Optimal. Leaf size=59 \[ -\frac {2 \left (a e^2+c d^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \sqrt {d+e x}}{e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ -\frac {2 \left (a e^2+c d^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \sqrt {d+e x}}{e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}}+\frac {c \sqrt {d+e x}}{e^2}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {4 c d \sqrt {d+e x}}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 43, normalized size = 0.73 \[ \frac {2 \left (c \left (-8 d^2-4 d e x+e^2 x^2\right )-3 a e^2\right )}{3 e^3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 49, normalized size = 0.83 \[ \frac {2 \, {\left (c e^{2} x^{2} - 4 \, c d e x - 8 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 54, normalized size = 0.92 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c e^{6} - 6 \, \sqrt {x e + d} c d e^{6}\right )} e^{\left (-9\right )} - \frac {2 \, {\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt {x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 41, normalized size = 0.69 \[ -\frac {2 \left (-c \,e^{2} x^{2}+4 c d e x +3 a \,e^{2}+8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 54, normalized size = 0.92 \[ \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} c - 6 \, \sqrt {e x + d} c d}{e^{2}} - \frac {3 \, {\left (c d^{2} + a e^{2}\right )}}{\sqrt {e x + d} e^{2}}\right )}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 44, normalized size = 0.75 \[ -\frac {6\,a\,e^2-2\,c\,{\left (d+e\,x\right )}^2+6\,c\,d^2+12\,c\,d\,\left (d+e\,x\right )}{3\,e^3\,\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 8.43, size = 58, normalized size = 0.98 \[ - \frac {4 c d \sqrt {d + e x}}{e^{3}} + \frac {2 c \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} - \frac {2 \left (a e^{2} + c d^{2}\right )}{e^{3} \sqrt {d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________