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

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

[Out]

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

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Rubi [A]  time = 0.190431, 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 \[ \frac{2 \left (a+b x^3\right )^{3/2} (A b-2 a B)}{9 b^3}-\frac{2 a \sqrt{a+b x^3} (A b-a B)}{3 b^3}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.6461, size = 68, normalized size = 0.93 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{3}} - \frac{2 a \sqrt{a + b x^{3}} \left (A b - B a\right )}{3 b^{3}} + \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}} \left (A b - 2 B a\right )}{9 b^{3}} \]

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

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Mathematica [A]  time = 0.0631038, size = 56, normalized size = 0.77 \[ \frac{2 \sqrt{a+b x^3} \left (8 a^2 B-2 a b \left (5 A+2 B x^3\right )+b^2 x^3 \left (5 A+3 B x^3\right )\right )}{45 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 53, normalized size = 0.7 \[ -{\frac{-6\,{b}^{2}B{x}^{6}-10\,A{x}^{3}{b}^{2}+8\,B{x}^{3}ab+20\,abA-16\,{a}^{2}B}{45\,{b}^{3}}\sqrt{b{x}^{3}+a}} \]

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]

-2/45*(b*x^3+a)^(1/2)*(-3*B*b^2*x^6-5*A*b^2*x^3+4*B*a*b*x^3+10*A*a*b-8*B*a^2)/b^
3

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Maxima [A]  time = 1.40538, size = 112, normalized size = 1.53 \[ \frac{2}{45} \, B{\left (\frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{3}} - \frac{10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{3}} + \frac{15 \, \sqrt{b x^{3} + a} a^{2}}{b^{3}}\right )} + \frac{2}{9} \, A{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b x^{3} + a} a}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*B*(3*(b*x^3 + a)^(5/2)/b^3 - 10*(b*x^3 + a)^(3/2)*a/b^3 + 15*sqrt(b*x^3 + a
)*a^2/b^3) + 2/9*A*((b*x^3 + a)^(3/2)/b^2 - 3*sqrt(b*x^3 + a)*a/b^2)

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Fricas [A]  time = 0.239409, size = 70, normalized size = 0.96 \[ \frac{2 \,{\left (3 \, B b^{2} x^{6} -{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 10 \, A a b\right )} \sqrt{b x^{3} + a}}{45 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.09375, size = 124, normalized size = 1.7 \[ \begin{cases} - \frac{4 A a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 A x^{3} \sqrt{a + b x^{3}}}{9 b} + \frac{16 B a^{2} \sqrt{a + b x^{3}}}{45 b^{3}} - \frac{8 B a x^{3} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 B x^{6} \sqrt{a + b x^{3}}}{15 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{9}}{9}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

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

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GIAC/XCAS [A]  time = 0.213418, size = 99, normalized size = 1.36 \[ \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} B - 10 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a + 15 \, \sqrt{b x^{3} + a} B a^{2} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b - 15 \, \sqrt{b x^{3} + a} A a b\right )}}{45 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(3*(b*x^3 + a)^(5/2)*B - 10*(b*x^3 + a)^(3/2)*B*a + 15*sqrt(b*x^3 + a)*B*a^
2 + 5*(b*x^3 + a)^(3/2)*A*b - 15*sqrt(b*x^3 + a)*A*a*b)/b^3