3.245 \(\int \frac{x^5 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.193813, 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 \[ -\frac{2 (A b-2 a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 a (A b-a B)}{9 b^3 \left (a+b x^3\right )^{3/2}}+\frac{2 B \sqrt{a+b x^3}}{3 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9223, size = 68, normalized size = 0.93 \[ \frac{2 B \sqrt{a + b x^{3}}}{3 b^{3}} + \frac{2 a \left (A b - B a\right )}{9 b^{3} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{2 \left (A b - 2 B a\right )}{3 b^{3} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.071057, size = 54, normalized size = 0.74 \[ \frac{16 a^2 B-4 a b \left (A-6 B x^3\right )+6 b^2 x^3 \left (B x^3-A\right )}{9 b^3 \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 53, normalized size = 0.7 \[ -{\frac{-6\,{b}^{2}B{x}^{6}+6\,A{x}^{3}{b}^{2}-24\,B{x}^{3}ab+4\,abA-16\,{a}^{2}B}{9\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9/(b*x^3+a)^(3/2)*(-3*B*b^2*x^6+3*A*b^2*x^3-12*B*a*b*x^3+2*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 1.37828, size = 113, normalized size = 1.55 \[ \frac{2}{9} \, B{\left (\frac{3 \, \sqrt{b x^{3} + a}}{b^{3}} + \frac{6 \, a}{\sqrt{b x^{3} + a} b^{3}} - \frac{a^{2}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{3}}\right )} - \frac{2}{9} \, A{\left (\frac{3}{\sqrt{b x^{3} + a} b^{2}} - \frac{a}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^5/(b*x^3 + a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.278463, size = 86, normalized size = 1.18 \[ \frac{2 \,{\left (3 \, B b^{2} x^{6} + 3 \,{\left (4 \, B a b - A b^{2}\right )} x^{3} + 8 \, B a^{2} - 2 \, A a b\right )}}{9 \,{\left (b^{4} x^{3} + a b^{3}\right )} \sqrt{b x^{3} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^5/(b*x^3 + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.64609, size = 240, normalized size = 3.29 \[ \begin{cases} - \frac{4 A a b}{9 a b^{3} \sqrt{a + b x^{3}} + 9 b^{4} x^{3} \sqrt{a + b x^{3}}} - \frac{6 A b^{2} x^{3}}{9 a b^{3} \sqrt{a + b x^{3}} + 9 b^{4} x^{3} \sqrt{a + b x^{3}}} + \frac{16 B a^{2}}{9 a b^{3} \sqrt{a + b x^{3}} + 9 b^{4} x^{3} \sqrt{a + b x^{3}}} + \frac{24 B a b x^{3}}{9 a b^{3} \sqrt{a + b x^{3}} + 9 b^{4} x^{3} \sqrt{a + b x^{3}}} + \frac{6 B b^{2} x^{6}}{9 a b^{3} \sqrt{a + b x^{3}} + 9 b^{4} x^{3} \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{9}}{9}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*A*a*b/(9*a*b**3*sqrt(a + b*x**3) + 9*b**4*x**3*sqrt(a + b*x**3)) -
6*A*b**2*x**3/(9*a*b**3*sqrt(a + b*x**3) + 9*b**4*x**3*sqrt(a + b*x**3)) + 16*B
*a**2/(9*a*b**3*sqrt(a + b*x**3) + 9*b**4*x**3*sqrt(a + b*x**3)) + 24*B*a*b*x**3
/(9*a*b**3*sqrt(a + b*x**3) + 9*b**4*x**3*sqrt(a + b*x**3)) + 6*B*b**2*x**6/(9*a
*b**3*sqrt(a + b*x**3) + 9*b**4*x**3*sqrt(a + b*x**3)), Ne(b, 0)), ((A*x**6/6 +
B*x**9/9)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.215931, size = 82, normalized size = 1.12 \[ \frac{2 \,{\left (3 \, \sqrt{b x^{3} + a} B + \frac{6 \,{\left (b x^{3} + a\right )} B a - B a^{2} - 3 \,{\left (b x^{3} + a\right )} A b + A a b}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\right )}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^5/(b*x^3 + a)^(5/2),x, algorithm="giac")

[Out]

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