3.1765 \(\int \frac{A+B x}{(d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=212 \[ \frac{(a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}-\frac{(a+b x) (B d-A e)}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac{b (a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b (a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.377514, antiderivative size = 212, 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 \[ \frac{(a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}-\frac{(a+b x) (B d-A e)}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac{b (a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b (a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.3563, size = 192, normalized size = 0.91 \[ - \frac{b \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{b \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{e \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{3}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right )}{4 e \left (d + e x\right )^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.304916, size = 131, normalized size = 0.62 \[ \frac{(a+b x) \left (\frac{2 (A b-a B)}{(d+e x) (b d-a e)^2}+\frac{B d-A e}{e (d+e x)^2 (a e-b d)}+\frac{2 b (A b-a B) \log (a+b x)}{(b d-a e)^3}-\frac{2 b (A b-a B) \log (d+e x)}{(b d-a e)^3}\right )}{2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.024, size = 321, normalized size = 1.5 \[ -{\frac{ \left ( bx+a \right ) \left ( 2\,A\ln \left ( bx+a \right ){x}^{2}{b}^{2}{e}^{3}-2\,A\ln \left ( ex+d \right ){x}^{2}{b}^{2}{e}^{3}-2\,B\ln \left ( bx+a \right ){x}^{2}ab{e}^{3}+2\,B\ln \left ( ex+d \right ){x}^{2}ab{e}^{3}+4\,A\ln \left ( bx+a \right ) x{b}^{2}d{e}^{2}-4\,A\ln \left ( ex+d \right ) x{b}^{2}d{e}^{2}-4\,B\ln \left ( bx+a \right ) xabd{e}^{2}+4\,B\ln \left ( ex+d \right ) xabd{e}^{2}+2\,A\ln \left ( bx+a \right ){b}^{2}{d}^{2}e-2\,A\ln \left ( ex+d \right ){b}^{2}{d}^{2}e-2\,Axab{e}^{3}+2\,Ax{b}^{2}d{e}^{2}-2\,B\ln \left ( bx+a \right ) ab{d}^{2}e+2\,B\ln \left ( ex+d \right ) ab{d}^{2}e+2\,Bx{a}^{2}{e}^{3}-2\,Bxabd{e}^{2}+A{a}^{2}{e}^{3}-4\,Aabd{e}^{2}+3\,A{b}^{2}{d}^{2}e+Bd{e}^{2}{a}^{2}-B{b}^{2}{d}^{3} \right ) }{2\, \left ( ae-bd \right ) ^{3}e \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*x+a)*(2*A*ln(b*x+a)*x^2*b^2*e^3-2*A*ln(e*x+d)*x^2*b^2*e^3-2*B*ln(b*x+a)*
x^2*a*b*e^3+2*B*ln(e*x+d)*x^2*a*b*e^3+4*A*ln(b*x+a)*x*b^2*d*e^2-4*A*ln(e*x+d)*x*
b^2*d*e^2-4*B*ln(b*x+a)*x*a*b*d*e^2+4*B*ln(e*x+d)*x*a*b*d*e^2+2*A*ln(b*x+a)*b^2*
d^2*e-2*A*ln(e*x+d)*b^2*d^2*e-2*A*x*a*b*e^3+2*A*x*b^2*d*e^2-2*B*ln(b*x+a)*a*b*d^
2*e+2*B*ln(e*x+d)*a*b*d^2*e+2*B*x*a^2*e^3-2*B*x*a*b*d*e^2+A*a^2*e^3-4*A*a*b*d*e^
2+3*A*b^2*d^2*e+B*d*e^2*a^2-B*b^2*d^3)/((b*x+a)^2)^(1/2)/(a*e-b*d)^3/e/(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((B*x + A)/(sqrt((b*x + a)^2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.292473, size = 463, normalized size = 2.18 \[ -\frac{B b^{2} d^{3} - 3 \, A b^{2} d^{2} e - A a^{2} e^{3} -{\left (B a^{2} - 4 \, A a b\right )} d e^{2} + 2 \,{\left ({\left (B a b - A b^{2}\right )} d e^{2} -{\left (B a^{2} - A a b\right )} e^{3}\right )} x + 2 \,{\left ({\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e^{2} x +{\left (B a b - A b^{2}\right )} d^{2} e\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e^{2} x +{\left (B a b - A b^{2}\right )} d^{2} e\right )} \log \left (e x + d\right )}{2 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(B*b^2*d^3 - 3*A*b^2*d^2*e - A*a^2*e^3 - (B*a^2 - 4*A*a*b)*d*e^2 + 2*((B*a*
b - A*b^2)*d*e^2 - (B*a^2 - A*a*b)*e^3)*x + 2*((B*a*b - A*b^2)*e^3*x^2 + 2*(B*a*
b - A*b^2)*d*e^2*x + (B*a*b - A*b^2)*d^2*e)*log(b*x + a) - 2*((B*a*b - A*b^2)*e^
3*x^2 + 2*(B*a*b - A*b^2)*d*e^2*x + (B*a*b - A*b^2)*d^2*e)*log(e*x + d))/(b^3*d^
5*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*e^4 + (b^3*d^3*e^3 - 3*a*b^2*d
^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*x^2 + 2*(b^3*d^4*e^2 - 3*a*b^2*d^3*e^3 + 3*a^2
*b*d^2*e^4 - a^3*d*e^5)*x)

