3.2.99 \(\int \frac {(a+b x^4)^{9/4}}{c+d x^4} \, dx\) [199]

3.2.99.1 Optimal result

Integrand size = 21, antiderivative size = 316 \[ \int \frac {\left (a+b x^4\right )^{9/4}}{c+d x^4} \, dx=-\frac {b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac {\sqrt {a} b^{3/2} (6 b c-11 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac {(b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2}+\frac {(b c-a d)^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2} \]

output

-1/12*b*(-11*a*d+6*b*c)*x*(b*x^4+a)^(1/4)/d^2+1/6*b*x*(b*x^4+a)^(5/4)/d+1/ 
12*b^(3/2)*(-11*a*d+6*b*c)*(1+a/b/x^4)^(3/4)*x^3*(cos(1/2*arccot(x^2*b^(1/ 
2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccot(x^2*b^(1/2)/a^(1/2)))*EllipticF(sin(1 
/2*arccot(x^2*b^(1/2)/a^(1/2))),2^(1/2))*a^(1/2)/d^2/(b*x^4+a)^(3/4)+1/2*( 
-a*d+b*c)^2*EllipticPi(b^(1/4)*x/(b*x^4+a)^(1/4),-(-a*d+b*c)^(1/2)/b^(1/2) 
/c^(1/2),I)*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2)/b^(1/4)/c/d^2+1/2*(-a*d+b* 
c)^2*EllipticPi(b^(1/4)*x/(b*x^4+a)^(1/4),(-a*d+b*c)^(1/2)/b^(1/2)/c^(1/2) 
,I)*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2)/b^(1/4)/c/d^2
 

3.2.99.2 Mathematica [C] (warning: unable to verify)

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

Time = 10.65 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^4\right )^{9/4}}{c+d x^4} \, dx=\frac {x \left (5 b \left (a+b x^4\right ) \left (-6 b c+13 a d+2 b d x^4\right )+\frac {b \left (12 b^2 c^2-30 a b c d+23 a^2 d^2\right ) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}-\frac {25 a^2 c \left (6 b^2 c^2-13 a b c d+12 a^2 d^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{\left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{60 d^2 \left (a+b x^4\right )^{3/4}} \]

input

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

output

(x*(5*b*(a + b*x^4)*(-6*b*c + 13*a*d + 2*b*d*x^4) + (b*(12*b^2*c^2 - 30*a* 
b*c*d + 23*a^2*d^2)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, - 
((b*x^4)/a), -((d*x^4)/c)])/c - (25*a^2*c*(6*b^2*c^2 - 13*a*b*c*d + 12*a^2 
*d^2)*AppellF1[1/4, 3/4, 1, 5/4, -((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)])))))/(60*d^2*(a + b*x^4)^(3 
/4))
 

3.2.99.3 Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.91, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {933, 25, 1025, 25, 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.

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

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int -\frac {\sqrt [4]{b x^4+a} \left (b (6 b c-11 a d) x^4+a (b c-6 a d)\right )}{d x^4+c}dx}{6 d}+\frac {b x \left (a+b x^4\right )^{5/4}}{6 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\int \frac {\sqrt [4]{b x^4+a} \left (b (6 b c-11 a d) x^4+a (b c-6 a d)\right )}{d x^4+c}dx}{6 d}\)

\(\Big \downarrow \) 1025

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {\int -\frac {b \left (12 b^2 c^2-30 a b d c+23 a^2 d^2\right ) x^4+a \left (6 b^2 c^2-13 a b d c+12 a^2 d^2\right )}{\left (b x^4+a\right )^{3/4} \left (d x^4+c\right )}dx}{2 d}+\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}}{6 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {\int \frac {b \left (12 b^2 c^2-30 a b d c+23 a^2 d^2\right ) x^4+a \left (6 b^2 c^2-13 a b d c+12 a^2 d^2\right )}{\left (b x^4+a\right )^{3/4} \left (d x^4+c\right )}dx}{2 d}}{6 d}\)

\(\Big \downarrow \) 404

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 (b c-a d)^2 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx-a b (6 b c-11 a d) \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx}{2 d}}{6 d}\)

\(\Big \downarrow \) 768

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 (b c-a d)^2 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx-\frac {a b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x^3}dx}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 858

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 (b c-a d)^2 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx+\frac {a b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x}d\frac {1}{x}}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 (b c-a d)^2 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx+\frac {a b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4}}d\frac {1}{x^2}}{2 \left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 229

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 (b c-a d)^2 \int \frac {\sqrt [4]{b x^4+a}}{d x^4+c}dx+\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 923

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \int \frac {1}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (c-\frac {(b c-a d) x^4}{b x^4+a}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \left (\frac {\int \frac {\sqrt {c}}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 c}+\frac {\int \frac {\sqrt {c}}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 c}\right )+\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \left (\frac {\int \frac {1}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\int \frac {1}{\sqrt {1-\frac {b x^4}{b x^4+a}} \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}\right )+\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac {\frac {b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{2 d}-\frac {12 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d)^2 \left (\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}+\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}\right )+\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\left (a+b x^4\right )^{3/4}}}{2 d}}{6 d}\)

input

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

output

(b*x*(a + b*x^4)^(5/4))/(6*d) - ((b*(6*b*c - 11*a*d)*x*(a + b*x^4)^(1/4))/ 
(2*d) - ((Sqrt[a]*b^(3/2)*(6*b*c - 11*a*d)*(1 + a/(b*x^4))^(3/4)*x^3*Ellip 
ticF[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(a + b*x^4)^(3/4) + 12*(b*c - a* 
d)^2*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*(EllipticPi[-(Sqrt[b*c - a*d]/(Sq 
rt[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)))/(2*d))/(6*d)
 

3.2.99.3.1 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 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 933

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

3.2.99.4 Maple [F]

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

input

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

output

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

3.2.99.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.2.99.6 Sympy [F]

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

input

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

output

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

3.2.99.7 Maxima [F]

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

input

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

output

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

3.2.99.8 Giac [F]

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

input

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

output

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

3.2.99.9 Mupad [F(-1)]

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

input

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

output

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