Optimal. Leaf size=110 \[ \frac {\left (a+b x^2\right )^{3/2} (2 a B+3 A b)}{6 a}+\frac {1}{2} \sqrt {a+b x^2} (2 a B+3 A b)-\frac {1}{2} \sqrt {a} (2 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 50, 63, 208} \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} (2 a B+3 A b)}{6 a}+\frac {1}{2} \sqrt {a+b x^2} (2 a B+3 A b)-\frac {1}{2} \sqrt {a} (2 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {\left (\frac {3 A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )}{2 a}\\ &=\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {1}{4} (3 A b+2 a B) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {1}{4} (a (3 A b+2 a B)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {(a (3 A b+2 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 b}\\ &=\frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {1}{2} \sqrt {a} (3 A b+2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 80, normalized size = 0.73 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (-3 a A+8 a B x^2+6 A b x^2+2 b B x^4\right )}{x^2}-3 \sqrt {a} (2 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 85, normalized size = 0.77 \begin {gather*} \frac {1}{2} \left (-2 a^{3/2} B-3 \sqrt {a} A b\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+b x^2} \left (-3 a A+8 a B x^2+6 A b x^2+2 b B x^4\right )}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 167, normalized size = 1.52 \begin {gather*} \left [\frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {a} x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B b x^{4} + 2 \, {\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x^{2}}, \frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, B b x^{4} + 2 \, {\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 103, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b + 6 \, \sqrt {b x^{2} + a} B a b + 6 \, \sqrt {b x^{2} + a} A b^{2} - \frac {3 \, \sqrt {b x^{2} + a} A a b}{x^{2}} + \frac {3 \, {\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 132, normalized size = 1.20 \begin {gather*} -\frac {3 A \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}-B \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {3 \sqrt {b \,x^{2}+a}\, A b}{2}+\sqrt {b \,x^{2}+a}\, B a +\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A b}{2 a}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B}{3}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.03, size = 109, normalized size = 0.99 \begin {gather*} -B a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {3}{2} \, A \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B + \sqrt {b x^{2} + a} B a + \frac {3}{2} \, \sqrt {b x^{2} + a} A b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.49, size = 94, normalized size = 0.85 \begin {gather*} \frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{3}-B\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+A\,b\,\sqrt {b\,x^2+a}+B\,a\,\sqrt {b\,x^2+a}-\frac {A\,a\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {3\,A\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 71.66, size = 184, normalized size = 1.67 \begin {gather*} - \frac {3 A \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {A a \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - B a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B a \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + B b \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________