3.292 \(\int \sqrt{x} (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{5} x^{5/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{7} b B x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0419955, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{2}{5} x^{5/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{7} b B x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.17717, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{7}{2}}}{7} + x^{\frac{5}{2}} \left (\frac{2 A b}{5} + \frac{2 B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0157931, size = 33, normalized size = 0.85 \[ \frac{2}{105} x^{3/2} (7 a (5 A+3 B x)+3 b x (7 A+5 B x)) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(7*a*(5*A + 3*B*x) + 3*b*x*(7*A + 5*B*x)))/105

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Maple [A]  time = 0.006, size = 28, normalized size = 0.7 \[{\frac{30\,bB{x}^{2}+42\,Abx+42\,Bax+70\,Aa}{105}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)*x^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.35339, size = 36, normalized size = 0.92 \[ \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} + \frac{2}{5} \,{\left (B a + A b\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205526, size = 41, normalized size = 1.05 \[ \frac{2}{105} \,{\left (15 \, B b x^{3} + 35 \, A a x + 21 \,{\left (B a + A b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.2038, size = 37, normalized size = 0.95 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{7}{2}}}{7} + \frac{2 x^{\frac{5}{2}} \left (A b + B a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249327, size = 39, normalized size = 1. \[ \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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