3.1420 \(\int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=76 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x}}{(a+b x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0851603, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x}}{(a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.5234, size = 61, normalized size = 0.8 \[ \frac{\sqrt{c + d x}}{\left (a + b x\right ) \left (a d - b c\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.126425, size = 77, normalized size = 1.01 \[ \frac{\frac{\sqrt{c+d x}}{a+b x}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{a d-b c} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]/(a + b*x) - (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/
(Sqrt[b]*Sqrt[b*c - a*d]))/(-(b*c) + a*d)

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Maple [A]  time = 0.012, size = 77, normalized size = 1. \[{\frac{d}{ \left ( ad-bc \right ) \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{d}{ad-bc}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(d*x+c)^(1/2)/(a*d-b*c)/(b*d*x+a*d)+d/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan((d*
x+c)^(1/2)*b/((a*d-b*c)*b)^(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(1/((b*x + a)^2*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233318, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d x + a d\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right ) + 2 \, \sqrt{b^{2} c - a b d} \sqrt{d x + c}}{2 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (b d x + a d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right ) - \sqrt{-b^{2} c + a b d} \sqrt{d x + c}}{{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*sqrt(d*x + c)),x, algorithm="fricas")

[Out]

[-1/2*((b*d*x + a*d)*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)) + 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/((a*b*c
- a^2*d + (b^2*c - a*b*d)*x)*sqrt(b^2*c - a*b*d)), ((b*d*x + a*d)*arctan(-(b*c
- a*d)/(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c))) - sqrt(-b^2*c + a*b*d)*sqrt(d*x + c
))/((a*b*c - a^2*d + (b^2*c - a*b*d)*x)*sqrt(-b^2*c + a*b*d))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.225117, size = 117, normalized size = 1.54 \[ -\frac{d \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} - \frac{\sqrt{d x + c} d}{{\left ({\left (d x + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*sqrt(d*x + c)),x, algorithm="giac")

[Out]

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