Integrand size = 30, antiderivative size = 168 \[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx=\frac {2 (b e-a f) x \left (c+d x^2\right )^{15/4}}{17 a (b c-a d) \left (a+b x^2\right )^{17/4}}+\frac {(15 b c e-17 a d e+2 a c f) x \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{17/4} \left (c+d x^2\right )^{15/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {17}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (c+d x^2\right )}\right )}{17 a c (b c-a d) \left (a+b x^2\right )^{17/4}} \] Output:
2/17*(-a*f+b*e)*x*(d*x^2+c)^(15/4)/a/(-a*d+b*c)/(b*x^2+a)^(17/4)+1/17*(2*a *c*f-17*a*d*e+15*b*c*e)*x*(c*(b*x^2+a)/a/(d*x^2+c))^(17/4)*(d*x^2+c)^(15/4 )*hypergeom([1/2, 17/4],[3/2],-(-a*d+b*c)*x^2/a/(d*x^2+c))/a/c/(-a*d+b*c)/ (b*x^2+a)^(17/4)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 12.58 (sec) , antiderivative size = 1387, normalized size of antiderivative = 8.26 \[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx =\text {Too large to display} \] Input:
Integrate[((c + d*x^2)^(11/4)*(e + f*x^2))/(a + b*x^2)^(21/4),x]
Output:
(-2*x*(10395*a*b^5*c^6*e*(a + b*x^2)^4*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x ^2)/a), -((d*x^2)/c)] - 1386*a^2*b^4*c^5*d*e*(a + b*x^2)^4*AppellF1[1/2, 1 /4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 11781*a^3*b^3*c^4*d^2*e*(a + b *x^2)^4*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 1386*a^ 2*b^4*c^6*f*(a + b*x^2)^4*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d* x^2)/c)] + 1386*a^3*b^3*c^5*d*f*(a + b*x^2)^4*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - (c + d*x^2)*(585*a^4*(b*c - a*d)^4*(b*e - a *f) + 45*a^3*(b*c - a*d)^3*(15*b^2*c*e - 39*a^2*d*f + 2*a*b*(11*d*e + c*f) )*(a + b*x^2) + 5*a^2*(b*c - a*d)^2*(165*b^3*c^2*e - 351*a^3*d^2*f + 9*a^2 *b*d*(5*d*e + 4*c*f) + a*b^2*c*(83*d*e + 22*c*f))*(a + b*x^2)^2 + a*(-(b*c ) + a*d)*(-1155*b^4*c^3*e + 585*a^4*d^3*f + 10*a^2*b^2*c*d*(26*d*e - 11*c* f) + 90*a^3*b*d^2*(2*d*e - c*f) - 22*a*b^3*c^2*(-22*d*e + 7*c*f))*(a + b*x ^2)^3 + 231*b^4*c^3*(15*b*c*e - 17*a*d*e + 2*a*c*f)*(a + b*x^2)^4)*(6*a*c* AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(a*d*Appell F1[3/2, 1/4, 5/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4 , 1/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) + 2310*b^5*c^4*d*e*x^2*(a + b*x^ 2)^4*(1 + (b*x^2)/a)^(1/4)*(1 + (d*x^2)/c)^(1/4)*AppellF1[3/2, 1/4, 1/4, 5 /2, -((b*x^2)/a), -((d*x^2)/c)]*(6*a*c*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x ^2)/a), -((d*x^2)/c)] - x^2*(a*d*AppellF1[3/2, 1/4, 5/4, 5/2, -((b*x^2)/a) , -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1/4, 5/2, -((b*x^2)/a), -((d*x...
Leaf count is larger than twice the leaf count of optimal. \(454\) vs. \(2(168)=336\).
