Optimal. Leaf size=103 \[ -\frac{2 a^2 (A b-a B)}{9 b^4 \left (a+b x^3\right )^{3/2}}+\frac{2 a (2 A b-3 a B)}{3 b^4 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (A b-3 a B)}{3 b^4}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^4} \]
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Rubi [A] time = 0.259859, antiderivative size = 103, 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 \[ -\frac{2 a^2 (A b-a B)}{9 b^4 \left (a+b x^3\right )^{3/2}}+\frac{2 a (2 A b-3 a B)}{3 b^4 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (A b-3 a B)}{3 b^4}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^4} \]
Antiderivative was successfully verified.
[In] Int[(x^8*(A + B*x^3))/(a + b*x^3)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 22.2459, size = 99, normalized size = 0.96 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{4}} - \frac{2 a^{2} \left (A b - B a\right )}{9 b^{4} \left (a + b x^{3}\right )^{\frac{3}{2}}} + \frac{2 a \left (2 A b - 3 B a\right )}{3 b^{4} \sqrt{a + b x^{3}}} + \frac{2 \sqrt{a + b x^{3}} \left (A b - 3 B a\right )}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(B*x**3+A)/(b*x**3+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0989231, size = 73, normalized size = 0.71 \[ \frac{2 \left (-16 a^3 B+8 a^2 b \left (A-3 B x^3\right )-6 a b^2 x^3 \left (B x^3-2 A\right )+b^3 x^6 \left (3 A+B x^3\right )\right )}{9 b^4 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^(5/2),x]
[Out]
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Maple [A] time = 0.01, size = 76, normalized size = 0.7 \[{\frac{2\,B{x}^{9}{b}^{3}+6\,A{b}^{3}{x}^{6}-12\,Ba{b}^{2}{x}^{6}+24\,Aa{b}^{2}{x}^{3}-48\,B{a}^{2}b{x}^{3}+16\,A{a}^{2}b-32\,B{a}^{3}}{9\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(B*x^3+A)/(b*x^3+a)^(5/2),x)
[Out]
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Maxima [A] time = 1.37989, size = 157, normalized size = 1.52 \[ \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{4}} - \frac{9 \, \sqrt{b x^{3} + a} a}{b^{4}} - \frac{9 \, a^{2}}{\sqrt{b x^{3} + a} b^{4}} + \frac{a^{3}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{4}}\right )} + \frac{2}{9} \, A{\left (\frac{3 \, \sqrt{b x^{3} + a}}{b^{3}} + \frac{6 \, a}{\sqrt{b x^{3} + a} b^{3}} - \frac{a^{2}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.297201, size = 117, normalized size = 1.14 \[ \frac{2 \,{\left (B b^{3} x^{9} - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} - 16 \, B a^{3} + 8 \, A a^{2} b - 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )}}{9 \,{\left (b^{5} x^{3} + a b^{4}\right )} \sqrt{b x^{3} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.4451, size = 338, normalized size = 3.28 \[ \begin{cases} \frac{16 A a^{2} b}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} + \frac{24 A a b^{2} x^{3}}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} + \frac{6 A b^{3} x^{6}}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} - \frac{32 B a^{3}}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} - \frac{48 B a^{2} b x^{3}}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} - \frac{12 B a b^{2} x^{6}}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} + \frac{2 B b^{3} x^{9}}{9 a b^{4} \sqrt{a + b x^{3}} + 9 b^{5} x^{3} \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{9}}{9} + \frac{B x^{12}}{12}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(B*x**3+A)/(b*x**3+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215964, size = 124, normalized size = 1.2 \[ \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 9 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b - \frac{9 \,{\left (b x^{3} + a\right )} B a^{2} - B a^{3} - 6 \,{\left (b x^{3} + a\right )} A a b + A a^{2} b}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\right )}}{9 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/(b*x^3 + a)^(5/2),x, algorithm="giac")
[Out]