3.1763 \(\int \frac{A+B x}{(d+e x) \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=107 \[ \frac{(a+b x) (A b-a B) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{(a+b x) (B d-A e) \log (d+e x)}{e \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.239343, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{(a+b x) (A b-a B) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{(a+b x) (B d-A e) \log (d+e x)}{e \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.0132, size = 99, normalized size = 0.93 \[ \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \log{\left (d + e x \right )}}{2 e \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{\left (a + b x\right ) \left (A b - B a\right ) \log{\left (a + b x \right )}}{b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.06505, size = 66, normalized size = 0.62 \[ \frac{(a+b x) (e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x))}{b e \sqrt{(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 75, normalized size = 0.7 \[{\frac{ \left ( bx+a \right ) \left ( A\ln \left ( ex+d \right ) be-A\ln \left ( bx+a \right ) be-B\ln \left ( ex+d \right ) bd+B\ln \left ( bx+a \right ) ae \right ) }{b \left ( ae-bd \right ) e}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.288939, size = 72, normalized size = 0.67 \[ -\frac{{\left (B a - A b\right )} e \log \left (b x + a\right ) -{\left (B b d - A b e\right )} \log \left (e x + d\right )}{b^{2} d e - a b e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.66532, size = 226, normalized size = 2.11 \[ - \frac{\left (- A e + B d\right ) \log{\left (x + \frac{- A a e - A b d + 2 B a d - \frac{a^{2} e \left (- A e + B d\right )}{a e - b d} + \frac{2 a b d \left (- A e + B d\right )}{a e - b d} - \frac{b^{2} d^{2} \left (- A e + B d\right )}{e \left (a e - b d\right )}}{- 2 A b e + B a e + B b d} \right )}}{e \left (a e - b d\right )} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a e - A b d + 2 B a d + \frac{a^{2} e^{2} \left (- A b + B a\right )}{b \left (a e - b d\right )} - \frac{2 a d e \left (- A b + B a\right )}{a e - b d} + \frac{b d^{2} \left (- A b + B a\right )}{a e - b d}}{- 2 A b e + B a e + B b d} \right )}}{b \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295831, size = 197, normalized size = 1.84 \[ \frac{B e^{\left (-1\right )}{\rm ln}\left ({\left | b x^{2} e + b d x + a x e + a d \right |}\right ){\rm sign}\left (b x + a\right )}{2 \, b} - \frac{{\left (B b d{\rm sign}\left (b x + a\right ) + B a e{\rm sign}\left (b x + a\right ) - 2 \, A b e{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{2 \, b{\left | b d - a e \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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