3.506 \(\int x^3 \sqrt{a+b x^2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=73 \[ \frac{\left (a+b x^2\right )^{5/2} (A b-2 a B)}{5 b^3}-\frac{a \left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^3}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^3} \]

[Out]

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

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Rubi [A]  time = 0.165699, 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 \[ \frac{\left (a+b x^2\right )^{5/2} (A b-2 a B)}{5 b^3}-\frac{a \left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^3}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.0767, size = 63, normalized size = 0.86 \[ \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{3}} - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b - 2 B a\right )}{5 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x**2+A)*(b*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0571598, size = 57, normalized size = 0.78 \[ \frac{\left (a+b x^2\right )^{3/2} \left (8 a^2 B-2 a b \left (7 A+6 B x^2\right )+3 b^2 x^2 \left (7 A+5 B x^2\right )\right )}{105 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^(3/2)*(8*a^2*B + 3*b^2*x^2*(7*A + 5*B*x^2) - 2*a*b*(7*A + 6*B*x^2))
)/(105*b^3)

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Maple [A]  time = 0.008, size = 53, normalized size = 0.7 \[ -{\frac{-15\,{b}^{2}B{x}^{4}-21\,A{x}^{2}{b}^{2}+12\,B{x}^{2}ab+14\,abA-8\,{a}^{2}B}{105\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x^2+A)*(b*x^2+a)^(1/2),x)

[Out]

-1/105*(b*x^2+a)^(3/2)*(-15*B*b^2*x^4-21*A*b^2*x^2+12*B*a*b*x^2+14*A*a*b-8*B*a^2
)/b^3

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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)*sqrt(b*x^2 + a)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227005, size = 101, normalized size = 1.38 \[ \frac{{\left (15 \, B b^{3} x^{6} + 3 \,{\left (B a b^{2} + 7 \, A b^{3}\right )} x^{4} + 8 \, B a^{3} - 14 \, A a^{2} b -{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*B*b^3*x^6 + 3*(B*a*b^2 + 7*A*b^3)*x^4 + 8*B*a^3 - 14*A*a^2*b - (4*B*a^
2*b - 7*A*a*b^2)*x^2)*sqrt(b*x^2 + a)/b^3

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Sympy [A]  time = 2.15881, size = 162, normalized size = 2.22 \[ \begin{cases} - \frac{2 A a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{A a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{A x^{4} \sqrt{a + b x^{2}}}{5} + \frac{8 B a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 B a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{B a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{B x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{4}}{4} + \frac{B x^{6}}{6}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x**2+A)*(b*x**2+a)**(1/2),x)

[Out]

Piecewise((-2*A*a**2*sqrt(a + b*x**2)/(15*b**2) + A*a*x**2*sqrt(a + b*x**2)/(15*
b) + A*x**4*sqrt(a + b*x**2)/5 + 8*B*a**3*sqrt(a + b*x**2)/(105*b**3) - 4*B*a**2
*x**2*sqrt(a + b*x**2)/(105*b**2) + B*a*x**4*sqrt(a + b*x**2)/(35*b) + B*x**6*sq
rt(a + b*x**2)/7, Ne(b, 0)), (sqrt(a)*(A*x**4/4 + B*x**6/6), True))

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GIAC/XCAS [A]  time = 0.228996, size = 107, normalized size = 1.47 \[ \frac{\frac{7 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} A}{b} + \frac{{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} B}{b^{2}}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(7*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*A/b + (15*(b*x^2 + a)^(7/
2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*B/b^2)/b