Integrand size = 29, antiderivative size = 78 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \]
[In]
[Out]
Rule 663
Rule 671
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e}+\frac {1}{3} (4 d) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx \\ & = -\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 (5 d+e x) \sqrt {c \left (d^2-e^2 x^2\right )}}{3 c e \sqrt {d+e x}} \]
[In]
[Out]
Time = 2.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, \left (e x +5 d \right )}{3 \sqrt {e x +d}\, c e}\) | \(39\) |
gosper | \(-\frac {2 \left (-e x +d \right ) \left (e x +5 d \right ) \sqrt {e x +d}}{3 e \sqrt {-c \,x^{2} e^{2}+c \,d^{2}}}\) | \(43\) |
risch | \(-\frac {2 \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, \left (e x +5 d \right ) \left (-e x +d \right )}{3 \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\) | \(81\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.59 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e x + 5 \, d\right )} \sqrt {e x + d}}{3 \, {\left (c e^{2} x + c d e\right )}} \]
[In]
[Out]
\[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.44 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\frac {2 \, {\left (e^{2} x^{2} + 4 \, d e x - 5 \, d^{2}\right )}}{3 \, \sqrt {-e x + d} \sqrt {c} e} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\frac {2 \, {\left (\frac {4 \, \sqrt {2} \sqrt {c d} d}{c} - \frac {6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} d}{c} + \frac {{\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c^{2}}\right )}}{3 \, e} \]
[In]
[Out]
Time = 10.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {10\,d\,\sqrt {d+e\,x}}{3\,c\,e^2}+\frac {2\,x\,\sqrt {d+e\,x}}{3\,c\,e}\right )}{x+\frac {d}{e}} \]
[In]
[Out]