3.4.67 \(\int \frac {a+b x^2}{x^5 \sqrt {-c+d x} \sqrt {c+d x}} \, dx\) [367]

Optimal. Leaf size=123 \[ \frac {a \sqrt {-c+d x} \sqrt {c+d x}}{4 c^2 x^4}+\frac {\left (4 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 c^4 x^2}+\frac {d^2 \left (4 b c^2+3 a d^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{8 c^5} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {465, 105, 12, 94, 211} \begin {gather*} \frac {d^2 \left (3 a d^2+4 b c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{8 c^5}+\frac {\sqrt {d x-c} \sqrt {c+d x} \left (3 a d^2+4 b c^2\right )}{8 c^4 x^2}+\frac {a \sqrt {d x-c} \sqrt {c+d x}}{4 c^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]

[Out]

(a*Sqrt[-c + d*x]*Sqrt[c + d*x])/(4*c^2*x^4) + ((4*b*c^2 + 3*a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(8*c^4*x^2)
+ (d^2*(4*b*c^2 + 3*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])/c])/(8*c^5)

Rule 12

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

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

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 465

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(a1*a2*e*(
m + 1))), x] + Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*(m + 1)), Int[(e*x)^(m + n)*(a1
+ b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && Eq
Q[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1
])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {a+b x^2}{x^5 \sqrt {-c+d x} \sqrt {c+d x}} \, dx &=\frac {a \sqrt {-c+d x} \sqrt {c+d x}}{4 c^2 x^4}+\frac {1}{4} \left (4 b+\frac {3 a d^2}{c^2}\right ) \int \frac {1}{x^3 \sqrt {-c+d x} \sqrt {c+d x}} \, dx\\ &=\frac {a \sqrt {-c+d x} \sqrt {c+d x}}{4 c^2 x^4}+\frac {\left (4 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 c^4 x^2}+\frac {\left (4 b c^2+3 a d^2\right ) \int \frac {d^2}{x \sqrt {-c+d x} \sqrt {c+d x}} \, dx}{8 c^4}\\ &=\frac {a \sqrt {-c+d x} \sqrt {c+d x}}{4 c^2 x^4}+\frac {\left (4 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 c^4 x^2}+\frac {\left (d^2 \left (4 b c^2+3 a d^2\right )\right ) \int \frac {1}{x \sqrt {-c+d x} \sqrt {c+d x}} \, dx}{8 c^4}\\ &=\frac {a \sqrt {-c+d x} \sqrt {c+d x}}{4 c^2 x^4}+\frac {\left (4 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 c^4 x^2}+\frac {\left (d^3 \left (4 b c^2+3 a d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2 d+d x^2} \, dx,x,\sqrt {-c+d x} \sqrt {c+d x}\right )}{8 c^4}\\ &=\frac {a \sqrt {-c+d x} \sqrt {c+d x}}{4 c^2 x^4}+\frac {\left (4 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 c^4 x^2}+\frac {d^2 \left (4 b c^2+3 a d^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{8 c^5}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 101, normalized size = 0.82 \begin {gather*} \frac {c \sqrt {-c+d x} \sqrt {c+d x} \left (2 a c^2+4 b c^2 x^2+3 a d^2 x^2\right )+2 d^2 \left (4 b c^2+3 a d^2\right ) x^4 \tan ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 c^5 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/(x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]

[Out]

(c*Sqrt[-c + d*x]*Sqrt[c + d*x]*(2*a*c^2 + 4*b*c^2*x^2 + 3*a*d^2*x^2) + 2*d^2*(4*b*c^2 + 3*a*d^2)*x^4*ArcTan[S
qrt[-c + d*x]/Sqrt[c + d*x]])/(8*c^5*x^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(105)=210\).
time = 0.30, size = 227, normalized size = 1.85

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (3 a \,d^{2} x^{2}+4 b \,c^{2} x^{2}+2 c^{2} a \right )}{8 c^{4} x^{4} \sqrt {d x -c}}+\frac {\left (-\frac {3 d^{4} \ln \left (\frac {-2 c^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}}{x}\right ) a}{8 c^{4} \sqrt {-c^{2}}}-\frac {d^{2} \ln \left (\frac {-2 c^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}}{x}\right ) b}{2 c^{2} \sqrt {-c^{2}}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{\sqrt {d x -c}\, \sqrt {d x +c}}\)	\(192\)
default	\(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (3 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,d^{4} x^{4}+4 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) b \,c^{2} d^{2} x^{4}-3 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, a \,d^{2} x^{2}-4 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, b \,c^{2} x^{2}-2 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, a \,c^{2}\right )}{8 c^{4} \sqrt {d^{2} x^{2}-c^{2}}\, x^{4} \sqrt {-c^{2}}}\)	\(227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.53, size = 114, normalized size = 0.93 \begin {gather*} -\frac {b d^{2} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{2 \, c^{3}} - \frac {3 \, a d^{4} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{8 \, c^{5}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} b}{2 \, c^{2} x^{2}} + \frac {3 \, \sqrt {d^{2} x^{2} - c^{2}} a d^{2}}{8 \, c^{4} x^{2}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a}{4 \, c^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

-1/2*b*d^2*arcsin(c/(d*abs(x)))/c^3 - 3/8*a*d^4*arcsin(c/(d*abs(x)))/c^5 + 1/2*sqrt(d^2*x^2 - c^2)*b/(c^2*x^2)
+ 3/8*sqrt(d^2*x^2 - c^2)*a*d^2/(c^4*x^2) + 1/4*sqrt(d^2*x^2 - c^2)*a/(c^2*x^4)

