Integrand size = 29, antiderivative size = 178 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} d^{3/2} e} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {677, 687, 675, 214} \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} d^{3/2} e}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}+\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]
[In]
[Out]
Rule 214
Rule 675
Rule 677
Rule 687
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac {1}{2} c \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx \\ & = \frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {1}{8} c^2 \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}} \, dx \\ & = \frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {c^2 \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{32 d} \\ & = \frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {\left (c^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 )}{16 d} \\ & = \frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac {c^{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 )}{16 \sqrt {2} d^{3/2} e} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=-\frac {\left (c \left (d^2-e^2 x^2\right )\right )^{3/2} \left (2 \sqrt {d} \sqrt {d^2-e^2 x^2} \left (7 d^2-22 d e x+3 e^2 x^2\right )+3 \sqrt {2} (d+e x)^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )\right )}{96 d^{3/2} e (d+e x)^{7/2} \left (d^2-e^2 x^2\right )^{3/2}} \]
[In]
[Out]
Time = 2.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.35
method | result | size |
default | \(-\frac {\sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, c \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c \,e^{3} x^{3}+9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c d \,e^{2} x^{2}+9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c \,d^{2} e x +3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) c \,d^{3}+6 e^{2} x^{2} \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}-44 d e x \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}+14 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\, d^{2}\right )}{96 \left (e x +d \right )^{\frac {7}{2}} \sqrt {c \left (-e x +d \right )}\, e d \sqrt {c d}}\) | \(241\) |
[In]
[Out]
none
Time = 0.45 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.49 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (3 \, c e^{2} x^{2} - 22 \, c d e x + 7 \, c d^{2}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{96 \, {\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}}, -\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) + {\left (3 \, c e^{2} x^{2} - 22 \, c d e x + 7 \, c d^{2}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{48 \, {\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\frac {\frac {3 \, \sqrt {2} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d} d} + \frac {2 \, {\left (12 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{4} d^{2} - 16 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{3} d - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2}\right )}}{{\left (e x + d\right )}^{3} c^{3} d}}{96 \, e} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]
[In]
[Out]