Optimal. Leaf size=73 \[ \frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3} \]
[Out]
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Rubi [A] time = 0.1899, antiderivative size = 73, 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 \[ \frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
[In] Int[(x^5*(A + B*x^3))/(a + b*x^3)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 16.5213, size = 68, normalized size = 0.93 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{3}} + \frac{2 a \left (A b - B a\right )}{3 b^{3} \sqrt{a + b x^{3}}} + \frac{2 \sqrt{a + b x^{3}} \left (A b - 2 B a\right )}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(B*x**3+A)/(b*x**3+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0676921, size = 55, normalized size = 0.75 \[ \frac{2 \left (-8 a^2 B+a \left (6 A b-4 b B x^3\right )+b^2 x^3 \left (3 A+B x^3\right )\right )}{9 b^3 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*(A + B*x^3))/(a + b*x^3)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 52, normalized size = 0.7 \[{\frac{2\,{b}^{2}B{x}^{6}+6\,A{x}^{3}{b}^{2}-8\,B{x}^{3}ab+12\,abA-16\,{a}^{2}B}{9\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(B*x^3+A)/(b*x^3+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.38358, size = 109, normalized size = 1.49 \[ \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{3}} - \frac{6 \, \sqrt{b x^{3} + a} a}{b^{3}} - \frac{3 \, a^{2}}{\sqrt{b x^{3} + a} b^{3}}\right )} + \frac{2}{3} \, A{\left (\frac{\sqrt{b x^{3} + a}}{b^{2}} + \frac{a}{\sqrt{b x^{3} + a} b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264255, size = 69, normalized size = 0.95 \[ \frac{2 \,{\left (B b^{2} x^{6} -{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{3} - 8 \, B a^{2} + 6 \, A a b\right )}}{9 \, \sqrt{b x^{3} + a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.7932, size = 124, normalized size = 1.7 \[ \begin{cases} \frac{4 A a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 A x^{3}}{3 b \sqrt{a + b x^{3}}} - \frac{16 B a^{2}}{9 b^{3} \sqrt{a + b x^{3}}} - \frac{8 B a x^{3}}{9 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{6}}{9 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{9}}{9}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(B*x**3+A)/(b*x**3+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216163, size = 88, normalized size = 1.21 \[ \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 6 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x^{3} + a}}\right )}}{9 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^5/(b*x^3 + a)^(3/2),x, algorithm="giac")
[Out]