Optimal. Leaf size=122 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b} \]
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Rubi [A] time = 0.171011, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}-\frac{a x \sqrt{a+b x^2} (6 A b-5 a B)}{16 b^3}+\frac{x^3 \sqrt{a+b x^2} (6 A b-5 a B)}{24 b^2}+\frac{B x^5 \sqrt{a+b x^2}}{6 b} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(A + B*x^2))/Sqrt[a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 18.1425, size = 114, normalized size = 0.93 \[ \frac{B x^{5} \sqrt{a + b x^{2}}}{6 b} + \frac{a^{2} \left (6 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{7}{2}}} - \frac{a x \sqrt{a + b x^{2}} \left (6 A b - 5 B a\right )}{16 b^{3}} + \frac{x^{3} \sqrt{a + b x^{2}} \left (6 A b - 5 B a\right )}{24 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(B*x**2+A)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.114659, size = 105, normalized size = 0.86 \[ \sqrt{a+b x^2} \left (\frac{a x (5 a B-6 A b)}{16 b^3}+\frac{x^3 (6 A b-5 a B)}{24 b^2}+\frac{B x^5}{6 b}\right )-\frac{a^2 (5 a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(A + B*x^2))/Sqrt[a + b*x^2],x]
[Out]
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Maple [A] time = 0.011, size = 143, normalized size = 1.2 \[{\frac{A{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,aAx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{{x}^{5}B}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,Ba{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,Bx{a}^{2}}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(B*x^2+A)/(b*x^2+a)^(1/2),x)
[Out]
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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)*x^4/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258052, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{5} - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 6 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, B b^{2} x^{5} - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 6 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 33.1091, size = 235, normalized size = 1.93 \[ - \frac{3 A a^{\frac{3}{2}} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A \sqrt{a} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{A x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{5}{2}} x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{3}{2}} x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B \sqrt{a} x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{B x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(B*x**2+A)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.234347, size = 144, normalized size = 1.18 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, B x^{2}}{b} - \frac{5 \, B a b^{3} - 6 \, A b^{4}}{b^{5}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{2} b^{2} - 6 \, A a b^{3}\right )}}{b^{5}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^4/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]