Optimal. Leaf size=119 \[ -\frac {64 d^2 \sqrt {c d^2-c e^2 x^2}}{15 c e \sqrt {d+e x}}-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {64 d^2 \sqrt {c d^2-c e^2 x^2}}{15 c e \sqrt {d+e x}}-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 663
Rule 671
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx &=-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e}+\frac {1}{5} (8 d) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx\\ &=-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e}+\frac {1}{15} \left (32 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx\\ &=-\frac {64 d^2 \sqrt {c d^2-c e^2 x^2}}{15 c e \sqrt {d+e x}}-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 56, normalized size = 0.47 \begin {gather*} -\frac {2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (43 d^2+14 d e x+3 e^2 x^2\right )}{15 c e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.54, size = 51, normalized size = 0.43
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (3 e^{2} x^{2}+14 d x e +43 d^{2}\right )}{15 \sqrt {e x +d}\, c e}\) | \(51\) |
gosper | \(-\frac {2 \left (-e x +d \right ) \left (3 e^{2} x^{2}+14 d x e +43 d^{2}\right ) \sqrt {e x +d}}{15 e \sqrt {-x^{2} c \,e^{2}+c \,d^{2}}}\) | \(55\) |
risch | \(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, \left (3 e^{2} x^{2}+14 d x e +43 d^{2}\right ) \left (-e x +d \right )}{15 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 57, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {c} x^{3} e^{3} + 11 \, \sqrt {c} d x^{2} e^{2} + 29 \, \sqrt {c} d^{2} x e - 43 \, \sqrt {c} d^{3}\right )} e^{\left (-1\right )}}{15 \, \sqrt {-x e + d} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.65, size = 58, normalized size = 0.49 \begin {gather*} -\frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (3 \, x^{2} e^{2} + 14 \, d x e + 43 \, d^{2}\right )} \sqrt {x e + d}}{15 \, {\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 {5}{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 = 3.84, size = 103, normalized size = 0.87 \begin {gather*} \frac {2}{15} \, {\left (\frac {32 \, \sqrt {2} \sqrt {c d} d^{2}}{c} - \frac {60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} d^{2}}{c} + \frac {20 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{3}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.61, size = 79, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{5\,c}+\frac {86\,d^2\,\sqrt {d+e\,x}}{15\,c\,e^2}+\frac {28\,d\,x\,\sqrt {d+e\,x}}{15\,c\,e}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________