Integrand size = 23, antiderivative size = 329 \[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {\left (\frac {b+a x}{d+c x}\right )^{3/4} \left (-96 a^2 c^2 d+45 b^2 c^2 d-6 a b c d^2-7 a^2 d^3-96 a^2 c^3 x+45 b^2 c^3 x-42 a b c^2 d x-3 a^2 c d^2 x-36 a b c^3 x^2+36 a^2 c^2 d x^2+32 a^2 c^3 x^3\right )}{96 a^3 c^2}+\frac {\left (-32 a^2 b c^3+15 b^3 c^3+32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {\left (32 a^2 b c^3-15 b^3 c^3-32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}} \]
1/96*((a*x+b)/(c*x+d))^(3/4)*(32*a^2*c^3*x^3+36*a^2*c^2*d*x^2-36*a*b*c^3*x ^2-96*a^2*c^3*x-3*a^2*c*d^2*x-42*a*b*c^2*d*x+45*b^2*c^3*x-96*a^2*c^2*d-7*a ^2*d^3-6*a*b*c*d^2+45*b^2*c^2*d)/a^3/c^2+1/64*(32*a^3*c^2*d-7*a^3*d^3-32*a ^2*b*c^3-3*a^2*b*c*d^2-5*a*b^2*c^2*d+15*b^3*c^3)*arctan(c^(1/4)*((a*x+b)/( c*x+d))^(1/4)/a^(1/4))/a^(13/4)/c^(11/4)+1/64*(-32*a^3*c^2*d+7*a^3*d^3+32* a^2*b*c^3+3*a^2*b*c*d^2+5*a*b^2*c^2*d-15*b^3*c^3)*arctanh(c^(1/4)*((a*x+b) /(c*x+d))^(1/4)/a^(1/4))/a^(13/4)/c^(11/4)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.16 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.52 \[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {4 \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x) \left ((b c-a d)^2 \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {3}{4},\frac {7}{4},\frac {c (b+a x)}{b c-a d}\right )+a \left (2 d (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {c (b+a x)}{b c-a d}\right )+a \left (-c^2+d^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},\frac {c (b+a x)}{b c-a d}\right )\right )\right )}{3 a^3 c^2 \sqrt [4]{\frac {a (d+c x)}{-b c+a d}}} \]
(4*((b + a*x)/(d + c*x))^(3/4)*(d + c*x)*((b*c - a*d)^2*Hypergeometric2F1[ -9/4, 3/4, 7/4, (c*(b + a*x))/(b*c - a*d)] + a*(2*d*(b*c - a*d)*Hypergeome tric2F1[-5/4, 3/4, 7/4, (c*(b + a*x))/(b*c - a*d)] + a*(-c^2 + d^2)*Hyperg eometric2F1[-1/4, 3/4, 7/4, (c*(b + a*x))/(b*c - a*d)])))/(3*a^3*c^2*((a*( d + c*x))/(-(b*c) + a*d))^(1/4))
Time = 0.87 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2055, 25, 7293, 2009}
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.
| \(\displaystyle \int \frac {x^2-1}{\sqrt [4]{\frac {a x+b}{c x+d}}} \, dx\) |
| \(\Big \downarrow \) 2055 |
| \(\displaystyle -4 (b c-a d) \int -\frac {\sqrt {\frac {b+a x}{d+c x}} \left (1-\frac {\left (b-\frac {d (b+a x)}{d+c x}\right )^2}{\left (a-\frac {c (b+a x)}{d+c x}\right )^2}\right )}{\left (a-\frac {c (b+a x)}{d+c x}\right )^2}d\sqrt [4]{\frac {b+a x}{d+c x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 (b c-a d) \int \frac {\sqrt {\frac {b+a x}{d+c x}} \left (1-\frac {\left (b-\frac {d (b+a x)}{d+c x}\right )^2}{\left (a-\frac {c (b+a x)}{d+c x}\right )^2}\right )}{\left (a-\frac {c (b+a x)}{d+c x}\right )^2}d\sqrt [4]{\frac {b+a x}{d+c x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 (b c-a d) \int \left (-\frac {\sqrt {\frac {b+a x}{d+c x}} (b c-a d)^2}{c^2 \left (\frac {c (b+a x)}{d+c x}-a\right )^4}+\frac {2 d \sqrt {\frac {b+a x}{d+c x}} (b c-a d)}{c^2 \left (\frac {c (b+a x)}{d+c x}-a\right )^3}+\frac {\left (c^2-d^2\right ) \sqrt {\frac {b+a x}{d+c x}}}{c^2 \left (\frac {c (b+a x)}{d+c x}-a\right )^2}\right )d\sqrt [4]{\frac {b+a x}{d+c x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 (b c-a d) \left (-\frac {15 (b c-a d)^2 \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{256 a^{13/4} c^{11/4}}-\frac {5 d (b c-a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{32 a^{9/4} c^{11/4}}+\frac {\left (c^2-d^2\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{8 a^{5/4} c^{11/4}}+\frac {15 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{256 a^{13/4} c^{11/4}}+\frac {5 d (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{32 a^{9/4} c^{11/4}}-\frac {\left (c^2-d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{8 a^{5/4} c^{11/4}}+\frac {15 (b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{128 a^3 c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )}+\frac {3 (b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{32 a^2 c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}+\frac {5 d (b c-a d) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{16 a^2 c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )}-\frac {\left (c^2-d^2\right ) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{4 a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )}+\frac {(b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{12 a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^3}+\frac {d (b c-a d) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{4 a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}\right )\) |
-4*(b*c - a*d)*(((b*c - a*d)^2*((b + a*x)/(d + c*x))^(3/4))/(12*a*c^2*(a - (c*(b + a*x))/(d + c*x))^3) + (d*(b*c - a*d)*((b + a*x)/(d + c*x))^(3/4)) /(4*a*c^2*(a - (c*(b + a*x))/(d + c*x))^2) + (3*(b*c - a*d)^2*((b + a*x)/( d + c*x))^(3/4))/(32*a^2*c^2*(a - (c*(b + a*x))/(d + c*x))^2) + (5*d*(b*c - a*d)*((b + a*x)/(d + c*x))^(3/4))/(16*a^2*c^2*(a - (c*(b + a*x))/(d + c* x))) + (15*(b*c - a*d)^2*((b + a*x)/(d + c*x))^(3/4))/(128*a^3*c^2*(a - (c *(b + a*x))/(d + c*x))) - ((c^2 - d^2)*((b + a*x)/(d + c*x))^(3/4))/(4*a*c ^2*(a - (c*(b + a*x))/(d + c*x))) - (5*d*(b*c - a*d)*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(32*a^(9/4)*c^(11/4)) - (15*(b*c - a*d)^ 2*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(256*a^(13/4)*c^( 11/4)) + ((c^2 - d^2)*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4) ])/(8*a^(5/4)*c^(11/4)) + (5*d*(b*c - a*d)*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(32*a^(9/4)*c^(11/4)) + (15*(b*c - a*d)^2*ArcTanh [(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(256*a^(13/4)*c^(11/4)) - ((c^2 - d^2)*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(8*a ^(5/4)*c^(11/4)))
3.30.14.3.1 Defintions of rubi rules used
Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/ (b*e - d*x^q)^(1/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1 /n))^r, x], x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b , c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/n] && Int egerQ[r]
\[\int \frac {x^{2}-1}{\left (\frac {a x +b}{c x +d}\right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 4866, normalized size of antiderivative = 14.79 \[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]
1/384*(3*a^3*c^2*((4116*a^11*b*c*d^11 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^12 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(1048576*a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6* b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 917504 *a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048 576*a^12 - 2228224*a^10*b^2 + 2297856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4 *b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 + 775*a^5 *b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6 *b^6)*c^6*d^6 - 24*(14336*a^11*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(448*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11) )^(1/4)*log(a^10*c^8*((4116*a^11*b*c*d^11 + 2401*a^12*d^12 + (1048576*a^8* b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^ 12 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^ 9 + 16875*a*b^11)*c^11*d + 6*(1048576*a^10*b^2 - 1245184*a^8*b^4 + 471040* a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 91 7504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + ( 1048576*a^12 - 2228224*a^10*b^2 + 2297856*a^8*b^4 - 874240*a^6*b^6 + 93775 *a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 + ...
