3.20 \(\int \frac{\sqrt{a+b x} (e+f x)}{x (c+d x)^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\left (2 a d^2 e-b c (c f+d e)\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{c^2 d^{3/2} \sqrt{b c-a d}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c^2}+\frac{\sqrt{a+b x} (d e-c f)}{c d (c+d x)} \]

[Out]

((d*e - c*f)*Sqrt[a + b*x])/(c*d*(c + d*x)) - ((2*a*d^2*e - b*c*(d*e + c*f))*Arc
Tan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c^2*d^(3/2)*Sqrt[b*c - a*d]) - (2
*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^2

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Rubi [A]  time = 0.389999, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (2 a d^2 e-b c (c f+d e)\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{c^2 d^{3/2} \sqrt{b c-a d}}-\frac{2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c^2}+\frac{\sqrt{a+b x} (d e-c f)}{c d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

((d*e - c*f)*Sqrt[a + b*x])/(c*d*(c + d*x)) - ((2*a*d^2*e - b*c*(d*e + c*f))*Arc
Tan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c^2*d^(3/2)*Sqrt[b*c - a*d]) - (2
*Sqrt[a]*e*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c^2

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Rubi in Sympy [A]  time = 40.9232, size = 114, normalized size = 0.89 \[ - \frac{2 \sqrt{a} e \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{c^{2}} - \frac{\sqrt{a + b x} \left (c f - d e\right )}{c d \left (c + d x\right )} + \frac{2 \left (a d^{2} e - \frac{b c \left (c f + d e\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}}{c^{2} d^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.374156, size = 123, normalized size = 0.96 \[ \frac{-\frac{\left (b c (c f+d e)-2 a d^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )}{d^{3/2} \sqrt{a d-b c}}+\frac{c \sqrt{a+b x} (d e-c f)}{d (c+d x)}-2 \sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 137, normalized size = 1.1 \[ 2\,b \left ( -{\frac{e\sqrt{a}}{b{c}^{2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }-{\frac{1}{b{c}^{2}} \left ( 1/2\,{\frac{bc \left ( cf-de \right ) \sqrt{bx+a}}{d \left ( \left ( bx+a \right ) d-ad+bc \right ) }}-1/2\,{\frac{2\,a{d}^{2}e-b{c}^{2}f-bcde}{d\sqrt{ \left ( ad-bc \right ) d}}{\it Artanh} \left ({\frac{\sqrt{bx+a}d}{\sqrt{ \left ( ad-bc \right ) d}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(-1/b*e*a^(1/2)/c^2*arctanh((b*x+a)^(1/2)/a^(1/2))-1/c^2/b*(1/2*b*c*(c*f-d*e
)/d*(b*x+a)^(1/2)/((b*x+a)*d-a*d+b*c)-1/2*(2*a*d^2*e-b*c^2*f-b*c*d*e)/d/((a*d-b*
c)*d)^(1/2)*arctanh((b*x+a)^(1/2)*d/((a*d-b*c)*d)^(1/2))))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.341979, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*(d^2*e*x + c*d*e)*sqrt(-b*c*d + a*d^2)*sqrt(a)*log((b*x - 2*sqrt(b*x + a
)*sqrt(a) + 2*a)/x) + 2*sqrt(-b*c*d + a*d^2)*(c*d*e - c^2*f)*sqrt(b*x + a) + (b*
c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*log((sq
rt(-b*c*d + a*d^2)*(b*d*x - b*c + 2*a*d) + 2*(b*c*d - a*d^2)*sqrt(b*x + a))/(d*x
+ c)))/((c^2*d^2*x + c^3*d)*sqrt(-b*c*d + a*d^2)), -1/2*(4*(d^2*e*x + c*d*e)*sq
rt(-b*c*d + a*d^2)*sqrt(-a)*arctan(sqrt(b*x + a)/sqrt(-a)) - 2*sqrt(-b*c*d + a*d
^2)*(c*d*e - c^2*f)*sqrt(b*x + a) - (b*c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*
d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*log((sqrt(-b*c*d + a*d^2)*(b*d*x - b*c + 2*a*d)
+ 2*(b*c*d - a*d^2)*sqrt(b*x + a))/(d*x + c)))/((c^2*d^2*x + c^3*d)*sqrt(-b*c*d
+ a*d^2)), ((d^2*e*x + c*d*e)*sqrt(b*c*d - a*d^2)*sqrt(a)*log((b*x - 2*sqrt(b*x
+ a)*sqrt(a) + 2*a)/x) + sqrt(b*c*d - a*d^2)*(c*d*e - c^2*f)*sqrt(b*x + a) + (b*
c^3*f + (b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*arctan(
-(b*c - a*d)/(sqrt(b*c*d - a*d^2)*sqrt(b*x + a))))/((c^2*d^2*x + c^3*d)*sqrt(b*c
*d - a*d^2)), -(2*(d^2*e*x + c*d*e)*sqrt(b*c*d - a*d^2)*sqrt(-a)*arctan(sqrt(b*x
+ a)/sqrt(-a)) - sqrt(b*c*d - a*d^2)*(c*d*e - c^2*f)*sqrt(b*x + a) - (b*c^3*f +
(b*c^2*d - 2*a*c*d^2)*e + (b*c^2*d*f + (b*c*d^2 - 2*a*d^3)*e)*x)*arctan(-(b*c -
a*d)/(sqrt(b*c*d - a*d^2)*sqrt(b*x + a))))/((c^2*d^2*x + c^3*d)*sqrt(b*c*d - a*
d^2))]

