3.6.0 ∫ √ ____ a+bx _ x√ ___

Integrand size = 22, antiderivative size = 85 \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}} \]

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

Rubi [A] (veriﬁed)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {132, 65, 223, 212, 12, 95, 214} \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}} \]

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

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin

ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^

(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[b*d^(m

+ n)*f^p, Int[(a + b*x)^(m - 1)/(c + d*x)^m, x], x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandTo

Sum[(a + b*x)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x

] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

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

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,

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

Mathematica [C] (veriﬁed)

Time = 1.94 (sec) , antiderivative size = 383, normalized size of antiderivative = 4.51 \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {a} \left (\left (\sqrt {a} \sqrt {d}-i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )+\left (\sqrt {a} \sqrt {d}+i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )\right )}{b c^{3/2}}-\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {d}} \]

(-2*Sqrt[a]*((Sqrt[a]*Sqrt[d] - I*Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*A

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

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

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

*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))]))/(b*c^(3/2)) - (4*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sq

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(61)=122\).

Time = 1.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56


method	result	size

default	\(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) \sqrt {b d}\, a +\sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}}\)	\(133\)

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

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

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (61) = 122\).

Time = 0.30 (sec) , antiderivative size = 711, normalized size of antiderivative = 8.36 \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=\left [\frac {1}{2} \, \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \frac {1}{2} \, \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), -\sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ), \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right )\right ] \]

[1/2*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)

*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + 1/2*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^

2*d^2)*x^2 - 4*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/

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

(b^2*c + a*b*d)*x)) + 1/2*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c^2 + (b*c^

2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2), sqrt(-a/c)*arctan(1/2*(2*

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

t(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*

x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x), sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(

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

Sympy [F]

\[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a + b x}}{x \sqrt {c + d x}}\, dx \]

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=\text {Exception raised: TypeError} \]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

Mupad [B] (veriﬁcation not implemented)

Time = 18.92 (sec) , antiderivative size = 4312, normalized size of antiderivative = 50.73 \[ \int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx=\text {Too large to display} \]

(4*atanh((64*b^10*c^2*(b*d)^(1/2))/(200*a^(3/2)*b^9*c^(1/2)*d^2 - 288*a^(1/2)*b^10*c^(3/2)*d + (544*a^(5/2)*b^

8*d^3)/c^(1/2) - (528*a^(7/2)*b^7*d^4)/c^(3/2) + (64*a^(9/2)*b^6*d^5)/c^(5/2) + (8*a^(11/2)*b^5*d^6)/c^(7/2) +

(64*b^10*c^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (864*a^2*b^8*d^3*((a + b*x)^(1/2) -

a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a^3*b^7*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c

^(1/2))) + (160*a^4*b^6*d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^5*b^5*d^6*(

(a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^9*c*d^2*((a + b*x)^(1/2) - a^(1/2)))/

((c + d*x)^(1/2) - c^(1/2))) - (864*b^8*(b*d)^(1/2))/((200*b^9*c^(1/2))/a^(1/2) + (544*a^(1/2)*b^8*d)/c^(1/2)

- (528*a^(3/2)*b^7*d^2)/c^(3/2) - (288*b^10*c^(3/2))/(a^(3/2)*d) + (64*a^(5/2)*b^6*d^3)/c^(5/2) + (8*a^(7/2)*b

^5*d^4)/c^(7/2) - (864*b^8*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*b^9*c*((a + b*x)^

(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^

(1/2) - c^(1/2))) + (160*a^2*b^6*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10

*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^4*((a + b*x)^(1/2) - a^(

1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2)))) + (320*a*b^7*(b*d)^(1/2))/(544*a^(1/2)*b^8*c^(1/2) - (528*a^(3/2)*b^

7*d)/c^(1/2) + (200*b^9*c^(3/2))/(a^(1/2)*d) + (64*a^(5/2)*b^6*d^2)/c^(3/2) - (288*b^10*c^(5/2))/(a^(3/2)*d^2)

