Optimal. Leaf size=103 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^4}+\frac{2 \left (a+b x^3\right )^{7/2} (A b-3 a B)}{21 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2} (2 A b-3 a B)}{15 b^4}+\frac{2 B \left (a+b x^3\right )^{9/2}}{27 b^4} \]
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Rubi [A] time = 0.255381, 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 \[ \frac{2 a^2 \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^4}+\frac{2 \left (a+b x^3\right )^{7/2} (A b-3 a B)}{21 b^4}-\frac{2 a \left (a+b x^3\right )^{5/2} (2 A b-3 a B)}{15 b^4}+\frac{2 B \left (a+b x^3\right )^{9/2}}{27 b^4} \]
Antiderivative was successfully verified.
[In] Int[x^8*Sqrt[a + b*x^3]*(A + B*x^3),x]
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Rubi in Sympy [A] time = 22.3575, size = 99, normalized size = 0.96 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{9}{2}}}{27 b^{4}} + \frac{2 a^{2} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (A b - B a\right )}{9 b^{4}} - \frac{2 a \left (a + b x^{3}\right )^{\frac{5}{2}} \left (2 A b - 3 B a\right )}{15 b^{4}} + \frac{2 \left (a + b x^{3}\right )^{\frac{7}{2}} \left (A b - 3 B a\right )}{21 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(B*x**3+A)*(b*x**3+a)**(1/2),x)
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Mathematica [A] time = 0.0931103, size = 75, normalized size = 0.73 \[ \frac{2 \left (a+b x^3\right )^{3/2} \left (-16 a^3 B+24 a^2 b \left (A+B x^3\right )-6 a b^2 x^3 \left (6 A+5 B x^3\right )+5 b^3 x^6 \left (9 A+7 B x^3\right )\right )}{945 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^8*Sqrt[a + b*x^3]*(A + B*x^3),x]
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Maple [A] time = 0.01, size = 77, normalized size = 0.8 \[{\frac{70\,B{x}^{9}{b}^{3}+90\,A{b}^{3}{x}^{6}-60\,Ba{b}^{2}{x}^{6}-72\,Aa{b}^{2}{x}^{3}+48\,B{a}^{2}b{x}^{3}+48\,A{a}^{2}b-32\,B{a}^{3}}{945\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(B*x^3+A)*(b*x^3+a)^(1/2),x)
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Maxima [A] time = 1.37814, size = 159, normalized size = 1.54 \[ \frac{2}{945} \, B{\left (\frac{35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )} + \frac{2}{315} \, A{\left (\frac{15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{b^{3}} - \frac{42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{b^{3}} + \frac{35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^8,x, algorithm="maxima")
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Fricas [A] time = 0.239531, size = 134, normalized size = 1.3 \[ \frac{2 \,{\left (35 \, B b^{4} x^{12} + 5 \,{\left (B a b^{3} + 9 \, A b^{4}\right )} x^{9} - 3 \,{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{6} - 16 \, B a^{4} + 24 \, A a^{3} b + 4 \,{\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{945 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^8,x, algorithm="fricas")
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Sympy [A] time = 11.2344, size = 219, normalized size = 2.13 \[ \begin{cases} \frac{16 A a^{3} \sqrt{a + b x^{3}}}{315 b^{3}} - \frac{8 A a^{2} x^{3} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 A a x^{6} \sqrt{a + b x^{3}}}{105 b} + \frac{2 A x^{9} \sqrt{a + b x^{3}}}{21} - \frac{32 B a^{4} \sqrt{a + b x^{3}}}{945 b^{4}} + \frac{16 B a^{3} x^{3} \sqrt{a + b x^{3}}}{945 b^{3}} - \frac{4 B a^{2} x^{6} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 B a x^{9} \sqrt{a + b x^{3}}}{189 b} + \frac{2 B x^{12} \sqrt{a + b x^{3}}}{27} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{9}}{9} + \frac{B x^{12}}{12}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(B*x**3+A)*(b*x**3+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.217445, size = 144, normalized size = 1.4 \[ \frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )} A}{b^{2}} + \frac{{\left (35 \,{\left (b x^{3} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}\right )} B}{b^{3}}\right )}}{945 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^8,x, algorithm="giac")
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