\(\int \frac {(a-b x^2)^{7/4}}{c+d x^2} \, dx\) [468]

Optimal result

Integrand size = 22, antiderivative size = 256 \[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=-\frac {2 b x \left (a-b x^2\right )^{3/4}}{5 d}-\frac {2 \sqrt {a} \sqrt {b} (5 b c+8 a d) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 d^2 \sqrt [4]{a-b x^2}}+\frac {\sqrt [4]{a} (b c+a d)^{3/2} \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{5/2} x}-\frac {\sqrt [4]{a} (b c+a d)^{3/2} \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{5/2} x} \] Output:

-2/5*b*x*(-b*x^2+a)^(3/4)/d-2/5*a^(1/2)*b^(1/2)*(8*a*d+5*b*c)*(1-b*x^2/a)^ 
(1/4)*EllipticE(sin(1/2*arcsin(b^(1/2)*x/a^(1/2))),2^(1/2))/d^2/(-b*x^2+a) 
^(1/4)+a^(1/4)*(a*d+b*c)^(3/2)*(b*x^2/a)^(1/2)*EllipticPi((-b*x^2+a)^(1/4) 
/a^(1/4),-a^(1/2)*d^(1/2)/(a*d+b*c)^(1/2),I)/d^(5/2)/x-a^(1/4)*(a*d+b*c)^( 
3/2)*(b*x^2/a)^(1/2)*EllipticPi((-b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/( 
a*d+b*c)^(1/2),I)/d^(5/2)/x
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 9.35 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=-\frac {x \left (\frac {b (5 b c+8 a d) x^2 \sqrt [4]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (3 a c \left (5 a^2 d-2 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (-a+b x^2\right ) \left (c+d x^2\right ) \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )-b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{15 d \sqrt [4]{a-b x^2}} \] Input:

Integrate[(a - b*x^2)^(7/4)/(c + d*x^2),x]
 

Output:

-1/15*(x*((b*(5*b*c + 8*a*d)*x^2*(1 - (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 
1, 5/2, (b*x^2)/a, -((d*x^2)/c)])/c + (6*(3*a*c*(5*a^2*d - 2*a*b*d*x^2 + 2 
*b^2*x^2*(c + d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, -((d*x^2)/c)] 
+ b*x^2*(-a + b*x^2)*(c + d*x^2)*(-4*a*d*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2 
)/a, -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, -((d*x^2)/c 
)])))/((c + d*x^2)*(-6*a*c*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, -((d*x^2) 
/c)] + x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, -((d*x^2)/c)] - b* 
c*AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, -((d*x^2)/c)])))))/(d*(a - b*x^2)^ 
(1/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {301, 211, 227, 226, 301, 227, 226, 310, 25, 993, 1542}

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.

\(\displaystyle \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {(a d+b c) \int \frac {\left (a-b x^2\right )^{3/4}}{d x^2+c}dx}{d}-\frac {b \int \left (a-b x^2\right )^{3/4}dx}{d}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(a d+b c) \int \frac {\left (a-b x^2\right )^{3/4}}{d x^2+c}dx}{d}-\frac {b \left (\frac {3}{5} a \int \frac {1}{\sqrt [4]{a-b x^2}}dx+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {(a d+b c) \int \frac {\left (a-b x^2\right )^{3/4}}{d x^2+c}dx}{d}-\frac {b \left (\frac {3 a \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{5 \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 226

\(\displaystyle \frac {(a d+b c) \int \frac {\left (a-b x^2\right )^{3/4}}{d x^2+c}dx}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {(a d+b c) \left (\frac {(a d+b c) \int \frac {1}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{d}-\frac {b \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{d}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {(a d+b c) \left (\frac {(a d+b c) \int \frac {1}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{d}-\frac {b \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{d \sqrt [4]{a-b x^2}}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 226