Time = 0.71 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.70, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {401, 27, 401, 27, 401, 27, 402, 27, 294}
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}-\frac {2 \int -\frac {\left (d x^2+c\right )^{7/4} \left (d (4 b e+13 a f) x^2+c (15 b e+2 a f)\right )}{2 \left (b x^2+a\right )^{17/4}}dx}{17 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (d x^2+c\right )^{7/4} \left (d (4 b e+13 a f) x^2+c (15 b e+2 a f)\right )}{\left (b x^2+a\right )^{17/4}}dx}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}-\frac {2 \int -\frac {\left (d x^2+c\right )^{3/4} \left (d \left (117 d f a^2+36 b d e a+8 b c f a+60 b^2 c e\right ) x^2+c (11 b c (15 b e+2 a f)+2 a d (4 b e+13 a f))\right )}{2 \left (b x^2+a\right )^{13/4}}dx}{13 a b}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d x^2+c\right )^{3/4} \left (d \left (117 d f a^2+36 b d e a+8 b c f a+60 b^2 c e\right ) x^2+c (11 b c (15 b e+2 a f)+2 a d (4 b e+13 a f))\right )}{\left (b x^2+a\right )^{13/4}}dx}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (c+d x^2\right )^{3/4} \left (-117 a^3 d^2 f-18 a^2 b d (2 d e-c f)-2 a b^2 c (26 d e-11 c f)+165 b^3 c^2 e\right )}{9 a b \left (a+b x^2\right )^{9/4}}-\frac {2 \int -\frac {d \left (585 d^2 f a^3+36 b d (5 d e+4 c f) a^2+4 b^2 c (83 d e+22 c f) a+660 b^3 c^2 e\right ) x^2+c \left (234 d^2 f a^3+18 b d (4 d e+11 c f) a^2+22 b^2 c (8 d e+7 c f) a+1155 b^3 c^2 e\right )}{2 \left (b x^2+a\right )^{9/4} \sqrt [4]{d x^2+c}}dx}{9 a b}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {d \left (585 d^2 f a^3+36 b d (5 d e+4 c f) a^2+4 b^2 c (83 d e+22 c f) a+660 b^3 c^2 e\right ) x^2+c \left (234 d^2 f a^3+18 b d (4 d e+11 c f) a^2+22 b^2 c (8 d e+7 c f) a+1155 b^3 c^2 e\right )}{\left (b x^2+a\right )^{9/4} \sqrt [4]{d x^2+c}}dx}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} \left (-117 a^3 d^2 f-18 a^2 b d (2 d e-c f)-2 a b^2 c (26 d e-11 c f)+165 b^3 c^2 e\right )}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 x \left (c+d x^2\right )^{3/4} \left (-585 a^4 d^3 f-90 a^3 b d^2 (2 d e-c f)-10 a^2 b^2 c d (26 d e-11 c f)-22 a b^3 c^2 (22 d e-7 c f)+1155 b^4 c^3 e\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}-\frac {2 \int -\frac {231 b^3 c^3 (15 b c e-17 a d e+2 a c f)}{2 \left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} \left (-117 a^3 d^2 f-18 a^2 b d (2 d e-c f)-2 a b^2 c (26 d e-11 c f)+165 b^3 c^2 e\right )}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {231 b^3 c^3 (2 a c f-17 a d e+15 b c e) \int \frac {1}{\left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}+\frac {2 x \left (c+d x^2\right )^{3/4} \left (-585 a^4 d^3 f-90 a^3 b d^2 (2 d e-c f)-10 a^2 b^2 c d (26 d e-11 c f)-22 a b^3 c^2 (22 d e-7 c f)+1155 b^4 c^3 e\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} \left (-117 a^3 d^2 f-18 a^2 b d (2 d e-c f)-2 a b^2 c (26 d e-11 c f)+165 b^3 c^2 e\right )}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
