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

Optimal. Leaf size=97 \[ -\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} (A b-3 a B)}{b^4}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^4} \]

[Out]

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

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Rubi [A]  time = 0.223697, antiderivative size = 97, 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 \[ -\frac{a^2 (A b-a B)}{3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{a (2 A b-3 a B)}{b^4 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2} (A b-3 a B)}{b^4}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.2659, size = 88, normalized size = 0.91 \[ \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{4}} - \frac{a^{2} \left (A b - B a\right )}{3 b^{4} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{a \left (2 A b - 3 B a\right )}{b^{4} \sqrt{a + b x^{2}}} + \frac{\sqrt{a + b x^{2}} \left (A b - 3 B a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0813512, size = 73, normalized size = 0.75 \[ \frac{-16 a^3 B+8 a^2 b \left (A-3 B x^2\right )-6 a b^2 x^2 \left (B x^2-2 A\right )+b^3 x^4 \left (3 A+B x^2\right )}{3 b^4 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-16*a^3*B + 8*a^2*b*(A - 3*B*x^2) - 6*a*b^2*x^2*(-2*A + B*x^2) + b^3*x^4*(3*A +
B*x^2))/(3*b^4*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.008, size = 76, normalized size = 0.8 \[{\frac{{x}^{6}B{b}^{3}+3\,A{b}^{3}{x}^{4}-6\,Ba{b}^{2}{x}^{4}+12\,Aa{b}^{2}{x}^{2}-24\,B{a}^{2}b{x}^{2}+8\,A{a}^{2}b-16\,B{a}^{3}}{3\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(B*b^3*x^6+3*A*b^3*x^4-6*B*a*b^2*x^4+12*A*a*b^2*x^2-24*B*a^2*b*x^2+8*A*a^2*b
-16*B*a^3)/(b*x^2+a)^(3/2)/b^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23518, size = 132, normalized size = 1.36 \[ \frac{{\left (B b^{3} x^{6} - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{4} - 16 \, B a^{3} + 8 \, A a^{2} b - 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(B*b^3*x^6 - 3*(2*B*a*b^2 - A*b^3)*x^4 - 16*B*a^3 + 8*A*a^2*b - 12*(2*B*a^2*
b - A*a*b^2)*x^2)*sqrt(b*x^2 + a)/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

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Sympy [A]  time = 6.3404, size = 337, normalized size = 3.47 \[ \begin{cases} \frac{8 A a^{2} b}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} + \frac{12 A a b^{2} x^{2}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} + \frac{3 A b^{3} x^{4}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} - \frac{16 B a^{3}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} - \frac{24 B a^{2} b x^{2}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} - \frac{6 B a b^{2} x^{4}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} + \frac{B b^{3} x^{6}}{3 a b^{4} \sqrt{a + b x^{2}} + 3 b^{5} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{8}}{8}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((8*A*a**2*b/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2))
+ 12*A*a*b**2*x**2/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) +
3*A*b**3*x**4/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) - 16*B
*a**3/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) - 24*B*a**2*b*x
**2/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) - 6*B*a*b**2*x**4
/(3*a*b**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)) + B*b**3*x**6/(3*a*b
**4*sqrt(a + b*x**2) + 3*b**5*x**2*sqrt(a + b*x**2)), Ne(b, 0)), ((A*x**6/6 + B*
x**8/8)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.231641, size = 124, normalized size = 1.28 \[ \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B - 9 \, \sqrt{b x^{2} + a} B a + 3 \, \sqrt{b x^{2} + a} A b - \frac{9 \,{\left (b x^{2} + a\right )} B a^{2} - B a^{3} - 6 \,{\left (b x^{2} + a\right )} A a b + A a^{2} b}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x^2 + a)^(3/2)*B - 9*sqrt(b*x^2 + a)*B*a + 3*sqrt(b*x^2 + a)*A*b - (9*(b
*x^2 + a)*B*a^2 - B*a^3 - 6*(b*x^2 + a)*A*a*b + A*a^2*b)/(b*x^2 + a)^(3/2))/b^4