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

Optimal. Leaf size=100 \[ \frac{a^2 \sqrt{a+b x^2} (A b-a B)}{b^4}+\frac{\left (a+b x^2\right )^{5/2} (A b-3 a B)}{5 b^4}-\frac{a \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^4} \]

[Out]

(a^2*(A*b - a*B)*Sqrt[a + b*x^2])/b^4 - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(3
*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^(5/2))/(5*b^4) + (B*(a + b*x^2)^(7/2))/(7*b^4
)

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Rubi [A]  time = 0.227582, antiderivative size = 100, 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 \[ \frac{a^2 \sqrt{a+b x^2} (A b-a B)}{b^4}+\frac{\left (a+b x^2\right )^{5/2} (A b-3 a B)}{5 b^4}-\frac{a \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(A*b - a*B)*Sqrt[a + b*x^2])/b^4 - (a*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(3
*b^4) + ((A*b - 3*a*B)*(a + b*x^2)^(5/2))/(5*b^4) + (B*(a + b*x^2)^(7/2))/(7*b^4
)

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Rubi in Sympy [A]  time = 25.0849, size = 90, normalized size = 0.9 \[ \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{a^{2} \sqrt{a + b x^{2}} \left (A b - B a\right )}{b^{4}} - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{3 b^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b - 3 B a\right )}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x**2)**(7/2)/(7*b**4) + a**2*sqrt(a + b*x**2)*(A*b - B*a)/b**4 - a*(a +
b*x**2)**(3/2)*(2*A*b - 3*B*a)/(3*b**4) + (a + b*x**2)**(5/2)*(A*b - 3*B*a)/(5*
b**4)

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Mathematica [A]  time = 0.0716602, size = 78, normalized size = 0.78 \[ \frac{\sqrt{a+b x^2} \left (-48 a^3 B+8 a^2 b \left (7 A+3 B x^2\right )-2 a b^2 x^2 \left (14 A+9 B x^2\right )+3 b^3 x^4 \left (7 A+5 B x^2\right )\right )}{105 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*(-48*a^3*B + 8*a^2*b*(7*A + 3*B*x^2) + 3*b^3*x^4*(7*A + 5*B*x^2
) - 2*a*b^2*x^2*(14*A + 9*B*x^2)))/(105*b^4)

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Maple [A]  time = 0.009, size = 77, normalized size = 0.8 \[{\frac{15\,{x}^{6}B{b}^{3}+21\,A{b}^{3}{x}^{4}-18\,Ba{b}^{2}{x}^{4}-28\,Aa{b}^{2}{x}^{2}+24\,B{a}^{2}b{x}^{2}+56\,A{a}^{2}b-48\,B{a}^{3}}{105\,{b}^{4}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(b*x^2+a)^(1/2)*(15*B*b^3*x^6+21*A*b^3*x^4-18*B*a*b^2*x^4-28*A*a*b^2*x^2+2
4*B*a^2*b*x^2+56*A*a^2*b-48*B*a^3)/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/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228859, size = 103, normalized size = 1.03 \[ \frac{{\left (15 \, B b^{3} x^{6} - 3 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} - 48 \, B a^{3} + 56 \, A a^{2} b + 4 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*B*b^3*x^6 - 3*(6*B*a*b^2 - 7*A*b^3)*x^4 - 48*B*a^3 + 56*A*a^2*b + 4*(6
*B*a^2*b - 7*A*a*b^2)*x^2)*sqrt(b*x^2 + a)/b^4

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Sympy [A]  time = 4.77603, size = 172, normalized size = 1.72 \[ \begin{cases} \frac{8 A a^{2} \sqrt{a + b x^{2}}}{15 b^{3}} - \frac{4 A a x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{A x^{4} \sqrt{a + b x^{2}}}{5 b} - \frac{16 B a^{3} \sqrt{a + b x^{2}}}{35 b^{4}} + \frac{8 B a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b^{3}} - \frac{6 B a x^{4} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{B x^{6} \sqrt{a + b x^{2}}}{7 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{8}}{8}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((8*A*a**2*sqrt(a + b*x**2)/(15*b**3) - 4*A*a*x**2*sqrt(a + b*x**2)/(15
*b**2) + A*x**4*sqrt(a + b*x**2)/(5*b) - 16*B*a**3*sqrt(a + b*x**2)/(35*b**4) +
8*B*a**2*x**2*sqrt(a + b*x**2)/(35*b**3) - 6*B*a*x**4*sqrt(a + b*x**2)/(35*b**2)
+ B*x**6*sqrt(a + b*x**2)/(7*b), Ne(b, 0)), ((A*x**6/6 + B*x**8/8)/sqrt(a), Tru
e))

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GIAC/XCAS [A]  time = 0.232575, size = 140, normalized size = 1.4 \[ \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B - 63 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a + 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} - 105 \, \sqrt{b x^{2} + a} B a^{3} + 21 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b - 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b + 105 \, \sqrt{b x^{2} + a} A a^{2} b}{105 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*(b*x^2 + a)^(7/2)*B - 63*(b*x^2 + a)^(5/2)*B*a + 105*(b*x^2 + a)^(3/2)
*B*a^2 - 105*sqrt(b*x^2 + a)*B*a^3 + 21*(b*x^2 + a)^(5/2)*A*b - 70*(b*x^2 + a)^(
3/2)*A*a*b + 105*sqrt(b*x^2 + a)*A*a^2*b)/b^4