3.2125 \(\int \frac{(a+b x) \sqrt{d+e x}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 (a+b x) (d+e x)^{3/2}}{3 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(a + b*x)*(d + e*x)^(3/2))/(3*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.1206, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 (a+b x) (d+e x)^{3/2}}{3 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*(d + e*x)^(3/2))/(3*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 15.934, size = 37, normalized size = 0.9 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}}{3 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(1/2)/((b*x+a)**2)**(1/2),x)

[Out]

2*(a + b*x)*(d + e*x)**(3/2)/(3*e*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0237514, size = 32, normalized size = 0.78 \[ \frac{2 (a+b x) (d+e x)^{3/2}}{3 e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*(d + e*x)^(3/2))/(3*e*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.004, size = 27, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{3\,e} \left ( ex+d \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*x+a)*(e*x+d)^(3/2)/e/((b*x+a)^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285023, size = 16, normalized size = 0.39 \[ \frac{2 \,{\left (e x + d\right )}^{\frac{3}{2}}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*sqrt(e*x + d)/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

2/3*(e*x + d)^(3/2)/e

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \sqrt{d + e x}}{\sqrt{\left (a + b x\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**(1/2)/((b*x+a)**2)**(1/2),x)

[Out]

Integral((a + b*x)*sqrt(d + e*x)/sqrt((a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.285804, size = 24, normalized size = 0.59 \[ \frac{2}{3} \,{\left (x e + d\right )}^{\frac{3}{2}} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x*e + d)^(3/2)*e^(-1)*sign(b*x + a)