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

Optimal. Leaf size=127 \[ \frac{\left (2 b^2 c e-a d (a f+b e)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x} (b e-a f)}{a b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.380165, antiderivative size = 127, 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 \[ \frac{\left (2 b^2 c e-a d (a f+b e)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{\sqrt{c+d x} (b e-a f)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.7122, size = 114, normalized size = 0.9 \[ - \frac{\sqrt{c + d x} \left (a f - b e\right )}{a b \left (a + b x\right )} - \frac{2 \sqrt{c} e \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{2}} + \frac{2 \left (\frac{a d \left (a f + b e\right )}{2} - b^{2} c e\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{2} b^{\frac{3}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.26824, size = 124, normalized size = 0.98 \[ \frac{-\frac{\left (a^2 d f+a b d e-2 b^2 c e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} \sqrt{b c-a d}}+\frac{a \sqrt{c+d x} (b e-a f)}{b (a+b x)}-2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 192, normalized size = 1.5 \[ -2\,{\frac{e\sqrt{c}}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{df}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{de}{a \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{df}{b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{de}{a}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-2\,{\frac{ceb}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.337477, 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(d*x + c)*(f*x + e)/((b*x + a)^2*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 81.722, size = 1467, normalized size = 11.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217718, size = 192, normalized size = 1.51 \[ \frac{2 \, c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{a^{2} \sqrt{-c}} + \frac{{\left (a^{2} d f - 2 \, b^{2} c e + a b d e\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b} - \frac{\sqrt{d x + c} a d f - \sqrt{d x + c} b d e}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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