3.1537 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{d+e x} \, dx\)

Optimal. Leaf size=80 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.118047, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0015, size = 63, normalized size = 0.79 \[ \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0329646, size = 42, normalized size = 0.52 \[ \frac{\sqrt{(a+b x)^2} ((a e-b d) \log (d+e x)+b e x)}{e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.031, size = 44, normalized size = 0.6 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( \ln \left ( bex+bd \right ) ae-\ln \left ( bex+bd \right ) bd+bex+ae \right ) }{{e}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206606, size = 34, normalized size = 0.42 \[ \frac{b e x -{\left (b d - a e\right )} \log \left (e x + d\right )}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*e*x - (b*d - a*e)*log(e*x + d))/e^2

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Sympy [A]  time = 1.30095, size = 20, normalized size = 0.25 \[ \frac{b x}{e} + \frac{\left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x/e + (a*e - b*d)*log(d + e*x)/e**2

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GIAC/XCAS [A]  time = 0.209285, size = 61, normalized size = 0.76 \[ b x e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) -{\left (b d{\rm sign}\left (b x + a\right ) - a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*e^(-1)*sign(b*x + a) - (b*d*sign(b*x + a) - a*e*sign(b*x + a))*e^(-2)*ln(abs
(x*e + d))