\(\int \frac {(a+b x^4)^{9/4}}{(c+d x^4)^2} \, dx\) [121]

Optimal result

Integrand size = 21, antiderivative size = 353 \[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac {\sqrt {a} b^{3/2} (3 b c-a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac {3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^2}-\frac {3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^2} \] Output:

1/4*b*(-a*d+3*b*c)*x*(b*x^4+a)^(1/4)/c/d^2-1/4*(-a*d+b*c)*x*(b*x^4+a)^(5/4 
)/c/d/(d*x^4+c)-1/4*a^(1/2)*b^(3/2)*(-a*d+3*b*c)*(1+a/b/x^4)^(3/4)*x^3*Inv 
erseJacobiAM(1/2*arccot(b^(1/2)*x^2/a^(1/2)),2^(1/2))/c/d^2/(b*x^4+a)^(3/4 
)-3/8*(-a*d+b*c)*(a*d+2*b*c)*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2)*EllipticP 
i(b^(1/4)*x/(b*x^4+a)^(1/4),-(-a*d+b*c)^(1/2)/b^(1/2)/c^(1/2),I)/b^(1/4)/c 
^2/d^2-3/8*(-a*d+b*c)*(a*d+2*b*c)*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2)*Elli 
pticPi(b^(1/4)*x/(b*x^4+a)^(1/4),(-a*d+b*c)^(1/2)/b^(1/2)/c^(1/2),I)/b^(1/ 
4)/c^2/d^2
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.54 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\frac {2 b \left (-3 b^2 c^2+3 a b c d+a^2 d^2\right ) x^5 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+\frac {5 c \left (-5 a c x \left (4 a^3 d^2+a^2 b d^2 x^4+b^3 c x^4 \left (3 c+2 d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^5 \left (a+b x^4\right ) \left (-2 a b c d+a^2 d^2+b^2 c \left (3 c+2 d x^4\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}}{20 c^2 d^2 \left (a+b x^4\right )^{3/4}} \] Input:

Integrate[(a + b*x^4)^(9/4)/(c + d*x^4)^2,x]
 

Output:

(2*b*(-3*b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^5*(1 + (b*x^4)/a)^(3/4)*AppellF1 
[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + (5*c*(-5*a*c*x*(4*a^3*d^2 
 + a^2*b*d^2*x^4 + b^3*c*x^4*(3*c + 2*d*x^4))*AppellF1[1/4, 3/4, 1, 5/4, - 
((b*x^4)/a), -((d*x^4)/c)] + x^5*(a + b*x^4)*(-2*a*b*c*d + a^2*d^2 + b^2*c 
*(3*c + 2*d*x^4))*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4 
)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/(( 
c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] 
+ x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b* 
c*AppellF1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))))/(20*c^2*d^2*( 
a + b*x^4)^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 1.10 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.88, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {930, 1025, 27, 404, 768, 858, 807, 229, 923, 925, 27, 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.

\(\displaystyle \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx\)

\(\Big \downarrow \) 930

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

\(\Big \downarrow \) 1025

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 404

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

\(\Big \downarrow \) 768

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

\(\Big \downarrow \) 858

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 229

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

\(\Big \downarrow \) 923

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1542

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

Input:

Int[(a + b*x^4)^(9/4)/(c + d*x^4)^2,x]
 

Output:

-1/4*((b*c - a*d)*x*(a + b*x^4)^(5/4))/(c*d*(c + d*x^4)) + ((b*(3*b*c - a* 
d)*x*(a + b*x^4)^(1/4))/d - ((Sqrt[a]*b^(3/2)*(3*b*c - a*d)*(1 + a/(b*x^4) 
)^(3/4)*x^3*EllipticF[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(a + b*x^4)^(3/ 
4) + 3*(b*c - a*d)*(2*b*c + a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*(Elli 
pticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4 
)^(1/4)], -1]/(2*b^(1/4)*c) + EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]) 
, ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1]/(2*b^(1/4)*c)))/d)/(4*c*d)
 

Defintions of rubi rules used

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 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 404

Int[((e_) + (f_.)*(x_)^4)/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4 
)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^4)^(3/4), x] 
, x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[(a + b*x^4)^(1/4)/(c + d*x^4), x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x]
 

rule 768

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 
/4)/(a + b*x^4)^(3/4))   Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ 
[{a, b}, x]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 858

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + 
b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int 
egerQ[m]
 

rule 923

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

rule 925

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

rule 930

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

rule 1025

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( 
f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( 
b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1))   Int[(a + b*x 
^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e 
- a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 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]

