∫ √ ____ c + dx_ x√ ___

Optimal. Leaf size=85 \[ -\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]

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

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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {132, 65, 223, 212, 12, 95, 214} \begin {gather*} \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}} \end {gather*}

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

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

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Mathematica [A]
time = 0.28, size = 130, normalized size = 1.53 \begin {gather*} -\frac {2 \left (\sqrt {b} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d} \left (-b x+\sqrt {\frac {b}{d}} \sqrt {a+b x} \sqrt {c+d x}\right )}{\sqrt {a} \sqrt {b} \sqrt {c}}\right )+\sqrt {a} \sqrt {d} \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )\right )}{\sqrt {a} \sqrt {\frac {b}{d}} \sqrt {d}} \end {gather*}

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

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(61)=122\).
time = 0.08, size = 133, normalized size = 1.56


method	result	size

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

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

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (61) = 122\).
time = 1.25, size = 711, normalized size = 8.36 \begin {gather*} \left [\frac {1}{2} \, \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), -\sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ), \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right )\right ] \end {gather*}

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

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

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

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

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

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

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

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

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

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x}}{x \sqrt {a + b x}}\, dx \end {gather*}

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Mupad [B]
time = 18.14, size = 2500, normalized size = 29.41 \begin {gather*} \text {Too large to display} \end {gather*}

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

((c + d*x)^(1/2) - c^(1/2))))/a^(1/2) + (4*atanh((64*a^2*b^2*(b*d)^(1/2))/((200*a^(1/2)*b^4*c^(3/2))/d - 288*a

^(3/2)*b^3*c^(1/2) + (544*b^5*c^(5/2))/(a^(1/2)*d^2) - (528*b^6*c^(7/2))/(a^(3/2)*d^3) + (64*b^7*c^(9/2))/(a^(

5/2)*d^4) + (8*b^8*c^(11/2))/(a^(7/2)*d^5) + (64*a^2*b^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(

1/2)) - (864*b^4*c^2*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*c*((a + b*x)^(1

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

) - c^(1/2))) + (160*b^6*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^5*

((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^4*((c + d*x)^(1/2) - c^(1/2)))) - (864*b^4*(b*d)^(1/2))/((544*b^5*c^(1/2))

/a^(1/2) + (200*a^(1/2)*b^4*d)/c^(1/2) - (288*a^(3/2)*b^3*d^2)/c^(3/2) - (528*b^6*c^(3/2))/(a^(3/2)*d) + (64*b

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

^(1/2) - c^(1/2)) + (320*b^5*c*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*d^2*(

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

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

- (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^2*((c + d*x)^(1/2) - c^(1/2)))) + (368*a*b^3*(b*d)^(1/2))/(

200*a^(1/2)*b^4*c^(1/2) - (288*a^(3/2)*b^3*d)/c^(1/2) + (544*b^5*c^(3/2))/(a^(1/2)*d) - (528*b^6*c^(5/2))/(a^(

3/2)*d^2) + (64*b^7*c^(7/2))/(a^(5/2)*d^3) + (8*b^8*c^(9/2))/(a^(7/2)*d^4) - (864*b^4*c*((a + b*x)^(1/2) - a^(

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

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

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

/2) - c^(1/2))) - (48*b^7*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^3*((c + d*x)^(1/2) - c^(1/2)))) + (320*b^5*c

*(b*d)^(1/2))/(544*a^(1/2)*b^5*c^(1/2)*d - (528*b^6*c^(3/2))/a^(1/2) + (200*a^(3/2)*b^4*d^2)/c^(1/2) - (288*a^

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

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

/2) - c^(1/2))) + (320*b^5*c*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*a^2*b^3*d^3*((a

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

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

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

544*a^(5/2)*b^5*c^(1/2)*d^3 + (200*a^(7/2)*b^4*d^4)/c^(1/2) - (288*a^(9/2)*b^3*d^5)/c^(3/2) - (48*b^7*c^3*d*((

a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (864*a^3*b^4*d^4*((a + b*x)^(1/2) - a^(1/2)))/((c + d

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

b^5*c*d^3*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*a^4*b^3*d^5*((a + b*x)^(1/2) - a^(1/

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

(1/2)))) + (160*b^6*c^2*(b*d)^(1/2))/((64*b^7*c^(5/2))/a^(1/2) - 528*a^(1/2)*b^6*c^(3/2)*d + 544*a^(3/2)*b^5*c

^(1/2)*d^2 + (200*a^(5/2)*b^4*d^3)/c^(1/2) - (288*a^(7/2)*b^3*d^4)/c^(3/2) + (8*b^8*c^(7/2))/(a^(3/2)*d) + (16

0*b^6*c^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2))

)/(a*((c + d*x)^(1/2) - c^(1/2))) - (864*a^2*b^4*d^3*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))

+ (368*a^3*b^3*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^4*b^2*d^5*((a + b*x)^(

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

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

/2))*((200*a^(1/2)*b^4)/(c^(1/2)*d) - (288*a^(3/2)*b^3)/c^(3/2) + (544*b^5*c^(1/2))/(a^(1/2)*d^2) - (528*b^6*c

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

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

- c^(1/2))) + (64*a^2*b^2*d*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (320*b^5*c*((a +

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

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3.8.2 \(\int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx\) [702]