Optimal. Leaf size=119 \[ -\frac {\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac {b x \sqrt {a+b x^2} (3 a B+2 A b)}{2 a}+\frac {1}{2} \sqrt {b} (3 a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {453, 277, 195, 217, 206} \begin {gather*} -\frac {\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac {b x \sqrt {a+b x^2} (3 a B+2 A b)}{2 a}+\frac {1}{2} \sqrt {b} (3 a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 277
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^4} \, dx &=-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}-\frac {(-2 A b-3 a B) \int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx}{3 a}\\ &=-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {(b (2 A b+3 a B)) \int \sqrt {a+b x^2} \, dx}{a}\\ &=\frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} (b (2 A b+3 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} (b (2 A b+3 a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} \sqrt {b} (2 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.08, size = 83, normalized size = 0.70 \begin {gather*} \frac {\sqrt {a+b x^2} (-3 a B-2 A b) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x \sqrt {\frac {b x^2}{a}+1}}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.23, size = 88, normalized size = 0.74 \begin {gather*} \frac {1}{2} \left (-3 a \sqrt {b} B-2 A b^{3/2}\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )+\frac {\sqrt {a+b x^2} \left (-2 a A-6 a B x^2-8 A b x^2+3 b B x^4\right )}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.98, size = 166, normalized size = 1.39 \begin {gather*} \left [\frac {3 \, {\left (3 \, B a + 2 \, A b\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (3 \, B b x^{4} - 2 \, {\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x^{3}}, -\frac {3 \, {\left (3 \, B a + 2 \, A b\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, B b x^{4} - 2 \, {\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.52, size = 207, normalized size = 1.74 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2} + a} B b x - \frac {1}{4} \, {\left (3 \, B a \sqrt {b} + 2 \, A b^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} \sqrt {b} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} \sqrt {b} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} b^{\frac {3}{2}} + 3 \, B a^{4} \sqrt {b} + 4 \, A a^{3} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 168, normalized size = 1.41 \begin {gather*} A \,b^{\frac {3}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {3 B a \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}+\frac {\sqrt {b \,x^{2}+a}\, A \,b^{2} x}{a}+\frac {3 \sqrt {b \,x^{2}+a}\, B b x}{2}+\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{2} x}{3 a^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B b x}{a}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {5}{2}} A b}{3 a^{2} x}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B}{a x}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 115, normalized size = 0.97 \begin {gather*} \frac {3}{2} \, \sqrt {b x^{2} + a} B b x + \frac {\sqrt {b x^{2} + a} A b^{2} x}{a} + \frac {3}{2} \, B a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + A b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{x} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{3 \, a x^{3}} \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 \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.75, size = 202, normalized size = 1.70 \begin {gather*} - \frac {A \sqrt {a} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + A b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {A b^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {B \sqrt {a} b x \sqrt {1 + \frac {b x^{2}}{a}}}{2} - \frac {B \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________