\[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [4]{\frac {a x + b}{c x + d}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.48 \[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {3 \, {\left (3 \, a^{2} b c^{3} d^{2} + 7 \, a^{3} c^{2} d^{3} + {\left (32 \, a^{2} b - 15 \, b^{3}\right )} c^{5} - {\left (32 \, a^{3} - 5 \, a b^{2}\right )} c^{4} d\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {11}{4}} - 6 \, {\left (17 \, a^{3} b c^{2} d^{2} - 3 \, a^{4} c d^{3} + {\left (32 \, a^{3} b - 21 \, a b^{3}\right )} c^{4} - {\left (32 \, a^{4} - 7 \, a^{2} b^{2}\right )} c^{3} d\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {7}{4}} - {\left (3 \, a^{4} b c d^{2} + 7 \, a^{5} d^{3} - {\left (96 \, a^{4} b - 113 \, a^{2} b^{3}\right )} c^{3} + 3 \, {\left (32 \, a^{5} - 41 \, a^{3} b^{2}\right )} c^{2} d\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {3}{4}}}{96 \, {\left (a^{6} c^{2} - \frac {3 \, {\left (a x + b\right )} a^{5} c^{3}}{c x + d} + \frac {3 \, {\left (a x + b\right )}^{2} a^{4} c^{4}}{{\left (c x + d\right )}^{2}} - \frac {{\left (a x + b\right )}^{3} a^{3} c^{5}}{{\left (c x + d\right )}^{3}}\right )}} - \frac {{\left (3 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3} + {\left (32 \, a^{2} b - 15 \, b^{3}\right )} c^{3} - {\left (32 \, a^{3} - 5 \, a b^{2}\right )} c^{2} d\right )} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {c} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {c} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} - \sqrt {\sqrt {a} \sqrt {c}}}{\sqrt {c} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} + \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}}\right )}}{128 \, a^{3} c^{2}} \]
1/96*(3*(3*a^2*b*c^3*d^2 + 7*a^3*c^2*d^3 + (32*a^2*b - 15*b^3)*c^5 - (32*a ^3 - 5*a*b^2)*c^4*d)*((a*x + b)/(c*x + d))^(11/4) - 6*(17*a^3*b*c^2*d^2 - 3*a^4*c*d^3 + (32*a^3*b - 21*a*b^3)*c^4 - (32*a^4 - 7*a^2*b^2)*c^3*d)*((a* x + b)/(c*x + d))^(7/4) - (3*a^4*b*c*d^2 + 7*a^5*d^3 - (96*a^4*b - 113*a^2 *b^3)*c^3 + 3*(32*a^5 - 41*a^3*b^2)*c^2*d)*((a*x + b)/(c*x + d))^(3/4))/(a ^6*c^2 - 3*(a*x + b)*a^5*c^3/(c*x + d) + 3*(a*x + b)^2*a^4*c^4/(c*x + d)^2 - (a*x + b)^3*a^3*c^5/(c*x + d)^3) - 1/128*(3*a^2*b*c*d^2 + 7*a^3*d^3 + ( 32*a^2*b - 15*b^3)*c^3 - (32*a^3 - 5*a*b^2)*c^2*d)*(2*arctan(sqrt(c)*((a*x + b)/(c*x + d))^(1/4)/sqrt(sqrt(a)*sqrt(c)))/(sqrt(sqrt(a)*sqrt(c))*sqrt( c)) + log((sqrt(c)*((a*x + b)/(c*x + d))^(1/4) - sqrt(sqrt(a)*sqrt(c)))/(s qrt(c)*((a*x + b)/(c*x + d))^(1/4) + sqrt(sqrt(a)*sqrt(c))))/(sqrt(sqrt(a) *sqrt(c))*sqrt(c)))/(a^3*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 1490 vs. \(2 (301) = 602\).
Time = 1.35 (sec) , antiderivative size = 1490, normalized size of antiderivative = 4.53 \[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]
-1/768*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)*(6*sqrt(2)*(32*a^2*b^2*c^4 - 15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a*b^3*c^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^ 2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/c)^ (1/4) + 2*((a*x + b)/(c*x + d))^(1/4))/(-a/c)^(1/4))/((-a*c^3)^(1/4)*a^3*c ^2) + 6*sqrt(2)*(32*a^2*b^2*c^4 - 15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a*b^3*c ^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*arc tan(-1/2*sqrt(2)*(sqrt(2)*(-a/c)^(1/4) - 2*((a*x + b)/(c*x + d))^(1/4))/(- a/c)^(1/4))/((-a*c^3)^(1/4)*a^3*c^2) - 3*sqrt(2)*(32*a^2*b^2*c^4 - 15*b^4* c^4 - 64*a^3*b*c^3*d + 20*a*b^3*c^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*log(sqrt(2)*((a*x + b)/(c*x + d))^(1/4)*(-a/ c)^(1/4) + sqrt((a*x + b)/(c*x + d)) + sqrt(-a/c))/((-a*c^3)^(1/4)*a^3*c^2 ) + 3*sqrt(2)*(32*a^2*b^2*c^4 - 15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a*b^3*c^3 *d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*log(- sqrt(2)*((a*x + b)/(c*x + d))^(1/4)*(-a/c)^(1/4) + sqrt((a*x + b)/(c*x + d )) + sqrt(-a/c))/((-a*c^3)^(1/4)*a^3*c^2) - 8*(96*a^4*b^2*c^4*((a*x + b)/( c*x + d))^(3/4) - 113*a^2*b^4*c^4*((a*x + b)/(c*x + d))^(3/4) - 192*(a*x + b)*a^3*b^2*c^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 126*(a*x + b)*a*b^ 4*c^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 96*(a*x + b)^2*a^2*b^2*c^6*( (a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 45*(a*x + b)^2*b^4*c^6*((a*x + b) /(c*x + d))^(3/4)/(c*x + d)^2 - 192*a^5*b*c^3*d*((a*x + b)/(c*x + d))^(...
