Optimal. Leaf size=78 \[ -\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 663
Rule 671
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx &=-\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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 44, normalized size = 0.56 \begin {gather*} -\frac {2 (5 d+e x) \sqrt {c \left (d^2-e^2 x^2\right )}}{3 c e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.50, size = 39, normalized size = 0.50
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (e x +5 d \right )}{3 \sqrt {e x +d}\, c e}\) | \(39\) |
gosper | \(-\frac {2 \left (-e x +d \right ) \left (e x +5 d \right ) \sqrt {e x +d}}{3 e \sqrt {-x^{2} c \,e^{2}+c \,d^{2}}}\) | \(43\) |
risch | \(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, \left (e x +5 d \right ) \left (-e x +d \right )}{3 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 34, normalized size = 0.44 \begin {gather*} \frac {2 \, {\left (x^{2} e^{2} + 4 \, d x e - 5 \, d^{2}\right )} e^{\left (-1\right )}}{3 \, \sqrt {-x e + d} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.71, size = 47, normalized size = 0.60 \begin {gather*} -\frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (x e + 5 \, d\right )} \sqrt {x e + d}}{3 \, {\left (c x e^{2} + c d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.20, size = 61, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, {\left (\frac {4 \, \sqrt {2} \sqrt {c d} d}{c} - \frac {6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} d}{c} + \frac {{\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c^{2}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.57, size = 61, normalized size = 0.78 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________