \left (39 c+32 a x^2\right )-2 a b^3 c x \left (105 c+74 a x^2\right )+10 b^5 \left (3 c x+2 a x^3\right )+3 b^4 \left (5 c^2-30 a c x^2+a^2 x^4\right )+16 a^2 c^2 \left (8 c^2+12 a c x^2+3 a^2 x^4\right )-4 a b^2 c \left (25 c^2-12 a c x^2+6 a^2 x^4\right )\right )}{\left (b^2-4 a c\right )^2}+15 b (c+x (b+a x))^{5/2} \log \left (a^3 \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )}{6 a^{7/2} x^5 \left (a+\frac {c+b x}{x^2}\right )^{5/2}} \]
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Time = 0.15 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.71
method | result | size |
default | \(-\frac {\left (a \,x^{2}+b x +c \right ) \left (-96 a^{\frac {13}{2}} c^{2} x^{4}+48 a^{\frac {11}{2}} b^{2} c \,x^{4}-512 a^{\frac {11}{2}} b \,c^{2} x^{3}-6 a^{\frac {9}{2}} b^{4} x^{4}-384 a^{\frac {11}{2}} c^{3} x^{2}+296 a^{\frac {9}{2}} b^{3} c \,x^{3}-96 a^{\frac {9}{2}} b^{2} c^{2} x^{2}-40 a^{\frac {7}{2}} b^{5} x^{3}-624 a^{\frac {9}{2}} b \,c^{3} x +180 a^{\frac {7}{2}} b^{4} c \,x^{2}-256 a^{\frac {9}{2}} c^{4}+420 a^{\frac {7}{2}} b^{3} c^{2} x -30 a^{\frac {5}{2}} b^{6} x^{2}+200 a^{\frac {7}{2}} b^{2} c^{3}-60 a^{\frac {5}{2}} b^{5} c x +240 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,c^{2}-120 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} c +15 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{5}-30 a^{\frac {5}{2}} b^{4} c^{2}\right )}{6 a^{\frac {11}{2}} \left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {5}{2}} x^{5} \left (4 a c -b^{2}\right )^{2}}\) | \(376\) |
risch | \(\text {Expression too large to display}\) | \(3225\) |
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Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (198) = 396\).
Time = 0.41 (sec) , antiderivative size = 1081, normalized size of antiderivative = 4.91 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{4} + 2 \, {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} x^{3} + {\left (b^{7} - 6 \, a b^{5} c + 32 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} x\right )} \sqrt {a} \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 ) + 4 \, {\left (3 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} x^{5} + 4 \, {\left (5 \, a^{2} b^{5} - 37 \, a^{3} b^{3} c + 64 \, a^{4} b c^{2}\right )} x^{4} + 3 \, {\left (5 \, a b^{6} - 30 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2} + 64 \, a^{4} c^{3}\right )} x^{3} + 6 \, {\left (5 \, a b^{5} c - 35 \, a^{2} b^{3} c^{2} + 52 \, a^{3} b c^{3}\right )} x^{2} + {\left (15 \, a b^{4} c^{2} - 100 \, a^{2} b^{2} c^{3} + 128 \, a^{3} c^{4}\right )} x\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{12 \, {\left (a^{4} b^{4} c^{2} - 8 \, a^{5} b^{2} c^{3} + 16 \, a^{6} c^{4} + {\left (a^{6} b^{4} - 8 \, a^{7} b^{2} c + 16 \, a^{8} c^{2}\right )} x^{4} + 2 \, {\left (a^{5} b^{5} - 8 \, a^{6} b^{3} c + 16 \, a^{7} b c^{2}\right )} x^{3} + {\left (a^{4} b^{6} - 6 \, a^{5} b^{4} c + 32 \, a^{7} c^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} c - 8 \, a^{5} b^{3} c^{2} + 16 \, a^{6} b c^{3}\right )} x\right )}}, \frac {15 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{4} + 2 \, {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} x^{3} + {\left (b^{7} - 6 \, a b^{5} c + 32 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} x\right )} \sqrt {-a} \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 ) + 2 \, {\left (3 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} x^{5} + 4 \, {\left (5 \, a^{2} b^{5} - 37 \, a^{3} b^{3} c + 64 \, a^{4} b c^{2}\right )} x^{4} + 3 \, {\left (5 \, a b^{6} - 30 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2} + 64 \, a^{4} c^{3}\right )} x^{3} + 6 \, {\left (5 \, a b^{5} c - 35 \, a^{2} b^{3} c^{2} + 52 \, a^{3} b c^{3}\right )} x^{2} + {\left (15 \, a b^{4} c^{2} - 100 \, a^{2} b^{2} c^{3} + 128 \, a^{3} c^{4}\right )} x\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{6 \, {\left (a^{4} b^{4} c^{2} - 8 \, a^{5} b^{2} c^{3} + 16 \, a^{6} c^{4} + {\left (a^{6} b^{4} - 8 \, a^{7} b^{2} c + 16 \, a^{8} c^{2}\right )} x^{4} + 2 \, {\left (a^{5} b^{5} - 8 \, a^{6} b^{3} c + 16 \, a^{7} b c^{2}\right )} x^{3} + {\left (a^{4} b^{6} - 6 \, a^{5} b^{4} c + 32 \, a^{7} c^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} c - 8 \, a^{5} b^{3} c^{2} + 16 \, a^{6} b c^{3}\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=\int \frac {1}{\left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (198) = 396\).
Time = 0.37 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.27 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {{\left (15 \, b^{5} \sqrt {c} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 120 \, a b^{3} c^{\frac {3}{2}} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 240 \, a^{2} b c^{\frac {5}{2}} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{4} c - 200 \, a^{\frac {3}{2}} b^{2} c^{2} + 256 \, a^{\frac {5}{2}} c^{3}\right )} \mathrm {sgn}\left (x\right )}{6 \, {\left (a^{\frac {7}{2}} b^{4} \sqrt {c} - 8 \, a^{\frac {9}{2}} b^{2} c^{\frac {3}{2}} + 16 \, a^{\frac {11}{2}} c^{\frac {5}{2}}\right )}} + \frac {{\left ({\left ({\left (\frac {3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} x}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )} + \frac {4 \, {\left (5 \, a b^{5} - 37 \, a^{2} b^{3} c + 64 \, a^{3} b c^{2}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {3 \, {\left (5 \, b^{6} - 30 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {6 \, {\left (5 \, b^{5} c - 35 \, a b^{3} c^{2} + 52 \, a^{2} b c^{3}\right )}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}\right )} x + \frac {15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}}{a^{3} b^{4} \mathrm {sgn}\left (x\right ) - 8 \, a^{4} b^{2} c \mathrm {sgn}\left (x\right ) + 16 \, a^{5} c^{2} \mathrm {sgn}\left (x\right )}}{3 \, {\left (a x^{2} + b x + c\right )}^{\frac {3}{2}}} + \frac {5 \, b \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{5/2}} \,d x \]
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