Integrand size = 29, antiderivative size = 136 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {4 \sqrt {2} c^{3/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {679, 675, 214} \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {4 \sqrt {2} c^{3/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e}+\frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
[In]
[Out]
Rule 214
Rule 675
Rule 679
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+(2 c d) \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx \\ & = \frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\left (4 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx \\ & = \frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\left (8 c^2 d^2 e\right ) \text {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right ) \\ & = \frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {4 \sqrt {2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 c \sqrt {c \left (d^2-e^2 x^2\right )} \left (\frac {7 d-e x}{\sqrt {d+e x}}-\frac {6 \sqrt {2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\right )}{3 e} \]
[In]
[Out]
Time = 2.59 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, c \left (6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c \,d^{2}+e x \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}-7 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\, d \right )}{3 \sqrt {e x +d}\, \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\) | \(112\) |
risch | \(\frac {2 \left (-e x +7 d \right ) \left (-e x +d \right ) \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2}}{3 e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}}-\frac {4 d^{2} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c e x +c d}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2}}{e \sqrt {c d}\, \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}}\) | \(177\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.98 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (c d e x + c d^{2}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (c e x - 7 \, c d\right )} \sqrt {e x + d}\right )}}{3 \, {\left (e^{2} x + d e\right )}}, -\frac {2 \, {\left (6 \, \sqrt {2} {\left (c d e x + c d^{2}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) + \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (c e x - 7 \, c d\right )} \sqrt {e x + d}\right )}}{3 \, {\left (e^{2} x + d e\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {6 \, \sqrt {2} c d^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{3} d + {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2}}{c^{3}}\right )} c}{3 \, e} - \frac {4 \, \sqrt {2} {\left (3 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {c d}}{\sqrt {-c d}}\right ) + 4 \, \sqrt {c d} \sqrt {-c d} c d\right )}}{3 \, \sqrt {-c d} e} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
[In]
[Out]