3.228 \(\int \frac{x^8 \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=103 \[ -\frac{2 a^2 (A b-a B)}{3 b^4 \sqrt{a+b x^3}}+\frac{2 \left (a+b x^3\right )^{3/2} (A b-3 a B)}{9 b^4}-\frac{2 a \sqrt{a+b x^3} (2 A b-3 a B)}{3 b^4}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^4} \]

[Out]

(-2*a^2*(A*b - a*B))/(3*b^4*Sqrt[a + b*x^3]) - (2*a*(2*A*b - 3*a*B)*Sqrt[a + b*x
^3])/(3*b^4) + (2*(A*b - 3*a*B)*(a + b*x^3)^(3/2))/(9*b^4) + (2*B*(a + b*x^3)^(5
/2))/(15*b^4)

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Rubi [A]  time = 0.257579, 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 \[ -\frac{2 a^2 (A b-a B)}{3 b^4 \sqrt{a+b x^3}}+\frac{2 \left (a+b x^3\right )^{3/2} (A b-3 a B)}{9 b^4}-\frac{2 a \sqrt{a+b x^3} (2 A b-3 a B)}{3 b^4}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(A + B*x^3))/(a + b*x^3)^(3/2),x]

[Out]

(-2*a^2*(A*b - a*B))/(3*b^4*Sqrt[a + b*x^3]) - (2*a*(2*A*b - 3*a*B)*Sqrt[a + b*x
^3])/(3*b^4) + (2*(A*b - 3*a*B)*(a + b*x^3)^(3/2))/(9*b^4) + (2*B*(a + b*x^3)^(5
/2))/(15*b^4)

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Rubi in Sympy [A]  time = 21.999, size = 99, normalized size = 0.96 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{4}} - \frac{2 a^{2} \left (A b - B a\right )}{3 b^{4} \sqrt{a + b x^{3}}} - \frac{2 a \sqrt{a + b x^{3}} \left (2 A b - 3 B a\right )}{3 b^{4}} + \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}} \left (A b - 3 B a\right )}{9 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(B*x**3+A)/(b*x**3+a)**(3/2),x)

[Out]

2*B*(a + b*x**3)**(5/2)/(15*b**4) - 2*a**2*(A*b - B*a)/(3*b**4*sqrt(a + b*x**3))
- 2*a*sqrt(a + b*x**3)*(2*A*b - 3*B*a)/(3*b**4) + 2*(a + b*x**3)**(3/2)*(A*b -
3*B*a)/(9*b**4)

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Mathematica [A]  time = 0.0921842, size = 77, normalized size = 0.75 \[ \frac{2 \left (48 a^3 B-8 a^2 b \left (5 A-3 B x^3\right )-2 a b^2 x^3 \left (10 A+3 B x^3\right )+b^3 x^6 \left (5 A+3 B x^3\right )\right )}{45 b^4 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(A + B*x^3))/(a + b*x^3)^(3/2),x]

[Out]

(2*(48*a^3*B - 8*a^2*b*(5*A - 3*B*x^3) + b^3*x^6*(5*A + 3*B*x^3) - 2*a*b^2*x^3*(
10*A + 3*B*x^3)))/(45*b^4*Sqrt[a + b*x^3])

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Maple [A]  time = 0.01, size = 77, normalized size = 0.8 \[ -{\frac{-6\,B{x}^{9}{b}^{3}-10\,A{b}^{3}{x}^{6}+12\,Ba{b}^{2}{x}^{6}+40\,Aa{b}^{2}{x}^{3}-48\,B{a}^{2}b{x}^{3}+80\,A{a}^{2}b-96\,B{a}^{3}}{45\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(B*x^3+A)/(b*x^3+a)^(3/2),x)

[Out]

-2/45/(b*x^3+a)^(1/2)*(-3*B*b^3*x^9-5*A*b^3*x^6+6*B*a*b^2*x^6+20*A*a*b^2*x^3-24*
B*a^2*b*x^3+40*A*a^2*b-48*B*a^3)/b^4

