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

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {247, 190, 43} \[ \frac {2 a}{b^2 d \left (a+b \sqrt {c+d x}\right )}+\frac {2 \log \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]

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)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \sqrt {c+d x}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, 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)

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 75, normalized size = 1.60 \[ \frac {2 \, {\left (\sqrt {d x + c} a b - a^{2} + {\left (b^{2} d x + b^{2} c - a^{2}\right )} \log \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2} x + {\left (b^{4} c - a^{2} b^{2}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.34, size = 44, normalized size = 0.94 \[ \frac {2 \, \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{2} d} + \frac {2 \, a}{{\left (\sqrt {d x + c} b + a\right )} b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.02, size = 142, normalized size = 3.02 \[ -\frac {2 a^{2}}{\left (b^{2} d x +b^{2} c -a^{2}\right ) b^{2} d}+\frac {a}{\left (-a +\sqrt {d x +c}\, b \right ) b^{2} d}+\frac {a}{\left (a +\sqrt {d x +c}\, b \right ) b^{2} d}-\frac {\ln \left (-a +\sqrt {d x +c}\, b \right )}{b^{2} d}+\frac {\ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d}+\frac {\ln \left (b^{2} d x +b^{2} c -a^{2}\right )}{b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 43, normalized size = 0.91 \[ \frac {2 \, {\left (\frac {a}{\sqrt {d x + c} b^{3} + a b^{2}} + \frac {\log \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 43, normalized size = 0.91 \[ \frac {2\,\ln \left (a+b\,\sqrt {c+d\,x}\right )}{b^2\,d}+\frac {2\,a}{b^2\,\left (a\,d+b\,d\,\sqrt {c+d\,x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*(c + d*x)^(1/2))^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.09, 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))

________________________________________________________________________________________