Integrand size = 21, antiderivative size = 193 \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=\frac {\sqrt {\frac {b+a x}{d+c x}} \left (15 b^2 c^2 d-4 a b c d^2-3 a^2 d^3+15 b^2 c^3 x-14 a b c^2 d x-a^2 c d^2 x-10 a b c^3 x^2+10 a^2 c^2 d x^2+8 a^2 c^3 x^3\right )}{24 a^3 c^2}+\frac {\left (-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2}} \]
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Time = 0.17 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.38, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1980, 424, 393, 205, 214} \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=-\frac {(3 a d+5 b c) (b c-a d)^2 \sqrt {\frac {a x+b}{c x+d}}}{12 a^2 c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}-\frac {\left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a x+b}{c x+d}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2}}+\frac {(c x+d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \sqrt {\frac {a x+b}{c x+d}}}{8 a^3 c^2}-\frac {(b c-a d)^2 \sqrt {\frac {a x+b}{c x+d}} \left (b-\frac {d (a x+b)}{c x+d}\right )}{3 a c \left (a-\frac {c (a x+b)}{c x+d}\right )^3} \]
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Rule 205
Rule 214
Rule 393
Rule 424
Rule 1980
Rubi steps \begin{align*} \text {integral}& = -\left ((2 (b c-a d)) \text {Subst}\left (\int \frac {\left (-b+d x^2\right )^2}{\left (a-c x^2\right )^4} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )\right ) \\ & = -\frac {(b c-a d)^2 \sqrt {\frac {b+a x}{d+c x}} \left (b-\frac {d (b+a x)}{d+c x}\right )}{3 a c \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d) \text {Subst}\left (\int \frac {-b (5 b c+a d)+3 d (b c+a d) x^2}{\left (a-c x^2\right )^3} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c} \\ & = -\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {\frac {b+a x}{d+c x}}}{12 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d)^2 \sqrt {\frac {b+a x}{d+c x}} \left (b-\frac {d (b+a x)}{d+c x}\right )}{3 a c \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (a-c x^2\right )^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{4 a^2 c^2} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)}{8 a^3 c^2}-\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {\frac {b+a x}{d+c x}}}{12 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d)^2 \sqrt {\frac {b+a x}{d+c x}} \left (b-\frac {d (b+a x)}{d+c x}\right )}{3 a c \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^2} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)}{8 a^3 c^2}-\frac {(b c-a d)^2 (5 b c+3 a d) \sqrt {\frac {b+a x}{d+c x}}}{12 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d)^2 \sqrt {\frac {b+a x}{d+c x}} \left (b-\frac {d (b+a x)}{d+c x}\right )}{3 a c \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2}} \\ \end{align*}
Time = 10.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04 \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=\frac {\sqrt {c} (b+a x) \sqrt {\frac {a (d+c x)}{-b c+a d}} \left (15 b^2 c^2-2 a b c (2 d+5 c x)+a^2 \left (-3 d^2+2 c d x+8 c^2 x^2\right )\right )+3 \sqrt {-b c+a d} \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {b+a x} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {b+a x}}{\sqrt {-b c+a d}}\right )}{24 a^3 c^{5/2} \sqrt {\frac {b+a x}{d+c x}} \sqrt {\frac {a (d+c x)}{-b c+a d}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(587\) vs. \(2(177)=354\).
