3.2469 \(\int \frac{A+B x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=150 \[ \frac{\sqrt{a+b x+c x^2} (B d-A e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{(-2 a B e+A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.386926, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt{a+b x+c x^2} (B d-A e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{(-2 a B e+A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.4961, size = 133, normalized size = 0.89 \[ \frac{\left (- A c d - B a e + \frac{b \left (A e + B d\right )}{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{\left (A e - B d\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.436203, size = 186, normalized size = 1.24 \[ \frac{2 \sqrt{a+x (b+c x)} (B d-A e) \sqrt{e (a e-b d)+c d^2}+(d+e x) (-2 a B e+A b e-2 A c d+b B d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )-(d+e x) \log (d+e x) (-2 a B e+A b e-2 A c d+b B d)}{2 (d+e x) \left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 1033, normalized size = 6.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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(c*x^2 + b*x + a)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 1.57623, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d^{2} - b d e + a e^{2}} \sqrt{c x^{2} + b x + a}{\left (B d - A e\right )} +{\left ({\left (B b - 2 \, A c\right )} d^{2} -{\left (2 \, B a - A b\right )} d e +{\left ({\left (B b - 2 \, A c\right )} d e -{\left (2 \, B a - A b\right )} e^{2}\right )} x\right )} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} + 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{4 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, \frac{2 \, \sqrt{-c d^{2} + b d e - a e^{2}} \sqrt{c x^{2} + b x + a}{\left (B d - A e\right )} +{\left ({\left (B b - 2 \, A c\right )} d^{2} -{\left (2 \, B a - A b\right )} d e +{\left ({\left (B b - 2 \, A c\right )} d e -{\left (2 \, B a - A b\right )} e^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{2 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(B*d - A*e) + ((B*b -
2*A*c)*d^2 - (2*B*a - A*b)*d*e + ((B*b - 2*A*c)*d*e - (2*B*a - A*b)*e^2)*x)*log(
((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*
a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 -
b*d*e + a*e^2) + 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2
*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a)
)/(e^2*x^2 + 2*d*e*x + d^2)))/((c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 +
a*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/2*(2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqr
t(c*x^2 + b*x + a)*(B*d - A*e) + ((B*b - 2*A*c)*d^2 - (2*B*a - A*b)*d*e + ((B*b
- 2*A*c)*d*e - (2*B*a - A*b)*e^2)*x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b
*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))))/
((c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt(-c*d^2 + b*d*e
- a*e^2))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (d + e x\right )^{2} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/((d + e*x)**2*sqrt(a + b*x + c*x**2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^2), x)