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

Optimal. Leaf size=85 \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2} \]

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(11
/2))/11 + (2*b^2*(A*b + 3*a*B)*x^(15/2))/15 + (2*b^3*B*x^(19/2))/19

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Rubi [A]  time = 0.112036, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{7} a^2 x^{7/2} (a B+3 A b)+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{19} b^3 B x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(3/2))/3 + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(11
/2))/11 + (2*b^2*(A*b + 3*a*B)*x^(15/2))/15 + (2*b^3*B*x^(19/2))/19

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Rubi in Sympy [A]  time = 16.9412, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B b^{3} x^{\frac{19}{2}}}{19} + \frac{2 a^{2} x^{\frac{7}{2}} \left (3 A b + B a\right )}{7} + \frac{6 a b x^{\frac{11}{2}} \left (A b + B a\right )}{11} + \frac{2 b^{2} x^{\frac{15}{2}} \left (A b + 3 B a\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(3/2)/3 + 2*B*b**3*x**(19/2)/19 + 2*a**2*x**(7/2)*(3*A*b + B*a)/7 +
6*a*b*x**(11/2)*(A*b + B*a)/11 + 2*b**2*x**(15/2)*(A*b + 3*B*a)/15

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Mathematica [A]  time = 0.0411249, size = 71, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (7315 a^3 A+3135 a^2 x^2 (a B+3 A b)+1463 b^2 x^6 (3 a B+A b)+5985 a b x^4 (a B+A b)+1155 b^3 B x^8\right )}{21945} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(7315*a^3*A + 3135*a^2*(3*A*b + a*B)*x^2 + 5985*a*b*(A*b + a*B)*x^4 +
1463*b^2*(A*b + 3*a*B)*x^6 + 1155*b^3*B*x^8))/21945

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Maple [A]  time = 0.009, size = 80, normalized size = 0.9 \[{\frac{2310\,{b}^{3}B{x}^{8}+2926\,{x}^{6}{b}^{3}A+8778\,{x}^{6}a{b}^{2}B+11970\,{x}^{4}a{b}^{2}A+11970\,{x}^{4}{a}^{2}bB+18810\,{x}^{2}A{a}^{2}b+6270\,{x}^{2}B{a}^{3}+14630\,{a}^{3}A}{21945}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*x^(3/2)*(1155*B*b^3*x^8+1463*A*b^3*x^6+4389*B*a*b^2*x^6+5985*A*a*b^2*x^4
+5985*B*a^2*b*x^4+9405*A*a^2*b*x^2+3135*B*a^3*x^2+7315*A*a^3)

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Maxima [A]  time = 1.32999, size = 99, normalized size = 1.16 \[ \frac{2}{19} \, B b^{3} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{15}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*B*b^3*x^(19/2) + 2/15*(3*B*a*b^2 + A*b^3)*x^(15/2) + 6/11*(B*a^2*b + A*a*b^
2)*x^(11/2) + 2/3*A*a^3*x^(3/2) + 2/7*(B*a^3 + 3*A*a^2*b)*x^(7/2)

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Fricas [A]  time = 0.212201, size = 103, normalized size = 1.21 \[ \frac{2}{21945} \,{\left (1155 \, B b^{3} x^{9} + 1463 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 5985 \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + 7315 \, A a^{3} x + 3135 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*(1155*B*b^3*x^9 + 1463*(3*B*a*b^2 + A*b^3)*x^7 + 5985*(B*a^2*b + A*a*b^2
)*x^5 + 7315*A*a^3*x + 3135*(B*a^3 + 3*A*a^2*b)*x^3)*sqrt(x)

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Sympy [A]  time = 10.2179, size = 95, normalized size = 1.12 \[ \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B b^{3} x^{\frac{19}{2}}}{19} + \frac{2 x^{\frac{15}{2}} \left (A b^{3} + 3 B a b^{2}\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 A a b^{2} + 3 B a^{2} b\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (3 A a^{2} b + B a^{3}\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(3/2)/3 + 2*B*b**3*x**(19/2)/19 + 2*x**(15/2)*(A*b**3 + 3*B*a*b**2)/
15 + 2*x**(11/2)*(3*A*a*b**2 + 3*B*a**2*b)/11 + 2*x**(7/2)*(3*A*a**2*b + B*a**3)
/7

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GIAC/XCAS [A]  time = 0.212652, size = 104, normalized size = 1.22 \[ \frac{2}{19} \, B b^{3} x^{\frac{19}{2}} + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} + \frac{2}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{6}{7} \, A a^{2} b x^{\frac{7}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*B*b^3*x^(19/2) + 2/5*B*a*b^2*x^(15/2) + 2/15*A*b^3*x^(15/2) + 6/11*B*a^2*b*
x^(11/2) + 6/11*A*a*b^2*x^(11/2) + 2/7*B*a^3*x^(7/2) + 6/7*A*a^2*b*x^(7/2) + 2/3
*A*a^3*x^(3/2)