Optimal. Leaf size=67 \[ \frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {655, 201, 223,
212} \begin {gather*} \frac {1}{2} d x \sqrt {a+c x^2}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 212
Rule 223
Rule 655
Rubi steps
\begin {align*} \int (d+e x) \sqrt {a+c x^2} \, dx &=\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+d \int \sqrt {a+c x^2} \, dx\\ &=\frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {1}{2} (a d) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {1}{2} (a d) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 68, normalized size = 1.01 \begin {gather*} \frac {\sqrt {a+c x^2} \left (2 a e+3 c d x+2 c e x^2\right )}{6 c}-\frac {a d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 54, normalized size = 0.81
method | result | size |
default | \(\frac {e \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c}+d \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )\) | \(54\) |
risch | \(\frac {\left (2 c e \,x^{2}+3 c d x +2 a e \right ) \sqrt {c \,x^{2}+a}}{6 c}+\frac {a d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 46, normalized size = 0.69 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + a} d x + \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.64, size = 128, normalized size = 1.91 \begin {gather*} \left [\frac {3 \, a \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, c d x + 2 \, {\left (c x^{2} + a\right )} e\right )} \sqrt {c x^{2} + a}}{12 \, c}, -\frac {3 \, a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, c d x + 2 \, {\left (c x^{2} + a\right )} e\right )} \sqrt {c x^{2} + a}}{6 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.52, size = 70, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a} d x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 \sqrt {c}} + e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.43, size = 57, normalized size = 0.85 \begin {gather*} -\frac {a d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, \sqrt {c}} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, x e + 3 \, d\right )} x + \frac {2 \, a e}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.53, size = 52, normalized size = 0.78 \begin {gather*} \frac {e\,{\left (c\,x^2+a\right )}^{3/2}}{3\,c}+\frac {d\,x\,\sqrt {c\,x^2+a}}{2}+\frac {a\,d\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{2\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________