Optimal. Leaf size=80 \[ -\frac {a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}+\frac {\left (a+c x^2\right )^{3/2} (4 A+3 B x)}{12 c}-\frac {a B x \sqrt {a+c x^2}}{8 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {780, 195, 217, 206} \begin {gather*} -\frac {a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}+\frac {\left (a+c x^2\right )^{3/2} (4 A+3 B x)}{12 c}-\frac {a B x \sqrt {a+c x^2}}{8 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 780
Rubi steps
\begin {align*} \int x (A+B x) \sqrt {a+c x^2} \, dx &=\frac {(4 A+3 B x) \left (a+c x^2\right )^{3/2}}{12 c}-\frac {(a B) \int \sqrt {a+c x^2} \, dx}{4 c}\\ &=-\frac {a B x \sqrt {a+c x^2}}{8 c}+\frac {(4 A+3 B x) \left (a+c x^2\right )^{3/2}}{12 c}-\frac {\left (a^2 B\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c}\\ &=-\frac {a B x \sqrt {a+c x^2}}{8 c}+\frac {(4 A+3 B x) \left (a+c x^2\right )^{3/2}}{12 c}-\frac {\left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c}\\ &=-\frac {a B x \sqrt {a+c x^2}}{8 c}+\frac {(4 A+3 B x) \left (a+c x^2\right )^{3/2}}{12 c}-\frac {a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 86, normalized size = 1.08 \begin {gather*} \frac {\sqrt {a+c x^2} \left (\sqrt {c} \left (8 a A+3 a B x+8 A c x^2+6 B c x^3\right )-\frac {3 a^{3/2} B \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}\right )}{24 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.20, size = 77, normalized size = 0.96 \begin {gather*} \frac {a^2 B \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 c^{3/2}}+\frac {\sqrt {a+c x^2} \left (8 a A+3 a B x+8 A c x^2+6 B c x^3\right )}{24 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 157, normalized size = 1.96 \begin {gather*} \left [\frac {3 \, B a^{2} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, B c^{2} x^{3} + 8 \, A c^{2} x^{2} + 3 \, B a c x + 8 \, A a c\right )} \sqrt {c x^{2} + a}}{48 \, c^{2}}, \frac {3 \, B a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (6 \, B c^{2} x^{3} + 8 \, A c^{2} x^{2} + 3 \, B a c x + 8 \, A a c\right )} \sqrt {c x^{2} + a}}{24 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 68, normalized size = 0.85 \begin {gather*} \frac {B a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, B x + 4 \, A\right )} x + \frac {3 \, B a}{c}\right )} x + \frac {8 \, A a}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 75, normalized size = 0.94 \begin {gather*} -\frac {B \,a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+a}\, B a x}{8 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B x}{4 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 67, normalized size = 0.84 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B x}{4 \, c} - \frac {\sqrt {c x^{2} + a} B a x}{8 \, c} - \frac {B a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.41, size = 124, normalized size = 1.55 \begin {gather*} A \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 ) + \frac {B a^{\frac {3}{2}} x}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 B \sqrt {a} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {3}{2}}} + \frac {B c x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________