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

Optimal. Leaf size=140 \[ -\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 \sqrt{c}}-\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{c+d x}}{a x (a+b x)} \]

[Out]

(-2*b*Sqrt[c + d*x])/(a^2*(a + b*x)) - Sqrt[c + d*x]/(a*x*(a + b*x)) + ((4*b*c -
a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a^3*Sqrt[c]) - (Sqrt[b]*(4*b*c - 3*a*d)*A
rcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.480811, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 \sqrt{c}}-\frac{2 b \sqrt{c+d x}}{a^2 (a+b x)}-\frac{\sqrt{c+d x}}{a x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*Sqrt[c + d*x])/(a^2*(a + b*x)) - Sqrt[c + d*x]/(a*x*(a + b*x)) + ((4*b*c -
a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a^3*Sqrt[c]) - (Sqrt[b]*(4*b*c - 3*a*d)*A
rcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 53.9792, size = 124, normalized size = 0.89 \[ - \frac{\sqrt{c + d x}}{a x \left (a + b x\right )} - \frac{2 b \sqrt{c + d x}}{a^{2} \left (a + b x\right )} - \frac{\sqrt{b} \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} \sqrt{a d - b c}} - \frac{\left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x)/(a*x*(a + b*x)) - 2*b*sqrt(c + d*x)/(a**2*(a + b*x)) - sqrt(b)*(3
*a*d - 4*b*c)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a**3*sqrt(a*d - b*c))
- (a*d - 4*b*c)*atanh(sqrt(c + d*x)/sqrt(c))/(a**3*sqrt(c))

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Mathematica [A]  time = 0.344843, size = 119, normalized size = 0.85 \[ \frac{-\frac{a (a+2 b x) \sqrt{c+d x}}{x (a+b x)}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+\frac{\sqrt{b} (3 a d-4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d}}}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

(-((a*(a + 2*b*x)*Sqrt[c + d*x])/(x*(a + b*x))) + ((4*b*c - a*d)*ArcTanh[Sqrt[c
+ d*x]/Sqrt[c]])/Sqrt[c] + (Sqrt[b]*(-4*b*c + 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d
*x])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/a^3

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Maple [A]  time = 0.029, size = 167, normalized size = 1.2 \[ -{\frac{1}{{a}^{2}x}\sqrt{dx+c}}-{\frac{d}{{a}^{2}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+4\,{\frac{\sqrt{c}b}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{bd}{{a}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-3\,{\frac{bd}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{b}^{2}c}{{a}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a^2*(d*x+c)^(1/2)/x-d/a^2/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))+4/a^3*c^(1/2
)*arctanh((d*x+c)^(1/2)/c^(1/2))*b-d/a^2*b*(d*x+c)^(1/2)/(b*d*x+a*d)-3*d/a^2*b/(
(a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))+4/a^3*b^2/((a*d-b
*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289208, 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)/((b*x + a)^2*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 154.485, size = 1114, normalized size = 7.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*c*d*sqrt(c + d*x)/(2*a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x - 2*a**2*
b**2*c*d*x) - 2*b*d**2*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) + b*d**2*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*d**2*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**2*c*d*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*a**2) + b**2*c*d*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) - 2*b*d*Piecewise((atan(sqrt(c + d*x)/s
qrt(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 - c*d*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c
+ d*x))/(2*a**2) + c*d*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x))/(2
*a**2) - 2*d*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(
sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/
sqrt(c))/sqrt(c), (-c < 0) & (c > c + d*x)))/a**2 - sqrt(c + d*x)/(a**2*x) + 4*b
**2*c*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*sq
rt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/a**3 + 4*b*c*Piecewi
se((-atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt(c + d*x)/sqrt(c
))/sqrt(c), (-c < 0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-
c < 0) & (c > c + d*x)))/a**3

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GIAC/XCAS [A]  time = 0.236075, size = 224, normalized size = 1.6 \[ \frac{{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3}} - \frac{{\left (4 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c}} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b d - 2 \, \sqrt{d x + c} b c d + \sqrt{d x + c} a d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*b^2*c - 3*a*b*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c +
a*b*d)*a^3) - (4*b*c - a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) - (2*(
d*x + c)^(3/2)*b*d - 2*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*a*d^2)/(((d*x + c)^2*
b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2)