3.1538 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]

[Out]

((b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^2*(a + b*x)*(d + e*x)) + (b*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 = 0.120931, antiderivative size = 85, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^2*(a + b*x)*(d + e*x)) + (b*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 = 22.8254, size = 76, normalized size = 0.89 \[ \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{2} \left (a + b x\right )} - \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2} \left (a + b x\right ) \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0369926, size = 50, normalized size = 0.59 \[ \frac{\sqrt{(a+b x)^2} (-a e+b (d+e x) \log (d+e x)+b d)}{e^2 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.015, size = 51, normalized size = 0.6 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( \ln \left ( bex+bd \right ) xbe+\ln \left ( bex+bd \right ) bd-ae+bd \right ) }{{e}^{2} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206633, size = 50, normalized size = 0.59 \[ \frac{b d - a e +{\left (b e x + b d\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.47269, size = 27, normalized size = 0.32 \[ \frac{b \log{\left (d + e x \right )}}{e^{2}} - \frac{a e - b d}{d e^{2} + e^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211564, size = 69, normalized size = 0.81 \[ b e^{\left (-2\right )}{\rm ln}\left ({\left | x e + d \right |}\right ){\rm sign}\left (b x + a\right ) + \frac{{\left (b d{\rm sign}\left (b x + a\right ) - a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{x e + d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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