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Fricas [A]
time = 2.35, size = 100, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (4 \, b c^{2} d^{2} + 3 \, a d^{4}\right )} x^{4} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right ) + {\left (2 \, a c^{3} + {\left (4 \, b c^{3} + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{8 \, c^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/x**5/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (105) = 210\).
time = 0.58, size = 325, normalized size = 2.64 \begin {gather*} -\frac {\frac {{\left (4 \, b c^{2} d^{3} + 3 \, a d^{5}\right )} \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right )}{c^{5}} + \frac {2 \, {\left (4 \, b c^{2} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 3 \, a d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 16 \, b c^{4} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} + 44 \, a c^{2} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} - 64 \, b c^{6} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 176 \, a c^{4} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 256 \, b c^{8} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} - 192 \, a c^{6} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{4}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-1/4*((4*b*c^2*d^3 + 3*a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^5 + 2*(4*b*c^2*d^3*(sqrt(d*x +
c) - sqrt(d*x - c))^14 + 3*a*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^14 + 16*b*c^4*d^3*(sqrt(d*x + c) - sqrt(d*x
- c))^10 + 44*a*c^2*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^10 - 64*b*c^6*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^6 -
176*a*c^4*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 256*b*c^8*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^2 - 192*a*c^6*
d^5*(sqrt(d*x + c) - sqrt(d*x - c))^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^4*c^4))/d

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Mupad [B]
time = 19.13, size = 1005, normalized size = 8.17 \begin {gather*} \frac {3\,a\,\sqrt {-c}\,d^4\,\ln \left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{8\,c^{11/2}}-\frac {\frac {b\,{\left (-c\right )}^{3/2}\,d^2}{32\,c^{9/2}}+\frac {b\,{\left (-c\right )}^{3/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{16\,c^{9/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}-\frac {15\,b\,{\left (-c\right )}^{3/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{32\,c^{9/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}}{\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}}-\frac {\frac {a\,\sqrt {-c}\,d^4}{1024\,c^{11/2}}-\frac {3\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{128\,c^{11/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}-\frac {53\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{512\,c^{11/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {87\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{256\,c^{11/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {657\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{1024\,c^{11/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}+\frac {121\,a\,\sqrt {-c}\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{256\,c^{11/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}}{\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}+\frac {4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}+\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{12}}}-\frac {b\,{\left (-c\right )}^{3/2}\,d^2\,\ln \left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{2\,c^{9/2}}-\frac {3\,a\,\sqrt {-c}\,d^4\,\ln \left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )}{8\,c^{11/2}}+\frac {b\,{\left (-c\right )}^{3/2}\,d^2\,\ln \left (\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+1\right )}{2\,c^{9/2}}-\frac {7\,a\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{256\,\sqrt {-c}\,c^{9/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {a\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{1024\,\sqrt {-c}\,c^{9/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}+\frac {b\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{32\,{\left (-c\right )}^{3/2}\,c^{3/2}\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a*(-c)^(1/2)*d^4*log(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(8*c^(11/2)) - ((b*(-c)^(
3/2)*d^2)/(32*c^(9/2)) + (b*(-c)^(3/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^2)/(16*c^(9/2)*((-c)^(1/2) - (d*x - c)^
(1/2))^2) - (15*b*(-c)^(3/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^4)/(32*c^(9/2)*((-c)^(1/2) - (d*x - c)^(1/2))^4))
/(((c + d*x)^(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (2*((c + d*x)^(1/2) - c^(1/2))^4)/((-c)^(1/
2) - (d*x - c)^(1/2))^4 + ((c + d*x)^(1/2) - c^(1/2))^6/((-c)^(1/2) - (d*x - c)^(1/2))^6) - ((a*(-c)^(1/2)*d^4
)/(1024*c^(11/2)) - (3*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^2)/(128*c^(11/2)*((-c)^(1/2) - (d*x - c)^(
1/2))^2) - (53*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^4)/(512*c^(11/2)*((-c)^(1/2) - (d*x - c)^(1/2))^4)
+ (87*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^6)/(256*c^(11/2)*((-c)^(1/2) - (d*x - c)^(1/2))^6) + (657*
a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^8)/(1024*c^(11/2)*((-c)^(1/2) - (d*x - c)^(1/2))^8) + (121*a*(-c)
^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^10)/(256*c^(11/2)*((-c)^(1/2) - (d*x - c)^(1/2))^10))/(((c + d*x)^(1/2)
- c^(1/2))^4/((-c)^(1/2) - (d*x - c)^(1/2))^4 + (4*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/
2))^6 + (6*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 + (4*((c + d*x)^(1/2) - c^(1/2))^10
)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + ((c + d*x)^(1/2) - c^(1/2))^12/((-c)^(1/2) - (d*x - c)^(1/2))^12) - (b*(
-c)^(3/2)*d^2*log(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(2*c^(9/2)) - (3*a*(-c)^(1/2)*d
^4*log(((c + d*x)^(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + 1))/(8*c^(11/2)) + (b*(-c)^(3/2)*d^2*l
og(((c + d*x)^(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + 1))/(2*c^(9/2)) - (7*a*d^4*((c + d*x)^(1/2
) - c^(1/2))^2)/(256*(-c)^(1/2)*c^(9/2)*((-c)^(1/2) - (d*x - c)^(1/2))^2) + (a*d^4*((c + d*x)^(1/2) - c^(1/2))
^4)/(1024*(-c)^(1/2)*c^(9/2)*((-c)^(1/2) - (d*x - c)^(1/2))^4) + (b*d^2*((c + d*x)^(1/2) - c^(1/2))^2)/(32*(-c
)^(3/2)*c^(3/2)*((-c)^(1/2) - (d*x - c)^(1/2))^2)

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