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Sympy [A]  time = 76.5002, size = 1409, normalized size = 11.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*b*d*e*sqrt(a + b*x)/(2*a*b*c**2*d + 2*a*b*c*d**2*x - 2*b**2*c**3 - 2*b**2*c*
*2*d*x) - a*b*f*sqrt(1/(d*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(1/(d*(a*d - b*c)*
*3)) + 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b**2*c**2*sqrt(1/(d*(a*d - b*c)**3
)) + sqrt(a + b*x))/2 + a*b*f*sqrt(1/(d*(a*d - b*c)**3))*log(a**2*d**2*sqrt(1/(d
*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d*(
a*d - b*c)**3)) + sqrt(a + b*x))/2 - 2*a*b*f*sqrt(a + b*x)/(2*a*b*c*d + 2*a*b*d*
*2*x - 2*b**2*c**2 - 2*b**2*c*d*x) + a*b*d*e*sqrt(1/(d*(a*d - b*c)**3))*log(-a**
2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b**2*
c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/(2*c) - a*b*d*e*sqrt(1/(d*(a*d
- b*c)**3))*log(a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(1/(d*(a*d
- b*c)**3)) + b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/(2*c) + 2*a*
d*e*Piecewise((-atan(sqrt(a + b*x)/sqrt(-a + b*c/d))/(d*sqrt(-a + b*c/d)), -a +
b*c/d > 0), (acoth(sqrt(a + b*x)/sqrt(a - b*c/d))/(d*sqrt(a - b*c/d)), (-a + b*c
/d < 0) & (a + b*x > a - b*c/d)), (atanh(sqrt(a + b*x)/sqrt(a - b*c/d))/(d*sqrt(
a - b*c/d)), (-a + b*c/d < 0) & (a + b*x < a - b*c/d)))/c**2 - 2*a*e*Piecewise((
-atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(a + b*x)/sqrt(a))/s
qrt(a), (-a < 0) & (a < a + b*x)), (atanh(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a <
0) & (a > a + b*x)))/c**2 + 2*b**2*c*f*sqrt(a + b*x)/(2*a*b*c*d**2 + 2*a*b*d**3*
x - 2*b**2*c**2*d - 2*b**2*c*d**2*x) + b**2*c*f*sqrt(1/(d*(a*d - b*c)**3))*log(-
a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b*
*2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/(2*d) - b**2*c*f*sqrt(1/(d*(
a*d - b*c)**3))*log(a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(1/(d*(
a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt(a + b*x))/(2*d) -
b**2*e*sqrt(1/(d*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(1/(d*(a*d - b*c)**3)) + 2*
a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) - b**2*c**2*sqrt(1/(d*(a*d - b*c)**3)) + sqrt
(a + b*x))/2 + b**2*e*sqrt(1/(d*(a*d - b*c)**3))*log(a**2*d**2*sqrt(1/(d*(a*d -
b*c)**3)) - 2*a*b*c*d*sqrt(1/(d*(a*d - b*c)**3)) + b**2*c**2*sqrt(1/(d*(a*d - b*
c)**3)) + sqrt(a + b*x))/2 - 2*b**2*e*sqrt(a + b*x)/(2*a*b*c*d + 2*a*b*d**2*x -
2*b**2*c**2 - 2*b**2*c*d*x) - 2*b*f*Piecewise((-atan(sqrt(a + b*x)/sqrt(-a + b*c
/d))/(d*sqrt(-a + b*c/d)), -a + b*c/d > 0), (acoth(sqrt(a + b*x)/sqrt(a - b*c/d)
)/(d*sqrt(a - b*c/d)), (-a + b*c/d < 0) & (a + b*x > a - b*c/d)), (atanh(sqrt(a
+ b*x)/sqrt(a - b*c/d))/(d*sqrt(a - b*c/d)), (-a + b*c/d < 0) & (a + b*x < a - b
*c/d)))/d

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GIAC/XCAS [A]  time = 0.221806, size = 192, normalized size = 1.5 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a} c^{2}} + \frac{{\left (b c^{2} f + b c d e - 2 \, a d^{2} e\right )} \arctan \left (\frac{\sqrt{b x + a} d}{\sqrt{b c d - a d^{2}}}\right )}{\sqrt{b c d - a d^{2}} c^{2} d} - \frac{\sqrt{b x + a} b c f - \sqrt{b x + a} b d e}{{\left (b c +{\left (b x + a\right )} d - a d\right )} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*arctan(sqrt(b*x + a)/sqrt(-a))*e/(sqrt(-a)*c^2) + (b*c^2*f + b*c*d*e - 2*a*d
^2*e)*arctan(sqrt(b*x + a)*d/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^2*d) -
(sqrt(b*x + a)*b*c*f - sqrt(b*x + a)*b*d*e)/((b*c + (b*x + a)*d - a*d)*c*d)