+ (8*a^(7/2)*b^5*d^3)/c^(5/2) - (864*b^8*c*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a*

b^7*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (160*a^2*b^6*d^2*((a + b*x)^(1/2) - a^(1/2)))

/(c*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a*d*((c + d*x)^(1/2) - c^(1/2)))

- (48*a^3*b^5*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^3*((a + b*x)^(1

/2) - a^(1/2)))/(a^2*d^2*((c + d*x)^(1/2) - c^(1/2)))) + (368*b^9*c*(b*d)^(1/2))/(200*a^(1/2)*b^9*c^(1/2)*d -

(288*b^10*c^(3/2))/a^(1/2) + (544*a^(3/2)*b^8*d^2)/c^(1/2) - (528*a^(5/2)*b^7*d^3)/c^(3/2) + (64*a^(7/2)*b^6*d

^4)/c^(5/2) + (8*a^(9/2)*b^5*d^5)/c^(7/2) - (864*a*b^8*d^2*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(

1/2)) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*d*((a + b*x)^(1

/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a^2*b^7*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2

) - c^(1/2))) + (160*a^3*b^6*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^4*b^5*

d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2)))) + (160*a^2*b^6*(b*d)^(1/2))/((64*a^(5/2)*b

^6*d)/c^(1/2) - 528*a^(3/2)*b^7*c^(1/2) + (544*a^(1/2)*b^8*c^(3/2))/d + (200*b^9*c^(5/2))/(a^(1/2)*d^2) + (8*a

^(7/2)*b^5*d^2)/c^(3/2) - (288*b^10*c^(7/2))/(a^(3/2)*d^3) + (160*a^2*b^6*d*((a + b*x)^(1/2) - a^(1/2)))/((c +

d*x)^(1/2) - c^(1/2)) - (864*b^8*c^2*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^

7*c*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*a^3*b^5*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c

*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x)^(1/2) - c^(1/2)))

+ (64*b^10*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2)))) - (48*a^3*b^5*(b*d)^(1/2))/

(64*a^(5/2)*b^6*c^(1/2) + (8*a^(7/2)*b^5*d)/c^(1/2) - (528*a^(3/2)*b^7*c^(3/2))/d + (544*a^(1/2)*b^8*c^(5/2))/

d^2 + (200*b^9*c^(7/2))/(a^(1/2)*d^3) - (288*b^10*c^(9/2))/(a^(3/2)*d^4) + (160*a^2*b^6*c*((a + b*x)^(1/2) - a

^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*a^3*b^5*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))

- (864*b^8*c^3*((a + b*x)^(1/2) - a^(1/2)))/(d^2*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*c^2*((a + b*x)^(1/2

) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a*d^3*((c + d*x)^(1

/2) - c^(1/2))) + (64*b^10*c^5*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^4*((c + d*x)^(1/2) - c^(1/2)))) + (8*a^(7/2

)*b^4*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(7/2)*((c + d*x)^(1/2) - c^(1/2))*((8*a^(7/2)*b^5)/c^(7/2) +

(544*a^(1/2)*b^8)/(c^(1/2)*d^3) + (200*b^9*c^(1/2))/(a^(1/2)*d^4) - (528*a^(3/2)*b^7)/(c^(3/2)*d^2) + (64*a^(

5/2)*b^6)/(c^(5/2)*d) - (288*b^10*c^(3/2))/(a^(3/2)*d^5) - (864*b^8*((a + b*x)^(1/2) - a^(1/2)))/(d^3*((c + d*

x)^(1/2) - c^(1/2))) - (48*a^3*b^5*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7

*((a + b*x)^(1/2) - a^(1/2)))/(c*d^2*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))/(a

*d^4*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*((a + b*x)^(1/2) - a^(1/2)))/(c^2*d*((c + d*x)^(1/2) - c^(1/2

))) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^5*((c + d*x)^(1/2) - c^(1/2))))) + (544*a^(1/2)*b^7*(b*

d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(1/2)*((c + d*x)^(1/2) - c^(1/2))*((544*a^(1/2)*b^8)/c^(1/2) - (864*b

