3.2126 \(\int \frac{a+b x}{\sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 (a+b x) \sqrt{d+e x}}{e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.117139, antiderivative size = 39, 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 \[ \frac{2 (a+b x) \sqrt{d+e x}}{e \sqrt{a^2+2 a b x+b^2 x^2}} \]

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

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Rubi in Sympy [A]  time = 16.0241, size = 36, normalized size = 0.92 \[ \frac{2 \left (a + b x\right ) \sqrt{d + e x}}{e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

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

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Mathematica [A]  time = 0.0206306, size = 30, normalized size = 0.77 \[ \frac{2 (a+b x) \sqrt{d+e x}}{e \sqrt{(a+b x)^2}} \]

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

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

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*(b*x+a)*(e*x+d)^(1/2)/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((b*x + a)^2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28539, size = 16, normalized size = 0.41 \[ \frac{2 \, \sqrt{e x + d}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(e*x + d)/e

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

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 = 0.2819, size = 24, normalized size = 0.62 \[ 2 \, \sqrt{x e + d} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x*e + d)*e^(-1)*sign(b*x + a)