Integrand size = 27, antiderivative size = 313 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {363 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.26 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6317, 6315, 100, 156, 162, 65, 214, 212} \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {363 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right ) \sqrt {c-\frac {c}{a x}}}{a^4 \sqrt {1-\frac {1}{a x}}}+\frac {149 x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {x^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{4 \sqrt {1-\frac {1}{a x}}}+\frac {17 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{24 a \sqrt {1-\frac {1}{a x}}} \]
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Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rule 214
Rule 6315
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a x}} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^3 \, dx}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^5 \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {17}{2 a}-\frac {15 x}{2 a^2}}{x^4 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\frac {107}{4 a^2}+\frac {85 x}{4 a^3}}{x^3 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{12 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {447}{8 a^3}-\frac {321 x}{8 a^4}}{x^2 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{24 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\frac {1089}{16 a^4}+\frac {447 x}{16 a^5}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{24 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^5 \sqrt {1-\frac {1}{a x}}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a^4 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{64 a^3 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {363 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 1.45 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.81 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (447+214 a x+136 a^2 x^2+48 a^3 x^3\right )}{-1+a x}-1089 \sqrt {c} \log (1-a x)+768 \sqrt {2} \sqrt {c} \log \left ((-1+a x)^2\right )+1089 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )-768 \sqrt {2} \sqrt {c} \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right )}{384 a^4} \]
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Time = 0.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (96 \sqrt {\left (a x +1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}+272 \sqrt {\left (a x +1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}+428 \sqrt {\left (a x +1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +894 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-768 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}+1089 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{384 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) a^{\frac {9}{2}} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(224\) |
risch | \(\frac {\left (48 a^{3} x^{3}+136 a^{2} x^{2}+214 a x +447\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{192 a^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {363 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{128 a^{3} \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{a^{4} \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(234\) |
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Time = 0.33 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.81 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\left [\frac {768 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 1089 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 350 \, a^{3} x^{3} + 661 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{768 \, {\left (a^{5} x - a^{4}\right )}}, \frac {768 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 1089 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 350 \, a^{3} x^{3} + 661 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} x - a^{4}\right )}}\right ] \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\text {Timed out} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int \frac {x^3\,\sqrt {c-\frac {c}{a\,x}}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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