| \(\Big \downarrow \) 294 |
| \(\displaystyle \frac {\frac {\frac {2 x \left (c+d x^2\right )^{3/4} \left (-117 a^3 d^2 f-18 a^2 b d (2 d e-c f)-2 a b^2 c (26 d e-11 c f)+165 b^3 c^2 e\right )}{9 a b \left (a+b x^2\right )^{9/4}}+\frac {\frac {2 x \left (c+d x^2\right )^{3/4} \left (-585 a^4 d^3 f-90 a^3 b d^2 (2 d e-c f)-10 a^2 b^2 c d (26 d e-11 c f)-22 a b^3 c^2 (22 d e-7 c f)+1155 b^4 c^3 e\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}+\frac {231 b^3 c^2 x \left (c+d x^2\right )^{3/4} \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{5/4} (2 a c f-17 a d e+15 b c e) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}}{9 a b}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b c (2 a f+15 b e)-a d (13 a f+4 b e))}{13 a b \left (a+b x^2\right )^{13/4}}}{17 a b}+\frac {2 x \left (c+d x^2\right )^{11/4} (b e-a f)}{17 a b \left (a+b x^2\right )^{17/4}}\) |
Input:
Int[((c + d*x^2)^(11/4)*(e + f*x^2))/(a + b*x^2)^(21/4),x]
Output:
(2*(b*e - a*f)*x*(c + d*x^2)^(11/4))/(17*a*b*(a + b*x^2)^(17/4)) + ((2*(b* c*(15*b*e + 2*a*f) - a*d*(4*b*e + 13*a*f))*x*(c + d*x^2)^(7/4))/(13*a*b*(a + b*x^2)^(13/4)) + ((2*(165*b^3*c^2*e - 117*a^3*d^2*f - 2*a*b^2*c*(26*d*e - 11*c*f) - 18*a^2*b*d*(2*d*e - c*f))*x*(c + d*x^2)^(3/4))/(9*a*b*(a + b* x^2)^(9/4)) + ((2*(1155*b^4*c^3*e - 585*a^4*d^3*f - 10*a^2*b^2*c*d*(26*d*e - 11*c*f) - 22*a*b^3*c^2*(22*d*e - 7*c*f) - 90*a^3*b*d^2*(2*d*e - c*f))*x *(c + d*x^2)^(3/4))/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)) + (231*b^3*c^2*(15 *b*c*e - 17*a*d*e + 2*a*c*f)*x*((c*(a + b*x^2))/(a*(c + d*x^2)))^(5/4)*(c + d*x^2)^(3/4)*Hypergeometric2F1[1/2, 5/4, 3/2, -(((b*c - a*d)*x^2)/(a*(c + d*x^2)))])/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)))/(9*a*b))/(13*a*b))/(17*a *b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[x*((a + b*x^2)^p/(c*(c*((a + b*x^2)/(a*(c + d*x^2))))^p*(c + d*x^2)^(1/2 + p)))*Hypergeometric2F1[1/2, -p, 3/2, (-(b*c - a*d))*(x^2/(a*(c + d*x^2))) ], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
\[\int \frac {\left (x^{2} d +c \right )^{\frac {11}{4}} \left (f \,x^{2}+e \right )}{\left (b \,x^{2}+a \right )^{\frac {21}{4}}}d x\]
Input:
int((d*x^2+c)^(11/4)*(f*x^2+e)/(b*x^2+a)^(21/4),x)
Output:
int((d*x^2+c)^(11/4)*(f*x^2+e)/(b*x^2+a)^(21/4),x)
\[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {11}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {21}{4}}} \,d x } \] Input:
integrate((d*x^2+c)^(11/4)*(f*x^2+e)/(b*x^2+a)^(21/4),x, algorithm="fricas ")
Output:
