3.1581 \(\int \frac{1}{(d+e x) \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.106982, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{(a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.9142, size = 76, normalized size = 0.88 \[ - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0335054, size = 42, normalized size = 0.49 \[ \frac{(a+b x) (\log (a+b x)-\log (d+e x))}{\sqrt{(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 42, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( \ln \left ( bx+a \right ) -\ln \left ( ex+d \right ) \right ) }{ae-bd}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.211808, size = 35, normalized size = 0.41 \[ \frac{\log \left (b x + a\right ) - \log \left (e x + d\right )}{b d - a e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.08141, size = 128, normalized size = 1.49 \[ \frac{\log{\left (x + \frac{- \frac{a^{2} e^{2}}{a e - b d} + \frac{2 a b d e}{a e - b d} + a e - \frac{b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} - \frac{\log{\left (x + \frac{\frac{a^{2} e^{2}}{a e - b d} - \frac{2 a b d e}{a e - b d} + a e + \frac{b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226185, size = 101, normalized size = 1.17 \[ \frac{{\rm ln}\left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right ){\rm sign}\left (b x + a\right )}{{\left | b d - a e \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)),x, algorithm="giac")

[Out]

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