3.540 \(\int x^3 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=73 \[ \frac{\left (a+b x^2\right )^{9/2} (A b-2 a B)}{9 b^3}-\frac{a \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^3}+\frac{B \left (a+b x^2\right )^{11/2}}{11 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^(9/2))/(
9*b^3) + (B*(a + b*x^2)^(11/2))/(11*b^3)

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Rubi [A]  time = 0.165741, 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 \[ \frac{\left (a+b x^2\right )^{9/2} (A b-2 a B)}{9 b^3}-\frac{a \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^3}+\frac{B \left (a+b x^2\right )^{11/2}}{11 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

-(a*(A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^3) + ((A*b - 2*a*B)*(a + b*x^2)^(9/2))/(
9*b^3) + (B*(a + b*x^2)^(11/2))/(11*b^3)

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Rubi in Sympy [A]  time = 19.7899, size = 63, normalized size = 0.86 \[ \frac{B \left (a + b x^{2}\right )^{\frac{11}{2}}}{11 b^{3}} - \frac{a \left (a + b x^{2}\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{9}{2}} \left (A b - 2 B a\right )}{9 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

B*(a + b*x**2)**(11/2)/(11*b**3) - a*(a + b*x**2)**(7/2)*(A*b - B*a)/(7*b**3) +
(a + b*x**2)**(9/2)*(A*b - 2*B*a)/(9*b**3)

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Mathematica [A]  time = 0.0784186, size = 57, normalized size = 0.78 \[ \frac{\left (a+b x^2\right )^{7/2} \left (8 a^2 B-2 a b \left (11 A+14 B x^2\right )+7 b^2 x^2 \left (11 A+9 B x^2\right )\right )}{693 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

((a + b*x^2)^(7/2)*(8*a^2*B + 7*b^2*x^2*(11*A + 9*B*x^2) - 2*a*b*(11*A + 14*B*x^
2)))/(693*b^3)

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Maple [A]  time = 0.009, size = 53, normalized size = 0.7 \[ -{\frac{-63\,{b}^{2}B{x}^{4}-77\,A{b}^{2}{x}^{2}+28\,Bab{x}^{2}+22\,abA-8\,{a}^{2}B}{693\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^(5/2)*(B*x^2+A),x)

[Out]

-1/693*(b*x^2+a)^(7/2)*(-63*B*b^2*x^4-77*A*b^2*x^2+28*B*a*b*x^2+22*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)^(5/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218933, size = 165, normalized size = 2.26 \[ \frac{{\left (63 \, B b^{5} x^{10} + 7 \,{\left (23 \, B a b^{4} + 11 \, A b^{5}\right )} x^{8} +{\left (113 \, B a^{2} b^{3} + 209 \, A a b^{4}\right )} x^{6} + 8 \, B a^{5} - 22 \, A a^{4} b + 3 \,{\left (B a^{3} b^{2} + 55 \, A a^{2} b^{3}\right )} x^{4} -{\left (4 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{693 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x^3,x, algorithm="fricas")

[Out]

1/693*(63*B*b^5*x^10 + 7*(23*B*a*b^4 + 11*A*b^5)*x^8 + (113*B*a^2*b^3 + 209*A*a*
b^4)*x^6 + 8*B*a^5 - 22*A*a^4*b + 3*(B*a^3*b^2 + 55*A*a^2*b^3)*x^4 - (4*B*a^4*b
- 11*A*a^3*b^2)*x^2)*sqrt(b*x^2 + a)/b^3

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Sympy [A]  time = 19.4106, size = 260, normalized size = 3.56 \[ \begin{cases} - \frac{2 A a^{4} \sqrt{a + b x^{2}}}{63 b^{2}} + \frac{A a^{3} x^{2} \sqrt{a + b x^{2}}}{63 b} + \frac{5 A a^{2} x^{4} \sqrt{a + b x^{2}}}{21} + \frac{19 A a b x^{6} \sqrt{a + b x^{2}}}{63} + \frac{A b^{2} x^{8} \sqrt{a + b x^{2}}}{9} + \frac{8 B a^{5} \sqrt{a + b x^{2}}}{693 b^{3}} - \frac{4 B a^{4} x^{2} \sqrt{a + b x^{2}}}{693 b^{2}} + \frac{B a^{3} x^{4} \sqrt{a + b x^{2}}}{231 b} + \frac{113 B a^{2} x^{6} \sqrt{a + b x^{2}}}{693} + \frac{23 B a b x^{8} \sqrt{a + b x^{2}}}{99} + \frac{B b^{2} x^{10} \sqrt{a + b x^{2}}}{11} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (\frac{A x^{4}}{4} + \frac{B x^{6}}{6}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

Piecewise((-2*A*a**4*sqrt(a + b*x**2)/(63*b**2) + A*a**3*x**2*sqrt(a + b*x**2)/(
63*b) + 5*A*a**2*x**4*sqrt(a + b*x**2)/21 + 19*A*a*b*x**6*sqrt(a + b*x**2)/63 +
A*b**2*x**8*sqrt(a + b*x**2)/9 + 8*B*a**5*sqrt(a + b*x**2)/(693*b**3) - 4*B*a**4
*x**2*sqrt(a + b*x**2)/(693*b**2) + B*a**3*x**4*sqrt(a + b*x**2)/(231*b) + 113*B
*a**2*x**6*sqrt(a + b*x**2)/693 + 23*B*a*b*x**8*sqrt(a + b*x**2)/99 + B*b**2*x**
10*sqrt(a + b*x**2)/11, Ne(b, 0)), (a**(5/2)*(A*x**4/4 + B*x**6/6), True))

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GIAC/XCAS [A]  time = 0.25041, size = 431, normalized size = 5.9 \[ \frac{\frac{231 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} A a^{2}}{b} + \frac{33 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} B a^{2}}{b^{2}} + \frac{66 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} A a}{b} + \frac{22 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} B a}{b^{2}} + \frac{11 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} A}{b} + \frac{{\left (315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}\right )} B}{b^{2}}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x^3,x, algorithm="giac")

[Out]

1/3465*(231*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*A*a^2/b + 33*(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^2/b^2 + 66
*(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*a/
b + 22*(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*a/b^2 + 11*(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)*A/b + (315
*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2970*(b*x^2 + a)^(7/2)*a^2 - 27
72*(b*x^2 + a)^(5/2)*a^3 + 1155*(b*x^2 + a)^(3/2)*a^4)*B/b^2)/b