Time = 0.09 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.05
method | result | size |
default | \(\frac {\left (a x +b \right ) \left (-12 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a^{2} c d x -36 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a b \,c^{2} x +3 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a^{3} d^{3}+3 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a^{2} b c \,d^{2}-15 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a \,b^{2} c^{2} d +9 \ln \left (\frac {2 a c x +2 \sqrt {a c \,x^{2}+a d x +c x b +b d}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b^{3} c^{3}+24 \ln \left (\frac {2 a c x +2 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) a \,b^{2} c^{2} d -24 \ln \left (\frac {2 a c x +2 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}+a d +b c}{2 \sqrt {a c}}\right ) b^{3} c^{3}+48 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}\, b^{2} c^{2}+16 \left (a c \,x^{2}+a d x +c x b +b d \right )^{\frac {3}{2}} a c \sqrt {a c}-6 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a^{2} d^{2}-24 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, a b c d -18 \sqrt {a c}\, \sqrt {a c \,x^{2}+a d x +c x b +b d}\, b^{2} c^{2}\right )}{48 a^{3} \sqrt {\frac {a x +b}{c x +d}}\, \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, c^{2} \sqrt {a c}}\) | \(588\) |
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Time = 0.32 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.17 \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} \log \left (-2 \, a c x - b c - a d - 2 \, \sqrt {a c} {\left (c x + d\right )} \sqrt {\frac {a x + b}{c x + d}}\right ) - 2 \, {\left (8 \, a^{3} c^{4} x^{3} + 15 \, a b^{2} c^{3} d - 4 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} - 10 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2} + {\left (15 \, a b^{2} c^{4} - 14 \, a^{2} b c^{3} d - a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{c x + d}}}{48 \, a^{4} c^{3}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} {\left (c x + d\right )} \sqrt {\frac {a x + b}{c x + d}}}{a c x + b c}\right ) + {\left (8 \, a^{3} c^{4} x^{3} + 15 \, a b^{2} c^{3} d - 4 \, a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3} - 10 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2} + {\left (15 \, a b^{2} c^{4} - 14 \, a^{2} b c^{3} d - a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{c x + d}}}{24 \, a^{4} c^{3}}\right ] \]
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\[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=\int \frac {x^{2}}{\sqrt {\frac {a x + b}{c x + d}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.84 \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=-\frac {3 \, {\left (5 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {5}{2}} - 8 \, {\left (5 \, a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {3}{2}} + 3 \, {\left (11 \, a^{2} b^{3} c^{3} - 13 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + a^{5} d^{3}\right )} \sqrt {\frac {a x + b}{c x + d}}}{24 \, {\left (a^{6} c^{2} - \frac {3 \, {\left (a x + b\right )} a^{5} c^{3}}{c x + d} + \frac {3 \, {\left (a x + b\right )}^{2} a^{4} c^{4}}{{\left (c x + d\right )}^{2}} - \frac {{\left (a x + b\right )}^{3} a^{3} c^{5}}{{\left (c x + d\right )}^{3}}\right )}} + \frac {{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\frac {c \sqrt {\frac {a x + b}{c x + d}} - \sqrt {a c}}{c \sqrt {\frac {a x + b}{c x + d}} + \sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=\frac {1}{24} \, \sqrt {a c x^{2} + b c x + a d x + b d} {\left (2 \, x {\left (\frac {4 \, x}{a \mathrm {sgn}\left (c x + d\right )} - \frac {5 \, a b c^{2} \mathrm {sgn}\left (c x + d\right ) - a^{2} c d \mathrm {sgn}\left (c x + d\right )}{a^{3} c^{2}}\right )} + \frac {15 \, b^{2} c^{2} \mathrm {sgn}\left (c x + d\right ) - 4 \, a b c d \mathrm {sgn}\left (c x + d\right ) - 3 \, a^{2} d^{2} \mathrm {sgn}\left (c x + d\right )}{a^{3} c^{2}}\right )} + \frac {{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | -b c - a d - 2 \, \sqrt {a c} {\left (\sqrt {a c} x - \sqrt {a c x^{2} + b c x + a d x + b d}\right )} \right |}\right )}{16 \, \sqrt {a c} a^{3} c^{2} \mathrm {sgn}\left (c x + d\right )} \]
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Time = 0.85 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.58 \[ \int \frac {x^2}{\sqrt {\frac {b+a x}{d+c x}}} \, dx=\frac {\frac {{\left (\frac {b+a\,x}{d+c\,x}\right )}^{5/2}\,\left (\frac {a^3\,d^3}{8}+\frac {a^2\,b\,c\,d^2}{8}+\frac {3\,a\,b^2\,c^2\,d}{8}-\frac {5\,b^3\,c^3}{8}\right )}{a^6}+\frac {{\left (\frac {b+a\,x}{d+c\,x}\right )}^{3/2}\,\left (\frac {a^3\,d^3}{3}-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+\frac {5\,b^3\,c^3}{3}\right )}{a^5\,c}-\frac {\sqrt {\frac {b+a\,x}{d+c\,x}}\,\left (\frac {a^3\,d^3}{8}+\frac {a^2\,b\,c\,d^2}{8}-\frac {13\,a\,b^2\,c^2\,d}{8}+\frac {11\,b^3\,c^3}{8}\right )}{a^4\,c^2}}{\frac {3\,c^2\,{\left (b+a\,x\right )}^2}{a^2\,{\left (d+c\,x\right )}^2}-\frac {c^3\,{\left (b+a\,x\right )}^3}{a^3\,{\left (d+c\,x\right )}^3}-\frac {3\,c\,\left (b+a\,x\right )}{a\,\left (d+c\,x\right )}+1}+\frac {\mathrm {atanh}\left (\frac {\sqrt {c}\,\sqrt {\frac {b+a\,x}{d+c\,x}}}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+5\,b^2\,c^2\right )}{8\,a^{7/2}\,c^{5/2}} \]
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