Optimal. Leaf size=46 \[ \frac{\left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.100122, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a + b*x^2]*(A + B*x^2),x]
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Rubi in Sympy [A] time = 13.1255, size = 37, normalized size = 0.8 \[ \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**2+A)*(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0392104, size = 34, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{3/2} \left (-2 a B+5 A b+3 b B x^2\right )}{15 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a + b*x^2]*(A + B*x^2),x]
[Out]
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Maple [A] time = 0.006, size = 31, normalized size = 0.7 \[{\frac{3\,bB{x}^{2}+5\,Ab-2\,Ba}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(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)*sqrt(b*x^2 + a)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212523, size = 68, normalized size = 1.48 \[ \frac{{\left (3 \, B b^{2} x^{4} - 2 \, B a^{2} + 5 \, A a b +{\left (B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.982783, size = 110, normalized size = 2.39 \[ \begin{cases} \frac{A a \sqrt{a + b x^{2}}}{3 b} + \frac{A x^{2} \sqrt{a + b x^{2}}}{3} - \frac{2 B a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{B a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{B x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{2}}{2} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**2+A)*(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.240444, size = 63, normalized size = 1.37 \[ \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A + \frac{{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} B}{b}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*x,x, algorithm="giac")
[Out]