3.28.64 \(\int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2764]

3.28.64.1 Optimal result

Integrand size = 31, antiderivative size = 261 \[ \int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {1}{c (d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {a \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {a \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \]

output

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

3.28.64.2 Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {1}{c (d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {a \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {a \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \]

input

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

output

-(1/(c*(d + c*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) - (a*ArcTan[(Sqrt[c]*Sq 
rt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt 
[c]*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (a*ArcT 
an[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2 
*d^2]]])/(Sqrt[c]*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^ 
2]])
 

3.28.64.3 Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2544, 2035, 2206, 27, 1406, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2544

\(\displaystyle 2 a \int \frac {b^2+\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}} \left (c b^2-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^2}d\left (a x+\sqrt {b^2+a^2 x^2}\right )\)

\(\Big \downarrow \) 2035

\(\displaystyle 4 a \int \frac {b^2+\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}{\left (c b^2-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^2}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}\)

\(\Big \downarrow \) 2206

\(\displaystyle 4 a \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{2 c \left (-c \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-2 a d \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2 c\right )}-\frac {\int -\frac {4 b^2 \left (b^2 c^2+a^2 d^2\right )}{c b^2-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{8 b^2 c \left (a^2 d^2+b^2 c^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 4 a \left (\frac {\int \frac {1}{c b^2-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 c}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{2 c \left (-c \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-2 a d \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2 c\right )}\right )\)

\(\Big \downarrow \) 1406

\(\displaystyle 4 a \left (\frac {\frac {c \int \frac {1}{-a d-c \left (a x+\sqrt {b^2+a^2 x^2}\right )+\sqrt {b^2 c^2+a^2 d^2}}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {a^2 d^2+b^2 c^2}}-\frac {c \int \frac {1}{-a d-c \left (a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {b^2 c^2+a^2 d^2}}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {a^2 d^2+b^2 c^2}}}{2 c}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{2 c \left (-c \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-2 a d \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2 c\right )}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle 4 a \left (\frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{2 \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{2 \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}}{2 c}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{2 c \left (-c \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-2 a d \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2 c\right )}\right )\)

input

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

output

4*a*(Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/(2*c*(b^2*c - 2*a*d*(a*x + Sqrt[b^2 + 
 a^2*x^2]) - c*(a*x + Sqrt[b^2 + a^2*x^2])^2)) + (-1/2*(Sqrt[c]*ArcTan[(Sq 
rt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]] 
])/(Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (Sqrt[c 
]*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 
 + a^2*d^2]]])/(2*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^ 
2]]))/(2*c))
 

3.28.64.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1406

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 
2 - 4*a*c, 2]}, Simp[c/q   Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q   I 
nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c 
, 0] && PosQ[b^2 - 4*a*c]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 2544

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^ 
2])^(n_.), x_Symbol] :> Simp[1/(2^(m + 1)*e^(m + 1))   Subst[Int[x^(n - m - 
 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt[a + 
 c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && I 
ntegerQ[m]
 

3.28.64.4 Maple [F]

\[\int \frac {1}{\left (c x +d \right )^{2} \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

input

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

output

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

3.28.64.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1174 vs. \(2 (221) = 442\).

Time = 0.30 (sec) , antiderivative size = 1174, normalized size of antiderivative = 4.50 \[ \int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {{\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} + {\left (a b^{2} c^{2} + a^{3} d^{2} - {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) - {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} - {\left (a b^{2} c^{2} + a^{3} d^{2} - {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) + {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} + {\left (a b^{2} c^{2} + a^{3} d^{2} + {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) - {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} - {\left (a b^{2} c^{2} + a^{3} d^{2} + {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{2 \, {\left (b^{2} c^{2} x + b^{2} c d\right )}} \]

input

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

output

1/2*((b^2*c^2*x + b^2*c*d)*sqrt((a^3*d + (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt( 
a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))*log(sqrt(a* 
x + sqrt(a^2*x^2 + b^2))*a^2 + (a*b^2*c^2 + a^3*d^2 - (b^4*c^5*d + a^2*b^2 
*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a^3*d + (b^4*c^5 + 
a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2 
*c^3*d^2))) - (b^2*c^2*x + b^2*c*d)*sqrt((a^3*d + (b^4*c^5 + a^2*b^2*c^3*d 
^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))*lo 
g(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a^2 - (a*b^2*c^2 + a^3*d^2 - (b^4*c^5*d 
+ a^2*b^2*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a^3*d + (b 
^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 
+ a^2*b^2*c^3*d^2))) + (b^2*c^2*x + b^2*c*d)*sqrt((a^3*d - (b^4*c^5 + a^2* 
b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3 
*d^2))*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a^2 + (a*b^2*c^2 + a^3*d^2 + (b 
^4*c^5*d + a^2*b^2*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a 
^3*d - (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/ 
(b^4*c^5 + a^2*b^2*c^3*d^2))) - (b^2*c^2*x + b^2*c*d)*sqrt((a^3*d - (b^4*c 
^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^ 
2*b^2*c^3*d^2))*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a^2 - (a*b^2*c^2 + a^3 
*d^2 + (b^4*c^5*d + a^2*b^2*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)) 
)*sqrt((a^3*d - (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4...
 

3.28.64.6 Sympy [F]

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

input

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

output

Integral(1/(sqrt(a*x + sqrt(a**2*x**2 + b**2))*(c*x + d)**2), x)
 

3.28.64.7 Maxima [F]

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

input

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

output

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

3.28.64.8 Giac [F]

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

input

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

output

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

3.28.64.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,{\left (d+c\,x\right )}^2} \,d x \]

input

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

output

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