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

Optimal. Leaf size=118 \[ \frac{2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.150112, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.4377, size = 122, normalized size = 1.03 \[ \frac{B \sqrt{x} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 b} + \frac{4 a \sqrt{x} \left (5 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 b \left (a + b x\right )} + \frac{2 \sqrt{x} \left (5 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*sqrt(x)*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*b) + 4*a*sqrt(x)*(5*
A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*b*(a + b*x)) + 2*sqrt(x)*(5*A*b
- B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*b)

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Mathematica [A]  time = 0.0392552, size = 49, normalized size = 0.42 \[ \frac{2 \sqrt{x} \sqrt{(a+b x)^2} (5 a (3 A+B x)+b x (5 A+3 B x))}{15 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 0.4 \[{\frac{6\,Bb{x}^{2}+10\,Abx+10\,aBx+30\,aA}{15\,bx+15\,a}\sqrt{x}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [A]  time = 0.70379, size = 46, normalized size = 0.39 \[ \frac{2}{15} \,{\left (3 \, b x^{2} + 5 \, a x\right )} B \sqrt{x} + \frac{2 \,{\left (b x^{2} + 3 \, a x\right )} A}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.301607, size = 36, normalized size = 0.31 \[ \frac{2}{15} \,{\left (3 \, B b x^{2} + 15 \, A a + 5 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{\left (a + b x\right )^{2}}}{\sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt((a + b*x)**2)/sqrt(x), x)

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GIAC/XCAS [A]  time = 0.27196, size = 72, normalized size = 0.61 \[ \frac{2}{5} \, B b x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A b x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, A a \sqrt{x}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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