3.1715 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=151 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) (d+e x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.264383, antiderivative size = 151, 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 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{2 e^3 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) (d+e x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^3,x]

[Out]

-((b*d - a*e)*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^3*(a + b*x)*(d + e
*x)^2) + ((2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x
)*(d + e*x)) + (b*B*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 = 40.6191, size = 133, normalized size = 0.88 \[ \frac{B b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{3} \left (a + b x\right )} - \frac{B \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{3} \left (a + b x\right ) \left (d + e x\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 e \left (d + e x\right )^{2} \left (a e - b d\right )} \]

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)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0897188, size = 89, normalized size = 0.59 \[ -\frac{\sqrt{(a+b x)^2} \left (a e (A e+B (d+2 e x))+b (A e (d+2 e x)-B d (3 d+4 e x))-2 b B (d+e x)^2 \log (d+e x)\right )}{2 e^3 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.02, size = 117, normalized size = 0.8 \[ -{\frac{{\it csgn} \left ( bx+a \right ) \left ( -2\,B\ln \left ( bex+bd \right ){x}^{2}b{e}^{2}-4\,B\ln \left ( bex+bd \right ) xbde+2\,Ab{e}^{2}x-2\,B\ln \left ( bex+bd \right ) b{d}^{2}+2\,aB{e}^{2}x-4\,Bbdex+A{e}^{2}a+Abde+aBde-3\,Bb{d}^{2} \right ) }{2\,{e}^{3} \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^3,x)

[Out]

-1/2*csgn(b*x+a)*(-2*B*ln(b*e*x+b*d)*x^2*b*e^2-4*B*ln(b*e*x+b*d)*x*b*d*e+2*A*b*e
^2*x-2*B*ln(b*e*x+b*d)*b*d^2+2*a*B*e^2*x-4*B*b*d*e*x+A*e^2*a+A*b*d*e+a*B*d*e-3*B
*b*d^2)/e^3/(e*x+d)^2

_______________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.296279, size = 142, normalized size = 0.94 \[ \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b e^{2} x^{2} + 2 \, B b d e x + B b d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/2*(3*B*b*d^2 - A*a*e^2 - (B*a + A*b)*d*e + 2*(2*B*b*d*e - (B*a + A*b)*e^2)*x +
2*(B*b*e^2*x^2 + 2*B*b*d*e*x + B*b*d^2)*log(e*x + d))/(e^5*x^2 + 2*d*e^4*x + d^
2*e^3)

_______________________________________________________________________________________

Sympy [A]  time = 4.39359, size = 94, normalized size = 0.62 \[ \frac{B b \log{\left (d + e x \right )}}{e^{3}} - \frac{A a e^{2} + A b d e + B a d e - 3 B b d^{2} + x \left (2 A b e^{2} + 2 B a e^{2} - 4 B b d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**3,x)

[Out]

B*b*log(d + e*x)/e**3 - (A*a*e**2 + A*b*d*e + B*a*d*e - 3*B*b*d**2 + x*(2*A*b*e*
*2 + 2*B*a*e**2 - 4*B*b*d*e))/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.277956, size = 171, normalized size = 1.13 \[ B b e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ){\rm sign}\left (b x + a\right ) + \frac{{\left (2 \,{\left (2 \, B b d{\rm sign}\left (b x + a\right ) - B a e{\rm sign}\left (b x + a\right ) - A b e{\rm sign}\left (b x + a\right )\right )} x +{\left (3 \, B b d^{2}{\rm sign}\left (b x + a\right ) - B a d e{\rm sign}\left (b x + a\right ) - A b d e{\rm sign}\left (b x + a\right ) - A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^3,x, algorithm="giac")

[Out]

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