_______________________________________________________________________________________

Sympy [A]  time = 8.00227, size = 558, normalized size = 2.63 \[ - \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} e - A b^{3} d + B a^{2} b e + B a b^{2} d - \frac{a^{4} b e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b^{2} d e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{3} d^{2} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac{4 a b^{4} d^{3} e \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac{b^{5} d^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}}}{- 2 A b^{3} e + 2 B a b^{2} e} \right )}}{\left (a e - b d\right )^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} e - A b^{3} d + B a^{2} b e + B a b^{2} d + \frac{a^{4} b e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b^{2} d e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{3} d^{2} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac{4 a b^{4} d^{3} e \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac{b^{5} d^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}}}{- 2 A b^{3} e + 2 B a b^{2} e} \right )}}{\left (a e - b d\right )^{3}} - \frac{A a e^{2} - 3 A b d e + B a d e + B b d^{2} + x \left (- 2 A b e^{2} + 2 B a e^{2}\right )}{2 a^{2} d^{2} e^{3} - 4 a b d^{3} e^{2} + 2 b^{2} d^{4} e + x^{2} \left (2 a^{2} e^{5} - 4 a b d e^{4} + 2 b^{2} d^{2} e^{3}\right ) + x \left (4 a^{2} d e^{4} - 8 a b d^{2} e^{3} + 4 b^{2} d^{3} e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(-A*b + B*a)*log(x + (-A*a*b**2*e - A*b**3*d + B*a**2*b*e + B*a*b**2*d - a**4
*b*e**4*(-A*b + B*a)/(a*e - b*d)**3 + 4*a**3*b**2*d*e**3*(-A*b + B*a)/(a*e - b*d
)**3 - 6*a**2*b**3*d**2*e**2*(-A*b + B*a)/(a*e - b*d)**3 + 4*a*b**4*d**3*e*(-A*b
+ B*a)/(a*e - b*d)**3 - b**5*d**4*(-A*b + B*a)/(a*e - b*d)**3)/(-2*A*b**3*e + 2
*B*a*b**2*e))/(a*e - b*d)**3 + b*(-A*b + B*a)*log(x + (-A*a*b**2*e - A*b**3*d +
B*a**2*b*e + B*a*b**2*d + a**4*b*e**4*(-A*b + B*a)/(a*e - b*d)**3 - 4*a**3*b**2*
d*e**3*(-A*b + B*a)/(a*e - b*d)**3 + 6*a**2*b**3*d**2*e**2*(-A*b + B*a)/(a*e - b
*d)**3 - 4*a*b**4*d**3*e*(-A*b + B*a)/(a*e - b*d)**3 + b**5*d**4*(-A*b + B*a)/(a
*e - b*d)**3)/(-2*A*b**3*e + 2*B*a*b**2*e))/(a*e - b*d)**3 - (A*a*e**2 - 3*A*b*d
*e + B*a*d*e + B*b*d**2 + x*(-2*A*b*e**2 + 2*B*a*e**2))/(2*a**2*d**2*e**3 - 4*a*
b*d**3*e**2 + 2*b**2*d**4*e + x**2*(2*a**2*e**5 - 4*a*b*d*e**4 + 2*b**2*d**2*e**
3) + x*(4*a**2*d*e**4 - 8*a*b*d**2*e**3 + 4*b**2*d**3*e**2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288097, size = 413, normalized size = 1.95 \[ -\frac{{\left (B a b^{2}{\rm sign}\left (b x + a\right ) - A b^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac{{\left (B a b e{\rm sign}\left (b x + a\right ) - A b^{2} e{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac{{\left (B b^{2} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, A b^{2} d^{2} e{\rm sign}\left (b x + a\right ) - B a^{2} d e^{2}{\rm sign}\left (b x + a\right ) + 4 \, A a b d e^{2}{\rm sign}\left (b x + a\right ) - A a^{2} e^{3}{\rm sign}\left (b x + a\right ) + 2 \,{\left (B a b d e^{2}{\rm sign}\left (b x + a\right ) - A b^{2} d e^{2}{\rm sign}\left (b x + a\right ) - B a^{2} e^{3}{\rm sign}\left (b x + a\right ) + A a b e^{3}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-1\right )}}{2 \,{\left (b d - a e\right )}^{3}{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(B*a*b^2*sign(b*x + a) - A*b^3*sign(b*x + a))*ln(abs(b*x + a))/(b^4*d^3 - 3*a*b
^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3) + (B*a*b*e*sign(b*x + a) - A*b^2*e*sign(
b*x + a))*ln(abs(x*e + d))/(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^
4) - 1/2*(B*b^2*d^3*sign(b*x + a) - 3*A*b^2*d^2*e*sign(b*x + a) - B*a^2*d*e^2*si
gn(b*x + a) + 4*A*a*b*d*e^2*sign(b*x + a) - A*a^2*e^3*sign(b*x + a) + 2*(B*a*b*d
*e^2*sign(b*x + a) - A*b^2*d*e^2*sign(b*x + a) - B*a^2*e^3*sign(b*x + a) + A*a*b
*e^3*sign(b*x + a))*x)*e^(-1)/((b*d - a*e)^3*(x*e + d)^2)