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

Optimal. Leaf size=73 \[ \frac{\left (a+b x^2\right )^{7/2} (A b-2 a B)}{7 b^3}-\frac{a \left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^3}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^(7/2))/(
7*b^3) + (B*(a + b*x^2)^(9/2))/(9*b^3)

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Rubi [A]  time = 0.167402, 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 \[ \frac{\left (a+b x^2\right )^{7/2} (A b-2 a B)}{7 b^3}-\frac{a \left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^3}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(A*b - a*B)*(a + b*x^2)^(5/2))/(5*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^(7/2))/(
7*b^3) + (B*(a + b*x^2)^(9/2))/(9*b^3)

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Rubi in Sympy [A]  time = 19.6681, size = 63, normalized size = 0.86 \[ \frac{B \left (a + b x^{2}\right )^{\frac{9}{2}}}{9 b^{3}} - \frac{a \left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{7}{2}} \left (A b - 2 B a\right )}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x**2)**(9/2)/(9*b**3) - a*(a + b*x**2)**(5/2)*(A*b - B*a)/(5*b**3) + (a
+ b*x**2)**(7/2)*(A*b - 2*B*a)/(7*b**3)

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Mathematica [A]  time = 0.0652535, size = 57, normalized size = 0.78 \[ \frac{\left (a+b x^2\right )^{5/2} \left (8 a^2 B-2 a b \left (9 A+10 B x^2\right )+5 b^2 x^2 \left (9 A+7 B x^2\right )\right )}{315 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^(5/2)*(8*a^2*B + 5*b^2*x^2*(9*A + 7*B*x^2) - 2*a*b*(9*A + 10*B*x^2)
))/(315*b^3)

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Maple [A]  time = 0.007, size = 53, normalized size = 0.7 \[ -{\frac{-35\,{b}^{2}B{x}^{4}-45\,A{b}^{2}{x}^{2}+20\,Bab{x}^{2}+18\,abA-8\,{a}^{2}B}{315\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/315*(b*x^2+a)^(5/2)*(-35*B*b^2*x^4-45*A*b^2*x^2+20*B*a*b*x^2+18*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)*(b*x^2 + a)^(3/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215094, size = 134, normalized size = 1.84 \[ \frac{{\left (35 \, B b^{4} x^{8} + 5 \,{\left (10 \, B a b^{3} + 9 \, A b^{4}\right )} x^{6} + 8 \, B a^{4} - 18 \, A a^{3} b + 3 \,{\left (B a^{2} b^{2} + 24 \, A a b^{3}\right )} x^{4} -{\left (4 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(35*B*b^4*x^8 + 5*(10*B*a*b^3 + 9*A*b^4)*x^6 + 8*B*a^4 - 18*A*a^3*b + 3*(B
*a^2*b^2 + 24*A*a*b^3)*x^4 - (4*B*a^3*b - 9*A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/b^3

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Sympy [A]  time = 7.30306, size = 209, normalized size = 2.86 \[ \begin{cases} - \frac{2 A a^{3} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{A a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b} + \frac{8 A a x^{4} \sqrt{a + b x^{2}}}{35} + \frac{A b x^{6} \sqrt{a + b x^{2}}}{7} + \frac{8 B a^{4} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{4 B a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{2}} + \frac{B a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b} + \frac{10 B a x^{6} \sqrt{a + b x^{2}}}{63} + \frac{B b x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\a^{\frac{3}{2}} \left (\frac{A x^{4}}{4} + \frac{B x^{6}}{6}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*A*a**3*sqrt(a + b*x**2)/(35*b**2) + A*a**2*x**2*sqrt(a + b*x**2)/(
35*b) + 8*A*a*x**4*sqrt(a + b*x**2)/35 + A*b*x**6*sqrt(a + b*x**2)/7 + 8*B*a**4*
sqrt(a + b*x**2)/(315*b**3) - 4*B*a**3*x**2*sqrt(a + b*x**2)/(315*b**2) + B*a**2
*x**4*sqrt(a + b*x**2)/(105*b) + 10*B*a*x**6*sqrt(a + b*x**2)/63 + B*b*x**8*sqrt
(a + b*x**2)/9, Ne(b, 0)), (a**(3/2)*(A*x**4/4 + B*x**6/6), True))

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GIAC/XCAS [A]  time = 0.22422, size = 247, normalized size = 3.38 \[ \frac{\frac{21 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} A a}{b} + \frac{3 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} B a}{b^{2}} + \frac{3 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} A}{b} + \frac{{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} B}{b^{2}}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(21*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*A*a/b + 3*(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*a/b^2 + 3*(15*(b*
x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*A/b + (35*(b
*x^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(b*x
^2 + a)^(3/2)*a^3)*B/b^2)/b