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Maxima [A]  time = 1.41615, size = 157, normalized size = 1.52 \[ \frac{2}{15} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{4}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{4}} + \frac{15 \, \sqrt{b x^{3} + a} a^{2}}{b^{4}} + \frac{5 \, a^{3}}{\sqrt{b x^{3} + a} b^{4}}\right )} + \frac{2}{9} \, A{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{3}} - \frac{6 \, \sqrt{b x^{3} + a} a}{b^{3}} - \frac{3 \, a^{2}}{\sqrt{b x^{3} + a} b^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^8/(b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

2/15*B*((b*x^3 + a)^(5/2)/b^4 - 5*(b*x^3 + a)^(3/2)*a/b^4 + 15*sqrt(b*x^3 + a)*a
^2/b^4 + 5*a^3/(sqrt(b*x^3 + a)*b^4)) + 2/9*A*((b*x^3 + a)^(3/2)/b^3 - 6*sqrt(b*
x^3 + a)*a/b^3 - 3*a^2/(sqrt(b*x^3 + a)*b^3))

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Fricas [A]  time = 0.251996, size = 103, normalized size = 1. \[ \frac{2 \,{\left (3 \, B b^{3} x^{9} -{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 48 \, B a^{3} - 40 \, A a^{2} b + 4 \,{\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )}}{45 \, \sqrt{b x^{3} + a} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^8/(b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

2/45*(3*B*b^3*x^9 - (6*B*a*b^2 - 5*A*b^3)*x^6 + 48*B*a^3 - 40*A*a^2*b + 4*(6*B*a
^2*b - 5*A*a*b^2)*x^3)/(sqrt(b*x^3 + a)*b^4)

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Sympy [A]  time = 11.6515, size = 175, normalized size = 1.7 \[ \begin{cases} - \frac{16 A a^{2}}{9 b^{3} \sqrt{a + b x^{3}}} - \frac{8 A a x^{3}}{9 b^{2} \sqrt{a + b x^{3}}} + \frac{2 A x^{6}}{9 b \sqrt{a + b x^{3}}} + \frac{32 B a^{3}}{15 b^{4} \sqrt{a + b x^{3}}} + \frac{16 B a^{2} x^{3}}{15 b^{3} \sqrt{a + b x^{3}}} - \frac{4 B a x^{6}}{15 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{9}}{15 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{9}}{9} + \frac{B x^{12}}{12}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(B*x**3+A)/(b*x**3+a)**(3/2),x)

[Out]

Piecewise((-16*A*a**2/(9*b**3*sqrt(a + b*x**3)) - 8*A*a*x**3/(9*b**2*sqrt(a + b*
x**3)) + 2*A*x**6/(9*b*sqrt(a + b*x**3)) + 32*B*a**3/(15*b**4*sqrt(a + b*x**3))
+ 16*B*a**2*x**3/(15*b**3*sqrt(a + b*x**3)) - 4*B*a*x**6/(15*b**2*sqrt(a + b*x**
3)) + 2*B*x**9/(15*b*sqrt(a + b*x**3)), Ne(b, 0)), ((A*x**9/9 + B*x**12/12)/a**(
3/2), True))

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GIAC/XCAS [A]  time = 0.215409, size = 131, normalized size = 1.27 \[ \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} B - 15 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a + 45 \, \sqrt{b x^{3} + a} B a^{2} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b - 30 \, \sqrt{b x^{3} + a} A a b + \frac{15 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x^{3} + a}}\right )}}{45 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^8/(b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

2/45*(3*(b*x^3 + a)^(5/2)*B - 15*(b*x^3 + a)^(3/2)*B*a + 45*sqrt(b*x^3 + a)*B*a^
2 + 5*(b*x^3 + a)^(3/2)*A*b - 30*sqrt(b*x^3 + a)*A*a*b + 15*(B*a^3 - A*a^2*b)/sq
rt(b*x^3 + a))/b^4