3.1582 \(\int \frac{1}{(d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=131 \[ \frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac{b (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{b (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.165761, antiderivative size = 131, 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 \[ \frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac{b (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{b (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.1413, size = 119, normalized size = 0.91 \[ \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{2}} - \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{2}} - \frac{e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0858076, size = 69, normalized size = 0.53 \[ \frac{(a+b x) (b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d)}{\sqrt{(a+b x)^2} (d+e x) (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 81, normalized size = 0.6 \[{\frac{ \left ( bx+a \right ) \left ( \ln \left ( bx+a \right ) xbe-\ln \left ( ex+d \right ) xbe+\ln \left ( bx+a \right ) bd-\ln \left ( ex+d \right ) bd-ae+bd \right ) }{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.212486, size = 124, normalized size = 0.95 \[ \frac{b d - a e +{\left (b e x + b d\right )} \log \left (b x + a\right ) -{\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} +{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.14052, size = 233, normalized size = 1.78 \[ - \frac{b \log{\left (x + \frac{- \frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} - \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e + \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} + \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e - \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} - \frac{1}{a d e - b d^{2} + x \left (a e^{2} - b d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*log(x + (-a**3*b*e**3/(a*e - b*d)**2 + 3*a**2*b**2*d*e**2/(a*e - b*d)**2 - 3*
a*b**3*d**2*e/(a*e - b*d)**2 + a*b*e + b**4*d**3/(a*e - b*d)**2 + b**2*d)/(2*b**
2*e))/(a*e - b*d)**2 + b*log(x + (a**3*b*e**3/(a*e - b*d)**2 - 3*a**2*b**2*d*e**
2/(a*e - b*d)**2 + 3*a*b**3*d**2*e/(a*e - b*d)**2 + a*b*e - b**4*d**3/(a*e - b*d
)**2 + b**2*d)/(2*b**2*e))/(a*e - b*d)**2 - 1/(a*d*e - b*d**2 + x*(a*e**2 - b*d*
e))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214043, size = 139, normalized size = 1.06 \[{\left (\frac{b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} - \frac{b e{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac{1}{{\left (b d - a e\right )}{\left (x e + d\right )}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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