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

Optimal. Leaf size=63 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2} \]

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(15/2
))/15 + (2*b^2*B*x^(21/2))/21

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Rubi [A]  time = 0.0962474, antiderivative size = 63, 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 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(15/2
))/15 + (2*b^2*B*x^(21/2))/21

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Rubi in Sympy [A]  time = 11.0932, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{21}{2}}}{21} + \frac{2 a x^{\frac{9}{2}} \left (2 A b + B a\right )}{9} + \frac{2 b x^{\frac{15}{2}} \left (A b + 2 B a\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(3/2)/3 + 2*B*b**2*x**(21/2)/21 + 2*a*x**(9/2)*(2*A*b + B*a)/9 + 2*b
*x**(15/2)*(A*b + 2*B*a)/15

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Mathematica [A]  time = 0.0360365, size = 53, normalized size = 0.84 \[ \frac{2}{315} x^{3/2} \left (105 a^2 A+21 b x^6 (2 a B+A b)+35 a x^3 (a B+2 A b)+15 b^2 B x^9\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(105*a^2*A + 35*a*(2*A*b + a*B)*x^3 + 21*b*(A*b + 2*a*B)*x^6 + 15*b^2
*B*x^9))/315

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Maple [A]  time = 0.01, size = 56, normalized size = 0.9 \[{\frac{30\,B{x}^{9}{b}^{2}+42\,A{b}^{2}{x}^{6}+84\,B{x}^{6}ab+140\,aAb{x}^{3}+70\,B{x}^{3}{a}^{2}+210\,A{a}^{2}}{315}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*x^(3/2)*(15*B*b^2*x^9+21*A*b^2*x^6+42*B*a*b*x^6+70*A*a*b*x^3+35*B*a^2*x^3+
105*A*a^2)

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Maxima [A]  time = 1.50146, size = 69, normalized size = 1.1 \[ \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{2}{15} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{15}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*B*b^2*x^(21/2) + 2/15*(2*B*a*b + A*b^2)*x^(15/2) + 2/9*(B*a^2 + 2*A*a*b)*x^
(9/2) + 2/3*A*a^2*x^(3/2)

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Fricas [A]  time = 0.229277, size = 73, normalized size = 1.16 \[ \frac{2}{315} \,{\left (15 \, B b^{2} x^{10} + 21 \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} x^{4} + 105 \, A a^{2} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(15*B*b^2*x^10 + 21*(2*B*a*b + A*b^2)*x^7 + 35*(B*a^2 + 2*A*a*b)*x^4 + 105
*A*a^2*x)*sqrt(x)

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Sympy [A]  time = 14.1647, size = 66, normalized size = 1.05 \[ \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{21}{2}}}{21} + \frac{2 x^{\frac{15}{2}} \left (A b^{2} + 2 B a b\right )}{15} + \frac{2 x^{\frac{9}{2}} \left (2 A a b + B a^{2}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(3/2)/3 + 2*B*b**2*x**(21/2)/21 + 2*x**(15/2)*(A*b**2 + 2*B*a*b)/15
+ 2*x**(9/2)*(2*A*a*b + B*a**2)/9

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GIAC/XCAS [A]  time = 0.211709, size = 72, normalized size = 1.14 \[ \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{4}{15} \, B a b x^{\frac{15}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*B*b^2*x^(21/2) + 4/15*B*a*b*x^(15/2) + 2/15*A*b^2*x^(15/2) + 2/9*B*a^2*x^(9
/2) + 4/9*A*a*b*x^(9/2) + 2/3*A*a^2*x^(3/2)