integral((d^2*f*x^6 + (d^2*e + 2*c*d*f)*x^4 + c^2*e + (2*c*d*e + c^2*f)*x^ 2)*(b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)/(b^6*x^12 + 6*a*b^5*x^10 + 15*a^2*b ^4*x^8 + 20*a^3*b^3*x^6 + 15*a^4*b^2*x^4 + 6*a^5*b*x^2 + a^6), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)**(11/4)*(f*x**2+e)/(b*x**2+a)**(21/4),x)
Output:
Timed out
\[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {11}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {21}{4}}} \,d x } \] Input:
integrate((d*x^2+c)^(11/4)*(f*x^2+e)/(b*x^2+a)^(21/4),x, algorithm="maxima ")
Output:
integrate((d*x^2 + c)^(11/4)*(f*x^2 + e)/(b*x^2 + a)^(21/4), x)
\[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {11}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {21}{4}}} \,d x } \] Input:
integrate((d*x^2+c)^(11/4)*(f*x^2+e)/(b*x^2+a)^(21/4),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(11/4)*(f*x^2 + e)/(b*x^2 + a)^(21/4), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{11/4}\,\left (f\,x^2+e\right )}{{\left (b\,x^2+a\right )}^{21/4}} \,d x \] Input:
int(((c + d*x^2)^(11/4)*(e + f*x^2))/(a + b*x^2)^(21/4),x)
Output:
int(((c + d*x^2)^(11/4)*(e + f*x^2))/(a + b*x^2)^(21/4), x)
\[ \int \frac {\left (c+d x^2\right )^{11/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{21/4}} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^(11/4)*(f*x^2+e)/(b*x^2+a)^(21/4),x)
Output:
int((c + d*x**2)**(3/4)/((a + b*x**2)**(1/4)*a**5 + 5*(a + b*x**2)**(1/4)* a**4*b*x**2 + 10*(a + b*x**2)**(1/4)*a**3*b**2*x**4 + 10*(a + b*x**2)**(1/ 4)*a**2*b**3*x**6 + 5*(a + b*x**2)**(1/4)*a*b**4*x**8 + (a + b*x**2)**(1/4 )*b**5*x**10),x)*c**2*e + int(((c + d*x**2)**(3/4)*x**6)/((a + b*x**2)**(1 /4)*a**5 + 5*(a + b*x**2)**(1/4)*a**4*b*x**2 + 10*(a + b*x**2)**(1/4)*a**3 *b**2*x**4 + 10*(a + b*x**2)**(1/4)*a**2*b**3*x**6 + 5*(a + b*x**2)**(1/4) *a*b**4*x**8 + (a + b*x**2)**(1/4)*b**5*x**10),x)*d**2*f + 2*int(((c + d*x **2)**(3/4)*x**4)/((a + b*x**2)**(1/4)*a**5 + 5*(a + b*x**2)**(1/4)*a**4*b *x**2 + 10*(a + b*x**2)**(1/4)*a**3*b**2*x**4 + 10*(a + b*x**2)**(1/4)*a** 2*b**3*x**6 + 5*(a + b*x**2)**(1/4)*a*b**4*x**8 + (a + b*x**2)**(1/4)*b**5 *x**10),x)*c*d*f + int(((c + d*x**2)**(3/4)*x**4)/((a + b*x**2)**(1/4)*a** 5 + 5*(a + b*x**2)**(1/4)*a**4*b*x**2 + 10*(a + b*x**2)**(1/4)*a**3*b**2*x **4 + 10*(a + b*x**2)**(1/4)*a**2*b**3*x**6 + 5*(a + b*x**2)**(1/4)*a*b**4 *x**8 + (a + b*x**2)**(1/4)*b**5*x**10),x)*d**2*e + int(((c + d*x**2)**(3/ 4)*x**2)/((a + b*x**2)**(1/4)*a**5 + 5*(a + b*x**2)**(1/4)*a**4*b*x**2 + 1 0*(a + b*x**2)**(1/4)*a**3*b**2*x**4 + 10*(a + b*x**2)**(1/4)*a**2*b**3*x* *6 + 5*(a + b*x**2)**(1/4)*a*b**4*x**8 + (a + b*x**2)**(1/4)*b**5*x**10),x )*c**2*f + 2*int(((c + d*x**2)**(3/4)*x**2)/((a + b*x**2)**(1/4)*a**5 + 5* (a + b*x**2)**(1/4)*a**4*b*x**2 + 10*(a + b*x**2)**(1/4)*a**3*b**2*x**4 + 10*(a + b*x**2)**(1/4)*a**2*b**3*x**6 + 5*(a + b*x**2)**(1/4)*a*b**4*x*...