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

Optimal. Leaf size=47 \[ \frac{2 a}{b^2 d \left (a+b \sqrt{c+d x}\right )}+\frac{2 \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

[Out]

(2*a)/(b^2*d*(a + b*Sqrt[c + d*x])) + (2*Log[a + b*Sqrt[c + d*x]])/(b^2*d)

_______________________________________________________________________________________

Rubi [A]  time = 0.0643198, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 a}{b^2 d \left (a+b \sqrt{c+d x}\right )}+\frac{2 \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*a)/(b^2*d*(a + b*Sqrt[c + d*x])) + (2*Log[a + b*Sqrt[c + d*x]])/(b^2*d)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.28379, size = 39, normalized size = 0.83 \[ \frac{2 a}{b^{2} d \left (a + b \sqrt{c + d x}\right )} + \frac{2 \log{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a/(b**2*d*(a + b*sqrt(c + d*x))) + 2*log(a + b*sqrt(c + d*x))/(b**2*d)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0294848, size = 40, normalized size = 0.85 \[ \frac{2 \left (\frac{a}{a+b \sqrt{c+d x}}+\log \left (a+b \sqrt{c+d x}\right )\right )}{b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a/(a + b*Sqrt[c + d*x]) + Log[a + b*Sqrt[c + d*x]]))/(b^2*d)

_______________________________________________________________________________________

Maple [B]  time = 0.027, size = 142, normalized size = 3. \[ -2\,{\frac{{a}^{2}}{ \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ){b}^{2}d}}+{\frac{\ln \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ) }{{b}^{2}d}}+{\frac{a}{{b}^{2}d} \left ( a+b\sqrt{dx+c} \right ) ^{-1}}+{\frac{1}{{b}^{2}d}\ln \left ( a+b\sqrt{dx+c} \right ) }+{\frac{a}{{b}^{2}d} \left ( -a+b\sqrt{dx+c} \right ) ^{-1}}-{\frac{1}{{b}^{2}d}\ln \left ( -a+b\sqrt{dx+c} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(d*x+c)^(1/2))^2,x)

[Out]

-2*a^2/(b^2*d*x+b^2*c-a^2)/b^2/d+ln(b^2*d*x+b^2*c-a^2)/b^2/d+a/b^2/d/(a+b*(d*x+c
)^(1/2))+ln(a+b*(d*x+c)^(1/2))/b^2/d+a/b^2/d/(-a+b*(d*x+c)^(1/2))-1/b^2/d*ln(-a+
b*(d*x+c)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.701641, size = 58, normalized size = 1.23 \[ \frac{2 \,{\left (\frac{a}{\sqrt{d x + c} b^{3} + a b^{2}} + \frac{\log \left (\sqrt{d x + c} b + a\right )}{b^{2}}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a/(sqrt(d*x + c)*b^3 + a*b^2) + log(sqrt(d*x + c)*b + a)/b^2)/d

_______________________________________________________________________________________

Fricas [A]  time = 0.276779, size = 66, normalized size = 1.4 \[ \frac{2 \,{\left ({\left (\sqrt{d x + c} b + a\right )} \log \left (\sqrt{d x + c} b + a\right ) + a\right )}}{\sqrt{d x + c} b^{3} d + a b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((sqrt(d*x + c)*b + a)*log(sqrt(d*x + c)*b + a) + a)/(sqrt(d*x + c)*b^3*d + a*
b^2*d)

_______________________________________________________________________________________

Sympy [A]  time = 3.43724, size = 124, normalized size = 2.64 \[ \begin{cases} \frac{x}{a^{2}} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x}{\left (a + b \sqrt{c}\right )^{2}} & \text{for}\: d = 0 \\\frac{2 a \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{a b^{2} d + b^{3} d \sqrt{c + d x}} + \frac{2 a}{a b^{2} d + b^{3} d \sqrt{c + d x}} + \frac{2 b \sqrt{c + d x} \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{a b^{2} d + b^{3} d \sqrt{c + d x}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/a**2, Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x/(a + b*sqrt(c))**2, Eq(
d, 0)), (2*a*log(a/b + sqrt(c + d*x))/(a*b**2*d + b**3*d*sqrt(c + d*x)) + 2*a/(a
*b**2*d + b**3*d*sqrt(c + d*x)) + 2*b*sqrt(c + d*x)*log(a/b + sqrt(c + d*x))/(a*
b**2*d + b**3*d*sqrt(c + d*x)), True))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef