Optimal. Leaf size=136 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{x \left (a+b x^2\right )^{5/2} (a B+6 A b)}{6 a}+\frac{5}{24} x \left (a+b x^2\right )^{3/2} (a B+6 A b)+\frac{5}{16} a x \sqrt{a+b x^2} (a B+6 A b)-\frac{A \left (a+b x^2\right )^{7/2}}{a x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.154543, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{x \left (a+b x^2\right )^{5/2} (a B+6 A b)}{6 a}+\frac{5}{24} x \left (a+b x^2\right )^{3/2} (a B+6 A b)+\frac{5}{16} a x \sqrt{a+b x^2} (a B+6 A b)-\frac{A \left (a+b x^2\right )^{7/2}}{a x} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.9891, size = 128, normalized size = 0.94 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{a x} + \frac{5 a^{2} \left (6 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 \sqrt{b}} + \frac{5 a x \sqrt{a + b x^{2}} \left (6 A b + B a\right )}{16} + x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (\frac{5 A b}{4} + \frac{5 B a}{24}\right ) + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (6 A b + B a\right )}{6 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.177886, size = 108, normalized size = 0.79 \[ \sqrt{a+b x^2} \left (-\frac{a^2 A}{x}+\frac{1}{24} b x^3 (13 a B+6 A b)+\frac{1}{16} a x (11 a B+18 A b)+\frac{1}{6} b^2 B x^5\right )+\frac{5 a^2 (a B+6 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 158, normalized size = 1.2 \[{\frac{Bx}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Bxa}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Bx{a}^{2}}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}-{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Axb}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Axb}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,Axab}{8}\sqrt{b{x}^{2}+a}}+{\frac{15\,A{a}^{2}}{8}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(5/2)*(B*x^2+A)/x^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.264099, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, B b^{2} x^{6} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{4} - 48 \, A a^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{96 \, \sqrt{b} x}, \frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, B b^{2} x^{6} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{4} - 48 \, A a^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{48 \, \sqrt{-b} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 57.1551, size = 306, normalized size = 2.25 \[ - \frac{A a^{\frac{5}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + A a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x^{2}}{a}} - \frac{7 A a^{\frac{3}{2}} b x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} b^{2} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 A a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8} + \frac{A b^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 B a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 B a^{\frac{3}{2}} b x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 B \sqrt{a} b^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{B b^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.250509, size = 197, normalized size = 1.45 \[ \frac{2 \, A a^{3} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} + \frac{1}{48} \,{\left (2 \,{\left (4 \, B b^{2} x^{2} + \frac{13 \, B a b^{5} + 6 \, A b^{6}}{b^{4}}\right )} x^{2} + \frac{3 \,{\left (11 \, B a^{2} b^{4} + 18 \, A a b^{5}\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x - \frac{5 \,{\left (B a^{3} \sqrt{b} + 6 \, A a^{2} b^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^2,x, algorithm="giac")
[Out]