3.1713 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{d+e x} \, dx\)

Optimal. Leaf size=132 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) \log (d+e x)}{e^3 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{e^2 (a+b x)}+\frac{B (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b e} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.241132, antiderivative size = 132, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) \log (d+e x)}{e^3 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{e^2 (a+b x)}+\frac{B (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(B*d - A*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^2*(a + b*x))) + (B*(a + b*x
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b*e) + ((b*d - a*e)*(B*d - A*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 = 24.7232, size = 114, normalized size = 0.86 \[ \frac{B \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 b e} + \frac{\left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2}} + \frac{\left (A e - B d\right ) \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{3} \left (a + b x\right )} \]

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),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0680754, size = 74, normalized size = 0.56 \[ \frac{\sqrt{(a+b x)^2} (e x (2 a B e+b (2 A e-2 B d+B e x))+2 (b d-a e) (B d-A e) \log (d+e x))}{2 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.022, size = 146, normalized size = 1.1 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( B{x}^{2}{b}^{2}{e}^{2}+2\,A\ln \left ( bex+bd \right ) ab{e}^{2}-2\,A\ln \left ( bex+bd \right ){b}^{2}de+2\,Ax{b}^{2}{e}^{2}-2\,B\ln \left ( bex+bd \right ) abde+2\,B\ln \left ( bex+bd \right ){b}^{2}{d}^{2}+2\,Bxab{e}^{2}-2\,Bx{b}^{2}de+2\,Aab{e}^{2}+{a}^{2}B{e}^{2}-2\,Bdabe \right ) }{2\,b{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.272014, size = 92, normalized size = 0.7 \[ \frac{B b e^{2} x^{2} - 2 \,{\left (B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.85187, size = 53, normalized size = 0.4 \[ \frac{B b x^{2}}{2 e} + \frac{x \left (A b e + B a e - B b d\right )}{e^{2}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278832, size = 161, normalized size = 1.22 \[{\left (B b d^{2}{\rm sign}\left (b x + a\right ) - B a d e{\rm sign}\left (b x + a\right ) - A b d e{\rm sign}\left (b x + a\right ) + A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b x^{2} e{\rm sign}\left (b x + a\right ) - 2 \, B b d x{\rm sign}\left (b x + a\right ) + 2 \, B a x e{\rm sign}\left (b x + a\right ) + 2 \, A b x e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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