Optimal. Leaf size=103 \[ \frac{2 a^2 \sqrt{a+b x^3} (A b-a B)}{3 b^4}+\frac{2 \left (a+b x^3\right )^{5/2} (A b-3 a B)}{15 b^4}-\frac{2 a \left (a+b x^3\right )^{3/2} (2 A b-3 a B)}{9 b^4}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.257071, 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 \[ \frac{2 a^2 \sqrt{a+b x^3} (A b-a B)}{3 b^4}+\frac{2 \left (a+b x^3\right )^{5/2} (A b-3 a B)}{15 b^4}-\frac{2 a \left (a+b x^3\right )^{3/2} (2 A b-3 a B)}{9 b^4}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^4} \]
Antiderivative was successfully verified.
[In] Int[(x^8*(A + B*x^3))/Sqrt[a + b*x^3],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.223, size = 99, normalized size = 0.96 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{7}{2}}}{21 b^{4}} + \frac{2 a^{2} \sqrt{a + b x^{3}} \left (A b - B a\right )}{3 b^{4}} - \frac{2 a \left (a + b x^{3}\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{9 b^{4}} + \frac{2 \left (a + b x^{3}\right )^{\frac{5}{2}} \left (A b - 3 B a\right )}{15 b^{4}} \]
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)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0832205, size = 78, normalized size = 0.76 \[ \frac{2 \sqrt{a+b x^3} \left (-48 a^3 B+8 a^2 b \left (7 A+3 B x^3\right )-2 a b^2 x^3 \left (14 A+9 B x^3\right )+3 b^3 x^6 \left (7 A+5 B x^3\right )\right )}{315 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(A + B*x^3))/Sqrt[a + b*x^3],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 77, normalized size = 0.8 \[{\frac{30\,B{x}^{9}{b}^{3}+42\,A{b}^{3}{x}^{6}-36\,Ba{b}^{2}{x}^{6}-56\,Aa{b}^{2}{x}^{3}+48\,B{a}^{2}b{x}^{3}+112\,A{a}^{2}b-96\,B{a}^{3}}{315\,{b}^{4}}\sqrt{b{x}^{3}+a}} \]
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)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.43124, size = 159, normalized size = 1.54 \[ \frac{2}{105} \, B{\left (\frac{5 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{b^{4}} - \frac{21 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{b^{4}} + \frac{35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{35 \, \sqrt{b x^{3} + a} a^{3}}{b^{4}}\right )} + \frac{2}{45} \, A{\left (\frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{3}} - \frac{10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{3}} + \frac{15 \, \sqrt{b x^{3} + a} a^{2}}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/sqrt(b*x^3 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.249319, size = 103, normalized size = 1. \[ \frac{2 \,{\left (15 \, B b^{3} x^{9} - 3 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} - 48 \, B a^{3} + 56 \, A a^{2} b + 4 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{315 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/sqrt(b*x^3 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 10.433, size = 175, normalized size = 1.7 \[ \begin{cases} \frac{16 A a^{2} \sqrt{a + b x^{3}}}{45 b^{3}} - \frac{8 A a x^{3} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 A x^{6} \sqrt{a + b x^{3}}}{15 b} - \frac{32 B a^{3} \sqrt{a + b x^{3}}}{105 b^{4}} + \frac{16 B a^{2} x^{3} \sqrt{a + b x^{3}}}{105 b^{3}} - \frac{4 B a x^{6} \sqrt{a + b x^{3}}}{35 b^{2}} + \frac{2 B x^{9} \sqrt{a + b x^{3}}}{21 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{9}}{9} + \frac{B x^{12}}{12}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
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)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.21427, size = 140, normalized size = 1.36 \[ \frac{2 \,{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} B - 63 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} B a + 105 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a^{2} - 105 \, \sqrt{b x^{3} + a} B a^{3} + 21 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} A b - 70 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A a b + 105 \, \sqrt{b x^{3} + a} A a^{2} b\right )}}{315 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^8/sqrt(b*x^3 + a),x, algorithm="giac")
[Out]