\(\int \frac {(a-b x^2)^{7/4}}{(c+d x^2)^2} \, dx\) [469]

Optimal result

Integrand size = 22, antiderivative size = 305 \[ \int \frac {\left (a-b x^2\right )^{7/4}}{\left (c+d x^2\right )^2} \, dx=\frac {(b c+a d) x \left (a-b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (5 b c+a d) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d^2 \sqrt [4]{a-b x^2}}-\frac {\sqrt [4]{a} (5 b c-2 a d) \sqrt {b c+a d} \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{5/2} x}+\frac {\sqrt [4]{a} (5 b c-2 a d) \sqrt {b c+a d} \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{5/2} x} \] Output:

1/2*(a*d+b*c)*x*(-b*x^2+a)^(3/4)/c/d/(d*x^2+c)+1/2*a^(1/2)*b^(1/2)*(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 
))/c/d^2/(-b*x^2+a)^(1/4)-1/4*a^(1/4)*(-2*a*d+5*b*c)*(a*d+b*c)^(1/2)*(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)/c/d^(5/2)/x+1/4*a^(1/4)*(-2*a*d+5*b*c)*(a*d+b*c)^(1/2)*(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)/c/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 = 10.25 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a-b x^2\right )^{7/4}}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (b (5 b c+a d) x^2 \sqrt [4]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {c \left (36 a c \left (2 a^2 d-b^2 c x^2-a b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 (b c+a d) x^2 \left (-a+b x^2\right ) \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 c^2 d \sqrt [4]{a-b x^2}} \] Input:

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

Output:

(x*(b*(5*b*c + 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*(36*a*c*(2*a^2*d - b^2*c*x^2 - a*b*d*x^2)*App 
ellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, -((d*x^2)/c)] - 6*(b*c + a*d)*x^2*(-a + 
 b*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*AppellF 
1[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)])))))/(12*c^2*d*(a - b*x^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {315, 27, 405, 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.

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 405

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

\(\Big \downarrow \) 227

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

\(\Big \downarrow \) 226

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

\(\Big \downarrow \) 310

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

((b*c + a*d)*x*(a - b*x^2)^(3/4))/(2*c*d*(c + d*x^2)) - ((-2*Sqrt[a]*Sqrt[ 
b]*(5*b*c + a*d)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a 
]]/2, 2])/(d*(a - b*x^2)^(1/4)) + (2*(5*b*c - 2*a*d)*(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)*Ell 
ipticPi[(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))/(4*c*d)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 405

Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 
), x_Symbol] :> Simp[f/d   Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d 
 Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
 

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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*(a - b*x**2)**(3/4)*a*b*x + 4*int((a - b*x**2)**(3/4)/(2*a**2*c**2*d + 
4*a**2*c*d**2*x**2 + 2*a**2*d**3*x**4 + 5*a*b*c**3 + 8*a*b*c**2*d*x**2 + a 
*b*c*d**2*x**4 - 2*a*b*d**3*x**6 - 5*b**2*c**3*x**2 - 10*b**2*c**2*d*x**4 
- 5*b**2*c*d**2*x**6),x)*a**4*c*d**2 + 4*int((a - b*x**2)**(3/4)/(2*a**2*c 
**2*d + 4*a**2*c*d**2*x**2 + 2*a**2*d**3*x**4 + 5*a*b*c**3 + 8*a*b*c**2*d* 
x**2 + a*b*c*d**2*x**4 - 2*a*b*d**3*x**6 - 5*b**2*c**3*x**2 - 10*b**2*c**2 
*d*x**4 - 5*b**2*c*d**2*x**6),x)*a**4*d**3*x**2 + 12*int((a - b*x**2)**(3/ 
4)/(2*a**2*c**2*d + 4*a**2*c*d**2*x**2 + 2*a**2*d**3*x**4 + 5*a*b*c**3 + 8 
*a*b*c**2*d*x**2 + a*b*c*d**2*x**4 - 2*a*b*d**3*x**6 - 5*b**2*c**3*x**2 - 
10*b**2*c**2*d*x**4 - 5*b**2*c*d**2*x**6),x)*a**3*b*c**2*d + 12*int((a - b 
*x**2)**(3/4)/(2*a**2*c**2*d + 4*a**2*c*d**2*x**2 + 2*a**2*d**3*x**4 + 5*a 
*b*c**3 + 8*a*b*c**2*d*x**2 + a*b*c*d**2*x**4 - 2*a*b*d**3*x**6 - 5*b**2*c 
**3*x**2 - 10*b**2*c**2*d*x**4 - 5*b**2*c*d**2*x**6),x)*a**3*b*c*d**2*x**2 
 + 5*int((a - b*x**2)**(3/4)/(2*a**2*c**2*d + 4*a**2*c*d**2*x**2 + 2*a**2* 
d**3*x**4 + 5*a*b*c**3 + 8*a*b*c**2*d*x**2 + a*b*c*d**2*x**4 - 2*a*b*d**3* 
x**6 - 5*b**2*c**3*x**2 - 10*b**2*c**2*d*x**4 - 5*b**2*c*d**2*x**6),x)*a** 
2*b**2*c**3 + 5*int((a - b*x**2)**(3/4)/(2*a**2*c**2*d + 4*a**2*c*d**2*x** 
2 + 2*a**2*d**3*x**4 + 5*a*b*c**3 + 8*a*b*c**2*d*x**2 + a*b*c*d**2*x**4 - 
2*a*b*d**3*x**6 - 5*b**2*c**3*x**2 - 10*b**2*c**2*d*x**4 - 5*b**2*c*d**2*x 
**6),x)*a**2*b**2*c**2*d*x**2 + 8*int(((a - b*x**2)**(3/4)*x**4)/(2*a**...