3.300 \(\int \sqrt{x} (a+b x)^2 (A+B x) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.0771655, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.02685, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} + \frac{2 a x^{\frac{5}{2}} \left (2 A b + B a\right )}{5} + \frac{2 b x^{\frac{7}{2}} \left (A b + 2 B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0271333, size = 52, normalized size = 0.83 \[ \frac{2}{315} x^{3/2} \left (21 a^2 (5 A+3 B x)+18 a b x (7 A+5 B x)+5 b^2 x^2 (9 A+7 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(21*a^2*(5*A + 3*B*x) + 18*a*b*x*(7*A + 5*B*x) + 5*b^2*x^2*(9*A + 7*B
*x)))/315

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Maple [A]  time = 0.007, size = 52, normalized size = 0.8 \[{\frac{70\,B{b}^{2}{x}^{3}+90\,A{b}^{2}{x}^{2}+180\,B{x}^{2}ab+252\,aAbx+126\,{a}^{2}Bx+210\,{a}^{2}A}{315}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*x^(3/2)*(35*B*b^2*x^3+45*A*b^2*x^2+90*B*a*b*x^2+126*A*a*b*x+63*B*a^2*x+105
*A*a^2)

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Maxima [A]  time = 1.35597, size = 69, normalized size = 1.1 \[ \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20871, size = 73, normalized size = 1.16 \[ \frac{2}{315} \,{\left (35 \, B b^{2} x^{4} + 105 \, A a^{2} x + 45 \,{\left (2 \, B a b + A b^{2}\right )} x^{3} + 63 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b^2*x^4 + 105*A*a^2*x + 45*(2*B*a*b + A*b^2)*x^3 + 63*(B*a^2 + 2*A*a
*b)*x^2)*sqrt(x)

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Sympy [A]  time = 4.23366, size = 66, normalized size = 1.05 \[ \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} + \frac{2 x^{\frac{7}{2}} \left (A b^{2} + 2 B a b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (2 A a b + B a^{2}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.248239, size = 72, normalized size = 1.14 \[ \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*B*b^2*x^(9/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 2/5*B*a^2*x^(5/2) +
4/5*A*a*b*x^(5/2) + 2/3*A*a^2*x^(3/2)