Integrand size = 21, antiderivative size = 140 \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {16 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6311, 6316, 47, 37} \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {16 x \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^3 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^2 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{5/2} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {\left (4 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 a \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {16 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.48 \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (8-4 a x+3 a^2 x^2+15 a^3 x^3\right )}{105 a^3 \sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (15 a^{2} x^{2}-12 a x +8\right ) \sqrt {-a c x +c}}{105 a^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(49\) |
default | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right ) \left (15 a^{2} x^{2}-12 a x +8\right )}{105 \sqrt {\frac {a x -1}{a x +1}}\, a^{3}}\) | \(50\) |
risch | \(-\frac {2 c \left (a x -1\right ) \left (15 a^{3} x^{3}+3 a^{2} x^{2}-4 a x +8\right )}{105 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a^{3}}\) | \(59\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.49 \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{4} x^{4} + 18 \, a^{3} x^{3} - a^{2} x^{2} + 4 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{4} x - a^{3}\right )}} \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\int \frac {x^{2} \sqrt {- c \left (a x - 1\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.39 \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (15 \, a^{3} \sqrt {-c} x^{3} + 3 \, a^{2} \sqrt {-c} x^{2} - 4 \, a \sqrt {-c} x + 8 \, \sqrt {-c}\right )} \sqrt {a x + 1}}{105 \, a^{3}} \]
[In]
[Out]
Exception generated. \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 4.65 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.41 \[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (15\,a^2\,x^2-12\,a\,x+8\right )}{105\,a^3\,\left (a\,x-1\right )} \]
[In]
[Out]