\(\int \frac {1}{(d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2556]

Optimal result

Integrand size = 31, antiderivative size = 215 \[ \int \frac {1}{(d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2144, 1642, 842, 840, 1180, 211} \[ \int \frac {1}{(d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {c} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {2}{c \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \]

[In]

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

[Out]

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

Rule 211

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

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[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] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{c x^{3/2}}+\frac {2 \left (b^2 c-a d x\right )}{c x^{3/2} \left (-b^2 c+2 a d x+c x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = -\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \text {Subst}\left (\int \frac {b^2 c-a d x}{x^{3/2} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c} \\ & = \frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {2 \text {Subst}\left (\int \frac {-a b^2 c d-b^2 c^2 x}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 c^2} \\ & = \frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {4 \text {Subst}\left (\int \frac {-a b^2 c d-b^2 c^2 x^2}{-b^2 c+2 a d x^2+c x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c^2} \\ & = \frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+2 \text {Subst}\left (\int \frac {1}{a d-\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )+2 \text {Subst}\left (\int \frac {1}{a d+\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right ) \\ & = \frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \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 {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {2 \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 {a d+\sqrt {b^2 c^2+a^2 d^2}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {2 \arctan \left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (179) = 358\).

Time = 0.31 (sec) , antiderivative size = 673, normalized size of antiderivative = 3.13 \[ \int \frac {1}{(d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {b^{2} c \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} \log \left (4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d\right )} \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - b^{2} c \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} \log \left (-4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d\right )} \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - b^{2} c \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} \log \left (4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d\right )} \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) + b^{2} c \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} \log \left (-4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d\right )} \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2} c} \]

[In]

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

[Out]

-(b^2*c*sqrt((b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) + a*d)/(b^2*c^3))*log(4*(b^2*c^3*sqrt((b^2*c^2 + a^2
*d^2)/(b^4*c^6)) - a*d)*sqrt((b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) + a*d)/(b^2*c^3)) + 4*sqrt(a*x + sqr
t(a^2*x^2 + b^2))) - b^2*c*sqrt((b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) + a*d)/(b^2*c^3))*log(-4*(b^2*c^3
*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) - a*d)*sqrt((b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) + a*d)/(b^2*c^3)
) + 4*sqrt(a*x + sqrt(a^2*x^2 + b^2))) - b^2*c*sqrt(-(b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) - a*d)/(b^2*
c^3))*log(4*(b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) + a*d)*sqrt(-(b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c
^6)) - a*d)/(b^2*c^3)) + 4*sqrt(a*x + sqrt(a^2*x^2 + b^2))) + b^2*c*sqrt(-(b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b
^4*c^6)) - a*d)/(b^2*c^3))*log(-4*(b^2*c^3*sqrt((b^2*c^2 + a^2*d^2)/(b^4*c^6)) + a*d)*sqrt(-(b^2*c^3*sqrt((b^2
*c^2 + a^2*d^2)/(b^4*c^6)) - a*d)/(b^2*c^3)) + 4*sqrt(a*x + sqrt(a^2*x^2 + b^2))) + 2*sqrt(a*x + sqrt(a^2*x^2
+ b^2))*(a*x - sqrt(a^2*x^2 + b^2)))/(b^2*c)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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