3.1811 \(\int \frac{(A+B x) \sqrt{d+e x}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=140 \[ -\frac{(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{b d-a e}}+\frac{\sqrt{d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

[Out]

((2*b*B*d + A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(b^2*(b*d - a*e)) - ((A*b - a*B)*(d
+ e*x)^(3/2))/(b*(b*d - a*e)*(a + b*x)) - ((2*b*B*d + A*b*e - 3*a*B*e)*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*Sqrt[b*d - a*e])

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Rubi [A]  time = 0.271377, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{b d-a e}}+\frac{\sqrt{d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*B*d + A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(b^2*(b*d - a*e)) - ((A*b - a*B)*(d
+ e*x)^(3/2))/(b*(b*d - a*e)*(a + b*x)) - ((2*b*B*d + A*b*e - 3*a*B*e)*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*Sqrt[b*d - a*e])

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Rubi in Sympy [A]  time = 54.8412, size = 124, normalized size = 0.89 \[ \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\sqrt{d + e x} \left (A b e - 3 B a e + 2 B b d\right )}{b^{2} \left (a e - b d\right )} + \frac{\left (A b e - 3 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{5}{2}} \sqrt{a e - b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.12541, size = 99, normalized size = 0.71 \[ \sqrt{d+e x} \left (\frac{a B-A b}{b^2 (a+b x)}+\frac{2 B}{b^2}\right )-\frac{(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*((2*B)/b^2 + (-(A*b) + a*B)/(b^2*(a + b*x))) - ((2*b*B*d + A*b*e -
3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*Sqrt[b*d -
a*e])

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Maple [A]  time = 0.026, size = 186, normalized size = 1.3 \[ 2\,{\frac{B\sqrt{ex+d}}{{b}^{2}}}-{\frac{Ae}{b \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{aBe}{{b}^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{Ae}{b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}-3\,{\frac{aBe}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+2\,{\frac{Bd}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295467, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} d - a b e}{\left (2 \, B b x + 3 \, B a - A b\right )} \sqrt{e x + d} +{\left (2 \, B a b d -{\left (3 \, B a^{2} - A a b\right )} e +{\left (2 \, B b^{2} d -{\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{2 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b^{2} d - a b e}}, \frac{\sqrt{-b^{2} d + a b e}{\left (2 \, B b x + 3 \, B a - A b\right )} \sqrt{e x + d} -{\left (2 \, B a b d -{\left (3 \, B a^{2} - A a b\right )} e +{\left (2 \, B b^{2} d -{\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{{\left (b^{3} x + a b^{2}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(b^2*d - a*b*e)*(2*B*b*x + 3*B*a - A*b)*sqrt(e*x + d) + (2*B*a*b*d -
(3*B*a^2 - A*a*b)*e + (2*B*b^2*d - (3*B*a*b - A*b^2)*e)*x)*log((sqrt(b^2*d - a*
b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((b^3*
x + a*b^2)*sqrt(b^2*d - a*b*e)), (sqrt(-b^2*d + a*b*e)*(2*B*b*x + 3*B*a - A*b)*s
qrt(e*x + d) - (2*B*a*b*d - (3*B*a^2 - A*a*b)*e + (2*B*b^2*d - (3*B*a*b - A*b^2)
*e)*x)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((b^3*x + a*b^
2)*sqrt(-b^2*d + a*b*e))]

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Sympy [A]  time = 28.5925, size = 1559, normalized size = 11.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b*
*3*d*e*x) + A*a*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e
- b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e
- b*d)**3)) + sqrt(d + e*x))/(2*b) - A*a*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a*
*2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b*
*2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - A*d*e*sqrt(-1/(b*(a
*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b
*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + A
*d*e*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a
*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqr
t(d + e*x))/2 + 2*A*d*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**2*x -
2*b**2*d*e*x) + 2*A*e*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e
/b - d)), a*e/b - d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b
+ d)), (a*e/b - d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a
*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b +
2*B*a**2*e**2*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**3*d*e + 2*a*b**3*e**2*x -
2*b**4*d*e*x) - B*a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/
(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/
(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) + B*a**2*e**2*sqrt(-1/(b*(a*e - b*
d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e -
b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) - 2*
B*a*d*e*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d
*e*x) + B*a*d*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d
)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d
)**3)) + sqrt(d + e*x))/(2*b) - B*a*d*e*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**
2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**
2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - 4*B*a*e*Piecewise((atan(s
qrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/b - d > 0), (-acoth(sqrt(
d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x > -a
*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b
- d < 0) & (d + e*x < -a*e/b + d)))/b**2 + 2*B*d*Piecewise((atan(sqrt(d + e*x)/
sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/b - d > 0), (-acoth(sqrt(d + e*x)/sqrt
(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x > -a*e/b + d)), (
-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (
d + e*x < -a*e/b + d)))/b + 2*B*sqrt(d + e*x)/b**2

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GIAC/XCAS [A]  time = 0.289141, size = 170, normalized size = 1.21 \[ \frac{2 \, \sqrt{x e + d} B}{b^{2}} + \frac{{\left (2 \, B b d - 3 \, B a e + A b e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{\sqrt{x e + d} B a e - \sqrt{x e + d} A b e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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