\(\displaystyle \frac {(a d+b c) \left (\frac {(a d+b c) \int \frac {1}{\sqrt [4]{a-b x^2} \left (d x^2+c\right )}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a-b x^2}}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {(a d+b c) \left (\frac {2 \sqrt {\frac {b x^2}{a}} (a d+b c) \int -\frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {a-b x^2}{a}} \left (b c+a d-d \left (a-b x^2\right )\right )}d\sqrt [4]{a-b x^2}}{d x}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a-b x^2}}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(a d+b c) \left (-\frac {2 \sqrt {\frac {b x^2}{a}} (a d+b c) \int \frac {\sqrt {a-b x^2}}{\sqrt {1-\frac {a-b x^2}{a}} \left (b c+a d-d \left (a-b x^2\right )\right )}d\sqrt [4]{a-b x^2}}{d x}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a-b x^2}}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {(a d+b c) \left (\frac {2 \sqrt {\frac {b x^2}{a}} (a d+b c) \left (\frac {\int \frac {1}{\left (\sqrt {b c+a d}+\sqrt {d} \sqrt {a-b x^2}\right ) \sqrt {1-\frac {a-b x^2}{a}}}d\sqrt [4]{a-b x^2}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {b c+a d}-\sqrt {d} \sqrt {a-b x^2}\right ) \sqrt {1-\frac {a-b x^2}{a}}}d\sqrt [4]{a-b x^2}}{2 \sqrt {d}}\right )}{d x}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a-b x^2}}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {(a d+b c) \left (\frac {2 \sqrt {\frac {b x^2}{a}} (a d+b c) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d+b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d+b c}}\right )}{d x}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a-b x^2}}\right )}{d}-\frac {b \left (\frac {6 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \sqrt [4]{a-b x^2}}+\frac {2}{5} x \left (a-b x^2\right )^{3/4}\right )}{d}\)

Input:

Int[(a - b*x^2)^(7/4)/(c + d*x^2),x]
 

Output:

-((b*((2*x*(a - b*x^2)^(3/4))/5 + (6*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*Ellipti 
cE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*Sqrt[b]*(a - b*x^2)^(1/4))))/d) + 
 ((b*c + a*d)*((-2*Sqrt[a]*Sqrt[b]*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[ 
(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d*(a - b*x^2)^(1/4)) + (2*(b*c + a*d)*Sqrt[(b 
*x^2)/a]*((a^(1/4)*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[b*c + a*d]), ArcSin 
[(a - b*x^2)^(1/4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqrt[b*c + a*d]) - (a^(1/4)*E 
llipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[b*c + a*d], ArcSin[(a - b*x^2)^(1/4)/a^(1 
/4)], -1])/(2*Sqrt[d]*Sqrt[b*c + a*d])))/(d*x)))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 226

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] 
))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ 
[a, 0] && NegQ[b/a]
 

rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a + b*x^2)^(1/4)   Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] 
&& PosQ[a]
 

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 310

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[2*(Sqrt[(-b)*(x^2/a)]/x)   Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* 
x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

Maple [F]

\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {7}{4}}}{x^{2} d +c}d x\]

Input:

int((-b*x^2+a)^(7/4)/(d*x^2+c),x)
 

Output:

int((-b*x^2+a)^(7/4)/(d*x^2+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=\text {Timed out} \] Input:

integrate((-b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {7}{4}}}{c + d x^{2}}\, dx \] Input:

integrate((-b*x**2+a)**(7/4)/(d*x**2+c),x)
 

Output:

Integral((a - b*x**2)**(7/4)/(c + d*x**2), x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {7}{4}}}{d x^{2} + c} \,d x } \] Input:

integrate((-b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="maxima")
 

Output:

integrate((-b*x^2 + a)^(7/4)/(d*x^2 + c), x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {7}{4}}}{d x^{2} + c} \,d x } \] Input:

integrate((-b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="giac")
 

Output:

integrate((-b*x^2 + a)^(7/4)/(d*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{7/4}}{d\,x^2+c} \,d x \] Input:

int((a - b*x^2)^(7/4)/(c + d*x^2),x)
 

Output:

int((a - b*x^2)^(7/4)/(c + d*x^2), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{7/4}}{c+d x^2} \, dx=\frac {-2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b x +5 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{-b d \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c}d x \right ) a^{2} d +2 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{-b d \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c}d x \right ) a b c -8 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}} x^{2}}{-b d \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c}d x \right ) a b d -5 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}} x^{2}}{-b d \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c}d x \right ) b^{2} c}{5 d} \] Input:

int((-b*x^2+a)^(7/4)/(d*x^2+c),x)
 

Output:

( - 2*(a - b*x**2)**(3/4)*b*x + 5*int((a - b*x**2)**(3/4)/(a*c + a*d*x**2 
- b*c*x**2 - b*d*x**4),x)*a**2*d + 2*int((a - b*x**2)**(3/4)/(a*c + a*d*x* 
*2 - b*c*x**2 - b*d*x**4),x)*a*b*c - 8*int(((a - b*x**2)**(3/4)*x**2)/(a*c 
 + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a*b*d - 5*int(((a - b*x**2)**(3/4)*x 
**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*b**2*c)/(5*d)