\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{\left (d \,x^{4}+c \right )^{2}}d x\]

Input:

int((b*x^4+a)^(9/4)/(d*x^4+c)^2,x)
 

Output:

int((b*x^4+a)^(9/4)/(d*x^4+c)^2,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\int \frac {\left (a + b x^{4}\right )^{\frac {9}{4}}}{\left (c + d x^{4}\right )^{2}}\, dx \] Input:

integrate((b*x**4+a)**(9/4)/(d*x**4+c)**2,x)
 

Output:

Integral((a + b*x**4)**(9/4)/(c + d*x**4)**2, x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{{\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:

integrate((b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^4 + a)^(9/4)/(d*x^4 + c)^2, x)
 

Giac [F]

\[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{{\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:

integrate((b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="giac")
 

Output:

integrate((b*x^4 + a)^(9/4)/(d*x^4 + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{9/4}}{{\left (d\,x^4+c\right )}^2} \,d x \] Input:

int((a + b*x^4)^(9/4)/(c + d*x^4)^2,x)
 

Output:

int((a + b*x^4)^(9/4)/(c + d*x^4)^2, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx=\text {too large to display} \] Input:

int((b*x^4+a)^(9/4)/(d*x^4+c)^2,x)
 

Output:

( - 6*(a + b*x**4)**(1/4)*a**2*b*d*x + 5*(a + b*x**4)**(1/4)*a*b**2*c*x + 
3*(a + b*x**4)**(1/4)*a*b**2*d*x**5 - 2*(a + b*x**4)**(1/4)*b**3*c*x**5 + 
18*int((a + b*x**4)**(1/4)/(3*a**2*c**2*d + 6*a**2*c*d**2*x**4 + 3*a**2*d* 
*3*x**8 - 2*a*b*c**3 - a*b*c**2*d*x**4 + 4*a*b*c*d**2*x**8 + 3*a*b*d**3*x* 
*12 - 2*b**2*c**3*x**4 - 4*b**2*c**2*d*x**8 - 2*b**2*c*d**2*x**12),x)*a**5 
*c*d**3 + 18*int((a + b*x**4)**(1/4)/(3*a**2*c**2*d + 6*a**2*c*d**2*x**4 + 
 3*a**2*d**3*x**8 - 2*a*b*c**3 - a*b*c**2*d*x**4 + 4*a*b*c*d**2*x**8 + 3*a 
*b*d**3*x**12 - 2*b**2*c**3*x**4 - 4*b**2*c**2*d*x**8 - 2*b**2*c*d**2*x**1 
2),x)*a**5*d**4*x**4 - 6*int((a + b*x**4)**(1/4)/(3*a**2*c**2*d + 6*a**2*c 
*d**2*x**4 + 3*a**2*d**3*x**8 - 2*a*b*c**3 - a*b*c**2*d*x**4 + 4*a*b*c*d** 
2*x**8 + 3*a*b*d**3*x**12 - 2*b**2*c**3*x**4 - 4*b**2*c**2*d*x**8 - 2*b**2 
*c*d**2*x**12),x)*a**4*b*c**2*d**2 - 6*int((a + b*x**4)**(1/4)/(3*a**2*c** 
2*d + 6*a**2*c*d**2*x**4 + 3*a**2*d**3*x**8 - 2*a*b*c**3 - a*b*c**2*d*x**4 
 + 4*a*b*c*d**2*x**8 + 3*a*b*d**3*x**12 - 2*b**2*c**3*x**4 - 4*b**2*c**2*d 
*x**8 - 2*b**2*c*d**2*x**12),x)*a**4*b*c*d**3*x**4 - 19*int((a + b*x**4)** 
(1/4)/(3*a**2*c**2*d + 6*a**2*c*d**2*x**4 + 3*a**2*d**3*x**8 - 2*a*b*c**3 
- a*b*c**2*d*x**4 + 4*a*b*c*d**2*x**8 + 3*a*b*d**3*x**12 - 2*b**2*c**3*x** 
4 - 4*b**2*c**2*d*x**8 - 2*b**2*c*d**2*x**12),x)*a**3*b**2*c**3*d - 19*int 
((a + b*x**4)**(1/4)/(3*a**2*c**2*d + 6*a**2*c*d**2*x**4 + 3*a**2*d**3*x** 
8 - 2*a*b*c**3 - a*b*c**2*d*x**4 + 4*a*b*c*d**2*x**8 + 3*a*b*d**3*x**12...