3.1714 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x) (-a B e-A b e+2 b B d)}{e^3 (a+b x)}+\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]

[Out]

(b*B*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^2*(a + b*x)) - ((b*d - a*e)*(B*d - A*e)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x)*(d + e*x)) - ((2*b*B*d - A*b*e -
a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^3*(a + b*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.269427, antiderivative size = 144, 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 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x) (-a B e-A b e+2 b B d)}{e^3 (a+b x)}+\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(b*B*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^2*(a + b*x)) - ((b*d - a*e)*(B*d - A*e)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x)*(d + e*x)) - ((2*b*B*d - A*b*e -
a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^3*(a + b*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.9614, size = 150, normalized size = 1.04 \[ - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e \left (d + e x\right ) \left (a e - b d\right )} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right )}{e^{2} \left (a e - b d\right )} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0830413, size = 96, normalized size = 0.67 \[ \frac{\sqrt{(a+b x)^2} \left (-(d+e x) \log (d+e x) (-a B e-A b e+2 b B d)+a e (B d-A e)+b \left (A d e-B d^2+B d e x+B e^2 x^2\right )\right )}{e^3 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(a*e*(B*d - A*e) + b*(-(B*d^2) + A*d*e + B*d*e*x + B*e^2*x^2)
- (2*b*B*d - A*b*e - a*B*e)*(d + e*x)*Log[d + e*x]))/(e^3*(a + b*x)*(d + e*x))

_______________________________________________________________________________________

Maple [C]  time = 0.021, size = 158, normalized size = 1.1 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( A\ln \left ( bex+bd \right ) xb{e}^{2}+B\ln \left ( bex+bd \right ) xa{e}^{2}-2\,B\ln \left ( bex+bd \right ) xbde+B{x}^{2}b{e}^{2}+A\ln \left ( bex+bd \right ) bde+B\ln \left ( bex+bd \right ) ade-2\,B\ln \left ( bex+bd \right ) b{d}^{2}+aB{e}^{2}x+Bbdex-A{e}^{2}a+Abde+2\,aBde-Bb{d}^{2} \right ) }{{e}^{3} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

csgn(b*x+a)*(A*ln(b*e*x+b*d)*x*b*e^2+B*ln(b*e*x+b*d)*x*a*e^2-2*B*ln(b*e*x+b*d)*x
*b*d*e+B*x^2*b*e^2+A*ln(b*e*x+b*d)*b*d*e+B*ln(b*e*x+b*d)*a*d*e-2*B*ln(b*e*x+b*d)
*b*d^2+a*B*e^2*x+B*b*d*e*x-A*e^2*a+A*b*d*e+2*a*B*d*e-B*b*d^2)/e^3/(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(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.272444, size = 138, normalized size = 0.96 \[ \frac{B b e^{2} x^{2} + B b d e x - B b d^{2} - A a e^{2} +{\left (B a + A b\right )} d e -{\left (2 \, B b d^{2} -{\left (B a + A b\right )} d e +{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b*e^2*x^2 + B*b*d*e*x - B*b*d^2 - A*a*e^2 + (B*a + A*b)*d*e - (2*B*b*d^2 - (B
*a + A*b)*d*e + (2*B*b*d*e - (B*a + A*b)*e^2)*x)*log(e*x + d))/(e^4*x + d*e^3)

_______________________________________________________________________________________

Sympy [A]  time = 2.76501, size = 71, normalized size = 0.49 \[ \frac{B b x}{e^{2}} + \frac{- A a e^{2} + A b d e + B a d e - B b d^{2}}{d e^{3} + e^{4} x} + \frac{\left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284098, size = 166, normalized size = 1.15 \[ B b x e^{\left (-2\right )}{\rm sign}\left (b x + a\right ) -{\left (2 \, B b d{\rm sign}\left (b x + a\right ) - B a e{\rm sign}\left (b x + a\right ) - A b e{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (B b d^{2}{\rm sign}\left (b x + a\right ) - B a d e{\rm sign}\left (b x + a\right ) - A b d e{\rm sign}\left (b x + a\right ) + A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{x e + d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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