3.180 \(\int x^5 \sqrt{a+b x^3} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=73 \[ \frac{2 \left (a+b x^3\right )^{5/2} (A b-2 a B)}{15 b^3}-\frac{2 a \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^3}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^3} \]

[Out]

(-2*a*(A*b - a*B)*(a + b*x^3)^(3/2))/(9*b^3) + (2*(A*b - 2*a*B)*(a + b*x^3)^(5/2
))/(15*b^3) + (2*B*(a + b*x^3)^(7/2))/(21*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.191061, 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 \[ \frac{2 \left (a+b x^3\right )^{5/2} (A b-2 a B)}{15 b^3}-\frac{2 a \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^3}+\frac{2 B \left (a+b x^3\right )^{7/2}}{21 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(-2*a*(A*b - a*B)*(a + b*x^3)^(3/2))/(9*b^3) + (2*(A*b - 2*a*B)*(a + b*x^3)^(5/2
))/(15*b^3) + (2*B*(a + b*x^3)^(7/2))/(21*b^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.6756, size = 68, normalized size = 0.93 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{7}{2}}}{21 b^{3}} - \frac{2 a \left (a + b x^{3}\right )^{\frac{3}{2}} \left (A b - B a\right )}{9 b^{3}} + \frac{2 \left (a + b x^{3}\right )^{\frac{5}{2}} \left (A b - 2 B a\right )}{15 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x**3+A)*(b*x**3+a)**(1/2),x)

[Out]

2*B*(a + b*x**3)**(7/2)/(21*b**3) - 2*a*(a + b*x**3)**(3/2)*(A*b - B*a)/(9*b**3)
+ 2*(a + b*x**3)**(5/2)*(A*b - 2*B*a)/(15*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0611116, size = 57, normalized size = 0.78 \[ \frac{2 \left (a+b x^3\right )^{3/2} \left (8 a^2 B-2 a b \left (7 A+6 B x^3\right )+3 b^2 x^3 \left (7 A+5 B x^3\right )\right )}{315 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(2*(a + b*x^3)^(3/2)*(8*a^2*B + 3*b^2*x^3*(7*A + 5*B*x^3) - 2*a*b*(7*A + 6*B*x^3
)))/(315*b^3)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 53, normalized size = 0.7 \[ -{\frac{-30\,{b}^{2}B{x}^{6}-42\,A{x}^{3}{b}^{2}+24\,B{x}^{3}ab+28\,abA-16\,{a}^{2}B}{315\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(B*x^3+A)*(b*x^3+a)^(1/2),x)

[Out]

-2/315*(b*x^3+a)^(3/2)*(-15*B*b^2*x^6-21*A*b^2*x^3+12*B*a*b*x^3+14*A*a*b-8*B*a^2
)/b^3

_______________________________________________________________________________________

Maxima [A]  time = 1.46076, size = 113, normalized size = 1.55 \[ \frac{2}{315} \, B{\left (\frac{15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}}}{b^{3}} - \frac{42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a}{b^{3}} + \frac{35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}}{b^{3}}\right )} + \frac{2}{45} \, A{\left (\frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^5,x, algorithm="maxima")

[Out]

2/315*B*(15*(b*x^3 + a)^(7/2)/b^3 - 42*(b*x^3 + a)^(5/2)*a/b^3 + 35*(b*x^3 + a)^
(3/2)*a^2/b^3) + 2/45*A*(3*(b*x^3 + a)^(5/2)/b^2 - 5*(b*x^3 + a)^(3/2)*a/b^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.248635, size = 101, normalized size = 1.38 \[ \frac{2 \,{\left (15 \, B b^{3} x^{9} + 3 \,{\left (B a b^{2} + 7 \, A b^{3}\right )} x^{6} + 8 \, B a^{3} - 14 \, A a^{2} b -{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^5,x, algorithm="fricas")

[Out]

2/315*(15*B*b^3*x^9 + 3*(B*a*b^2 + 7*A*b^3)*x^6 + 8*B*a^3 - 14*A*a^2*b - (4*B*a^
2*b - 7*A*a*b^2)*x^3)*sqrt(b*x^3 + a)/b^3

_______________________________________________________________________________________

Sympy [A]  time = 4.71401, size = 168, normalized size = 2.3 \[ \begin{cases} - \frac{4 A a^{2} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 A a x^{3} \sqrt{a + b x^{3}}}{45 b} + \frac{2 A x^{6} \sqrt{a + b x^{3}}}{15} + \frac{16 B a^{3} \sqrt{a + b x^{3}}}{315 b^{3}} - \frac{8 B a^{2} x^{3} \sqrt{a + b x^{3}}}{315 b^{2}} + \frac{2 B a x^{6} \sqrt{a + b x^{3}}}{105 b} + \frac{2 B x^{9} \sqrt{a + b x^{3}}}{21} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{6}}{6} + \frac{B x^{9}}{9}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x**3+A)*(b*x**3+a)**(1/2),x)

[Out]

Piecewise((-4*A*a**2*sqrt(a + b*x**3)/(45*b**2) + 2*A*a*x**3*sqrt(a + b*x**3)/(4
5*b) + 2*A*x**6*sqrt(a + b*x**3)/15 + 16*B*a**3*sqrt(a + b*x**3)/(315*b**3) - 8*
B*a**2*x**3*sqrt(a + b*x**3)/(315*b**2) + 2*B*a*x**6*sqrt(a + b*x**3)/(105*b) +
2*B*x**9*sqrt(a + b*x**3)/21, Ne(b, 0)), (sqrt(a)*(A*x**6/6 + B*x**9/9), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218476, size = 107, normalized size = 1.47 \[ \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} A}{b} + \frac{{\left (15 \,{\left (b x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )} B}{b^{2}}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^5,x, algorithm="giac")

[Out]

2/315*(7*(3*(b*x^3 + a)^(5/2) - 5*(b*x^3 + a)^(3/2)*a)*A/b + (15*(b*x^3 + a)^(7/
2) - 42*(b*x^3 + a)^(5/2)*a + 35*(b*x^3 + a)^(3/2)*a^2)*B/b^2)/b