Time = 8.91 (sec) , antiderivative size = 1566, normalized size of antiderivative = 4.76 \[ \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]
((((b + a*x)/(d + c*x))^(11/4)*((7*a^3*d^3)/32 - (15*b^3*c^3)/32 + a^2*b*c ^3 - a^3*c^2*d + (5*a*b^2*c^2*d)/32 + (3*a^2*b*c*d^2)/32))/a^6 + (((b + a* x)/(d + c*x))^(7/4)*((3*a^3*d^3)/16 + (21*b^3*c^3)/16 - 2*a^2*b*c^3 + 2*a^ 3*c^2*d - (7*a*b^2*c^2*d)/16 - (17*a^2*b*c*d^2)/16))/(a^5*c) - (((b + a*x) /(d + c*x))^(3/4)*((7*a^3*d^3)/96 + (113*b^3*c^3)/96 - a^2*b*c^3 + a^3*c^2 *d - (41*a*b^2*c^2*d)/32 + (a^2*b*c*d^2)/32))/(a^4*c^2))/((3*c^2*(b + a*x) ^2)/(a^2*(d + c*x)^2) - (c^3*(b + a*x)^3)/(a^3*(d + c*x)^3) - (3*c*(b + a* x))/(a*(d + c*x)) + 1) - (atan((c^(1/2)*(a*d - b*c)*((b + a*x)/(d + c*x))^ (1/4)*(7*a^2*d^2 - 32*a^2*c^2 + 15*b^2*c^2 + 10*a*b*c*d)*(225*b^6*c^(13/2) - 960*a^2*b^4*c^(13/2) + 1024*a^4*b^2*c^(13/2) + 49*a^6*c^(1/2)*d^6 - 448 *a^6*c^(5/2)*d^4 + 1024*a^6*c^(9/2)*d^2 + 42*a^5*b*c^(3/2)*d^5 + 256*a^5*b *c^(7/2)*d^3 + 1280*a^3*b^3*c^(11/2)*d + 79*a^4*b^2*c^(5/2)*d^4 - 180*a^3* b^3*c^(7/2)*d^3 - 65*a^2*b^4*c^(9/2)*d^2 - 128*a^4*b^2*c^(9/2)*d^2 - 150*a *b^5*c^(11/2)*d - 2048*a^5*b*c^(11/2)*d))/(a^(1/4)*(21600*a^2*b^7*c^(39/4) - 3375*b^9*c^(39/4) - 46080*a^4*b^5*c^(39/4) + 32768*a^6*b^3*c^(39/4) + 3 43*a^9*c^(3/4)*d^9 - 4704*a^9*c^(11/4)*d^7 + 21504*a^9*c^(19/4)*d^5 - 3276 8*a^9*c^(27/4)*d^3 + 441*a^8*b*c^(7/4)*d^8 + 672*a^8*b*c^(15/4)*d^6 - 3379 2*a^8*b*c^(23/4)*d^4 + 98304*a^8*b*c^(31/4)*d^2 - 36000*a^3*b^6*c^(35/4)*d + 107520*a^5*b^4*c^(35/4)*d - 98304*a^7*b^2*c^(35/4)*d + 924*a^7*b^2*c^(1 1/4)*d^7 - 1548*a^6*b^3*c^(15/4)*d^6 - 1230*a^5*b^4*c^(19/4)*d^5 - 3552...