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

Optimal. Leaf size=37 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} b B x^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.0422918, antiderivative size = 37, 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 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} b B x^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.31753, size = 39, normalized size = 1.05 \[ 2 A a \sqrt{x} + \frac{2 B b x^{\frac{5}{2}}}{5} + x^{\frac{3}{2}} \left (\frac{2 A b}{3} + \frac{2 B a}{3}\right ) \]

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

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Mathematica [A]  time = 0.0153496, size = 31, normalized size = 0.84 \[ \frac{2}{15} \sqrt{x} (5 a (3 A+B x)+b x (5 A+3 B x)) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(5*a*(3*A + B*x) + b*x*(5*A + 3*B*x)))/15

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Maple [A]  time = 0.006, size = 28, normalized size = 0.8 \[{\frac{6\,bB{x}^{2}+10\,Abx+10\,Bax+30\,Aa}{15}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^(1/2),x)

[Out]

2/15*x^(1/2)*(3*B*b*x^2+5*A*b*x+5*B*a*x+15*A*a)

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Maxima [A]  time = 1.32874, size = 36, normalized size = 0.97 \[ \frac{2}{5} \, B b x^{\frac{5}{2}} + 2 \, A a \sqrt{x} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206699, size = 36, normalized size = 0.97 \[ \frac{2}{15} \,{\left (3 \, B b x^{2} + 15 \, A a + 5 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*B*b*x^2 + 15*A*a + 5*(B*a + A*b)*x)*sqrt(x)

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Sympy [A]  time = 5.04412, size = 44, normalized size = 1.19 \[ 2 A a \sqrt{x} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{5}{2}}}{5} \]

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

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GIAC/XCAS [A]  time = 0.261892, size = 39, normalized size = 1.05 \[ \frac{2}{5} \, B b x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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