Optimal. Leaf size=122 \[ -\frac{a^2 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}+\frac{a x \sqrt{a+b x^2} (2 A b-a B)}{16 b^2}+\frac{x^3 \sqrt{a+b x^2} (2 A b-a B)}{8 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b} \]
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Rubi [A] time = 0.182169, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{a^2 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}+\frac{a x \sqrt{a+b x^2} (2 A b-a B)}{16 b^2}+\frac{x^3 \sqrt{a+b x^2} (2 A b-a B)}{8 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b} \]
Antiderivative was successfully verified.
[In] Int[x^2*Sqrt[a + b*x^2]*(A + B*x^2),x]
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Rubi in Sympy [A] time = 19.4402, size = 107, normalized size = 0.88 \[ \frac{B x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{6 b} - \frac{a^{2} \left (2 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{5}{2}}} + \frac{a x \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{16 b^{2}} + \frac{x^{3} \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x**2+A)*(b*x**2+a)**(1/2),x)
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Mathematica [A] time = 0.0965216, size = 99, normalized size = 0.81 \[ \frac{a^2 (a B-2 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{a x (a B-2 A b)}{16 b^2}+\frac{x^3 (a B+6 A b)}{24 b}+\frac{B x^5}{6}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2*Sqrt[a + b*x^2]*(A + B*x^2),x]
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Maple [A] time = 0.01, size = 139, normalized size = 1.1 \[{\frac{Ax}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aAx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{3}}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bxa}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bx{a}^{2}}{16\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x^2+A)*(b*x^2+a)^(1/2),x)
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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)*sqrt(b*x^2 + a)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257115, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{5} + 2 \,{\left (B a b + 6 \, A b^{2}\right )} x^{3} - 3 \,{\left (B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{5}{2}}}, \frac{{\left (8 \, B b^{2} x^{5} + 2 \,{\left (B a b + 6 \, A b^{2}\right )} x^{3} - 3 \,{\left (B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 29.6152, size = 226, normalized size = 1.85 \[ \frac{A a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{A b x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x}{16 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{3}{2}} x^{3}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{5}{2}}} + \frac{B b x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x**2+A)*(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.252508, size = 135, normalized size = 1.11 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, B x^{2} + \frac{B a b^{3} + 6 \, A b^{4}}{b^{4}}\right )} x^{2} - \frac{3 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (B a^{3} - 2 \, A a^{2} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*x^2,x, algorithm="giac")
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