Optimal. Leaf size=99 \[ -\frac{a^2 (A b-a B)}{b^4 \sqrt{a+b x^2}}+\frac{\left (a+b x^2\right )^{3/2} (A b-3 a B)}{3 b^4}-\frac{a \sqrt{a+b x^2} (2 A b-3 a B)}{b^4}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^4} \]
[Out]
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Rubi [A] time = 0.226423, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^2 (A b-a B)}{b^4 \sqrt{a+b x^2}}+\frac{\left (a+b x^2\right )^{3/2} (A b-3 a B)}{3 b^4}-\frac{a \sqrt{a+b x^2} (2 A b-3 a B)}{b^4}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 25.0018, size = 88, normalized size = 0.89 \[ \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{a^{2} \left (A b - B a\right )}{b^{4} \sqrt{a + b x^{2}}} - \frac{a \sqrt{a + b x^{2}} \left (2 A b - 3 B a\right )}{b^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - 3 B a\right )}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0747605, size = 77, normalized size = 0.78 \[ \frac{48 a^3 B-8 a^2 b \left (5 A-3 B x^2\right )-2 a b^2 x^2 \left (10 A+3 B x^2\right )+b^3 x^4 \left (5 A+3 B x^2\right )}{15 b^4 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 77, normalized size = 0.8 \[ -{\frac{-3\,{x}^{6}B{b}^{3}-5\,A{b}^{3}{x}^{4}+6\,Ba{b}^{2}{x}^{4}+20\,Aa{b}^{2}{x}^{2}-24\,B{a}^{2}b{x}^{2}+40\,A{a}^{2}b-48\,B{a}^{3}}{15\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^2+A)/(b*x^2+a)^(3/2),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^5/(b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215972, size = 119, normalized size = 1.2 \[ \frac{{\left (3 \, B b^{3} x^{6} -{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{4} + 48 \, B a^{3} - 40 \, A a^{2} b + 4 \,{\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.34624, size = 172, normalized size = 1.74 \[ \begin{cases} - \frac{8 A a^{2}}{3 b^{3} \sqrt{a + b x^{2}}} - \frac{4 A a x^{2}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{A x^{4}}{3 b \sqrt{a + b x^{2}}} + \frac{16 B a^{3}}{5 b^{4} \sqrt{a + b x^{2}}} + \frac{8 B a^{2} x^{2}}{5 b^{3} \sqrt{a + b x^{2}}} - \frac{2 B a x^{4}}{5 b^{2} \sqrt{a + b x^{2}}} + \frac{B x^{6}}{5 b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{8}}{8}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.240234, size = 131, normalized size = 1.32 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B - 15 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a + 45 \, \sqrt{b x^{2} + a} B a^{2} + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b - 30 \, \sqrt{b x^{2} + a} A a b + \frac{15 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x^{2} + a}}}{15 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]