Optimal. Leaf size=66 \[ -\frac {x (A+B x)}{c \sqrt {a+c x^2}}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}+\frac {2 B \sqrt {a+c x^2}}{c^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {819, 641, 217, 206} \begin {gather*} -\frac {x (A+B x)}{c \sqrt {a+c x^2}}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}+\frac {2 B \sqrt {a+c x^2}}{c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 641
Rule 819
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {x (A+B x)}{c \sqrt {a+c x^2}}+\frac {\int \frac {a A+2 a B x}{\sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {x (A+B x)}{c \sqrt {a+c x^2}}+\frac {2 B \sqrt {a+c x^2}}{c^2}+\frac {A \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c}\\ &=-\frac {x (A+B x)}{c \sqrt {a+c x^2}}+\frac {2 B \sqrt {a+c x^2}}{c^2}+\frac {A \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c}\\ &=-\frac {x (A+B x)}{c \sqrt {a+c x^2}}+\frac {2 B \sqrt {a+c x^2}}{c^2}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 67, normalized size = 1.02 \begin {gather*} \frac {A \sqrt {c} \sqrt {a+c x^2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+2 a B+c x (B x-A)}{c^2 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.36, size = 61, normalized size = 0.92 \begin {gather*} \frac {2 a B-A c x+B c x^2}{c^2 \sqrt {a+c x^2}}-\frac {A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 164, normalized size = 2.48 \begin {gather*} \left [\frac {{\left (A c x^{2} + A a\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (B c x^{2} - A c x + 2 \, B a\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} x^{2} + a c^{2}\right )}}, -\frac {{\left (A c x^{2} + A a\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (B c x^{2} - A c x + 2 \, B a\right )} \sqrt {c x^{2} + a}}{c^{3} x^{2} + a c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 58, normalized size = 0.88 \begin {gather*} \frac {{\left (\frac {B x}{c} - \frac {A}{c}\right )} x + \frac {2 \, B a}{c^{2}}}{\sqrt {c x^{2} + a}} - \frac {A \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 72, normalized size = 1.09 \begin {gather*} \frac {B \,x^{2}}{\sqrt {c \,x^{2}+a}\, c}-\frac {A x}{\sqrt {c \,x^{2}+a}\, c}+\frac {A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}+\frac {2 B a}{\sqrt {c \,x^{2}+a}\, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 64, normalized size = 0.97 \begin {gather*} \frac {B x^{2}}{\sqrt {c x^{2} + a} c} - \frac {A x}{\sqrt {c x^{2} + a} c} + \frac {A \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, B a}{\sqrt {c x^{2} + a} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.47, size = 61, normalized size = 0.92 \begin {gather*} \frac {A\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{c^{3/2}}-\frac {A\,x}{c\,\sqrt {c\,x^2+a}}+\frac {B\,\left (c\,x^2+2\,a\right )}{c^2\,\sqrt {c\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 8.87, size = 83, normalized size = 1.26 \begin {gather*} A \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{c^{\frac {3}{2}}} - \frac {x}{\sqrt {a} c \sqrt {1 + \frac {c x^{2}}{a}}}\right ) + B \left (\begin {cases} \frac {2 a}{c^{2} \sqrt {a + c x^{2}}} + \frac {x^{2}}{c \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________