^8*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (528*a^(3/2)*b^7*d)/c^(3/2) + (200*b^9*c^(1/2))/

(a^(1/2)*d) - (288*b^10*c^(3/2))/(a^(3/2)*d^2) + (64*a^(5/2)*b^6*d^2)/c^(5/2) + (8*a^(7/2)*b^5*d^3)/c^(7/2) +

(368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))/(a*d*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*d^2*((a + b*x)^(1/2)

- a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(

1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^2*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^

7*d*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))))) - (528*a^(3/2)*b^6*(b*d)^(1/2)*((a + b*x)^(

1/2) - a^(1/2)))/(c^(3/2)*((c + d*x)^(1/2) - c^(1/2))*((64*a^(5/2)*b^6*d)/c^(5/2) - (528*a^(3/2)*b^7)/c^(3/2)

+ (544*a^(1/2)*b^8)/(c^(1/2)*d) + (200*b^9*c^(1/2))/(a^(1/2)*d^2) - (288*b^10*c^(3/2))/(a^(3/2)*d^3) + (8*a^(7

/2)*b^5*d^2)/c^(7/2) - (864*b^8*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*((a

+ b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*d*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((

c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x)^(1/2) - c^(1/2))) - (48

*a^3*b^5*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^(1/2) -

a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2))))) + (200*b^8*c^(1/2)*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/

(a^(1/2)*((c + d*x)^(1/2) - c^(1/2))*((200*b^9*c^(1/2))/a^(1/2) + (544*a^(1/2)*b^8*d)/c^(1/2) - (528*a^(3/2)*b

^7*d^2)/c^(3/2) - (288*b^10*c^(3/2))/(a^(3/2)*d) + (64*a^(5/2)*b^6*d^3)/c^(5/2) + (8*a^(7/2)*b^5*d^4)/c^(7/2)

- (864*b^8*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2))

)/(a*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))

) + (160*a^2*b^6*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^

(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c

+ d*x)^(1/2) - c^(1/2))))) + (64*a^(5/2)*b^5*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(5/2)*((c + d*x)^(1/2

) - c^(1/2))*((64*a^(5/2)*b^6)/c^(5/2) + (8*a^(7/2)*b^5*d)/c^(7/2) + (544*a^(1/2)*b^8)/(c^(1/2)*d^2) + (200*b^

9*c^(1/2))/(a^(1/2)*d^3) - (528*a^(3/2)*b^7)/(c^(3/2)*d) - (288*b^10*c^(3/2))/(a^(3/2)*d^4) - (864*b^8*((a + b

*x)^(1/2) - a^(1/2)))/(d^2*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c +

d*x)^(1/2) - c^(1/2))) + (320*a*b^7*((a + b*x)^(1/2) - a^(1/2)))/(c*d*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*

b^5*d*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))

/(a*d^3*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^4*((c + d*x)^(1/2) - c

^(1/2))))) - (288*b^9*c^(3/2)*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(a^(3/2)*((c + d*x)^(1/2) - c^(1/2))*((

200*b^9*c^(1/2)*d)/a^(1/2) - (288*b^10*c^(3/2))/a^(3/2) + (544*a^(1/2)*b^8*d^2)/c^(1/2) - (528*a^(3/2)*b^7*d^3

)/c^(3/2) + (64*a^(5/2)*b^6*d^4)/c^(5/2) + (8*a^(7/2)*b^5*d^5)/c^(7/2) - (864*b^8*d^2*((a + b*x)^(1/2) - a^(1/

2)))/((c + d*x)^(1/2) - c^(1/2)) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*((c + d*x)^(1/2) - c^(1/2)))

+ (320*a*b^7*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*d^4*((a + b*x)^(

1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d

*x)^(1/2) - c^(1/2))) + (368*b^9*c*d*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))))))*(b*d)^(1/

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

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

\(\int \frac {\sqrt {a+b x}}{x \sqrt {c+d x}} \, dx\) [580]

Optimal result