Optimal. Leaf size=100 \[ \frac{a^2 \sqrt{a+b x^2} (A b-a B)}{b^4}+\frac{\left (a+b x^2\right )^{5/2} (A b-3 a B)}{5 b^4}-\frac{a \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^4} \]
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Rubi [A] time = 0.227582, antiderivative size = 100, 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 \[ \frac{a^2 \sqrt{a+b x^2} (A b-a B)}{b^4}+\frac{\left (a+b x^2\right )^{5/2} (A b-3 a B)}{5 b^4}-\frac{a \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^4} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^2))/Sqrt[a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 25.0849, size = 90, normalized size = 0.9 \[ \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{a^{2} \sqrt{a + b x^{2}} \left (A b - B a\right )}{b^{4}} - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{3 b^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b - 3 B a\right )}{5 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**2+A)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0716602, size = 78, normalized size = 0.78 \[ \frac{\sqrt{a+b x^2} \left (-48 a^3 B+8 a^2 b \left (7 A+3 B x^2\right )-2 a b^2 x^2 \left (14 A+9 B x^2\right )+3 b^3 x^4 \left (7 A+5 B x^2\right )\right )}{105 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^2))/Sqrt[a + b*x^2],x]
[Out]
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Maple [A] time = 0.009, size = 77, normalized size = 0.8 \[{\frac{15\,{x}^{6}B{b}^{3}+21\,A{b}^{3}{x}^{4}-18\,Ba{b}^{2}{x}^{4}-28\,Aa{b}^{2}{x}^{2}+24\,B{a}^{2}b{x}^{2}+56\,A{a}^{2}b-48\,B{a}^{3}}{105\,{b}^{4}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^2+A)/(b*x^2+a)^(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^5/sqrt(b*x^2 + a),x, algorithm="maxima")
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Fricas [A] time = 0.228859, size = 103, normalized size = 1.03 \[ \frac{{\left (15 \, B b^{3} x^{6} - 3 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} - 48 \, B a^{3} + 56 \, A a^{2} b + 4 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.77603, size = 172, normalized size = 1.72 \[ \begin{cases} \frac{8 A a^{2} \sqrt{a + b x^{2}}}{15 b^{3}} - \frac{4 A a x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{A x^{4} \sqrt{a + b x^{2}}}{5 b} - \frac{16 B a^{3} \sqrt{a + b x^{2}}}{35 b^{4}} + \frac{8 B a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b^{3}} - \frac{6 B a x^{4} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{B x^{6} \sqrt{a + b x^{2}}}{7 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{8}}{8}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**2+A)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232575, size = 140, normalized size = 1.4 \[ \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B - 63 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a + 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} - 105 \, \sqrt{b x^{2} + a} B a^{3} + 21 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b - 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b + 105 \, \sqrt{b x^{2} + a} A a^{2} b}{105 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^5/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]