\(\int \frac {(a+b x^4)^{7/4}}{c+d x^4} \, dx\) [103]

Optimal result

Integrand size = 21, antiderivative size = 211 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{c+d x^4} \, dx=\frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {b^{3/4} (4 b c-7 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac {(b c-a d)^{7/4} \arctan \left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}-\frac {b^{3/4} (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac {(b c-a d)^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2} \] Output:

1/4*b*x*(b*x^4+a)^(3/4)/d-1/8*b^(3/4)*(-7*a*d+4*b*c)*arctan(b^(1/4)*x/(b*x 
^4+a)^(1/4))/d^2+1/2*(-a*d+b*c)^(7/4)*arctan((-a*d+b*c)^(1/4)*x/c^(1/4)/(b 
*x^4+a)^(1/4))/c^(3/4)/d^2-1/8*b^(3/4)*(-7*a*d+4*b*c)*arctanh(b^(1/4)*x/(b 
*x^4+a)^(1/4))/d^2+1/2*(-a*d+b*c)^(7/4)*arctanh((-a*d+b*c)^(1/4)*x/c^(1/4) 
/(b*x^4+a)^(1/4))/c^(3/4)/d^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.91 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{c+d x^4} \, dx=\frac {2 b d x \left (a+b x^4\right )^{3/4}-b^{3/4} (4 b c-7 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {(2+2 i) (b c-a d)^{7/4} \arctan \left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}-\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{c^{3/4}}-b^{3/4} (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {(2+2 i) (b c-a d)^{7/4} \text {arctanh}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}+\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{c^{3/4}}}{8 d^2} \] Input:

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

Output:

(2*b*d*x*(a + b*x^4)^(3/4) - b^(3/4)*(4*b*c - 7*a*d)*ArcTan[(b^(1/4)*x)/(a 
 + b*x^4)^(1/4)] + ((2 + 2*I)*(b*c - a*d)^(7/4)*ArcTan[(((1 - I)*(b*c - a* 
d)^(1/4)*x^2)/(c^(1/4)*(a + b*x^4)^(1/4)) - ((1 + I)*c^(1/4)*(a + b*x^4)^( 
1/4))/(b*c - a*d)^(1/4))/(2*x)])/c^(3/4) - b^(3/4)*(4*b*c - 7*a*d)*ArcTanh 
[(b^(1/4)*x)/(a + b*x^4)^(1/4)] + ((2 + 2*I)*(b*c - a*d)^(7/4)*ArcTanh[((( 
1 - I)*(b*c - a*d)^(1/4)*x^2)/(c^(1/4)*(a + b*x^4)^(1/4)) + ((1 + I)*c^(1/ 
4)*(a + b*x^4)^(1/4))/(b*c - a*d)^(1/4))/(2*x)])/c^(3/4))/(8*d^2)
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {933, 25, 1026, 770, 756, 216, 219, 902, 756, 218, 221}

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.

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

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int -\frac {b (4 b c-7 a d) x^4+a (b c-4 a d)}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{4 d}+\frac {b x \left (a+b x^4\right )^{3/4}}{4 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\int \frac {b (4 b c-7 a d) x^4+a (b c-4 a d)}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{4 d}\)

\(\Big \downarrow \) 1026

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \int \frac {1}{\sqrt [4]{b x^4+a}}dx}{d}-\frac {4 (b c-a d)^2 \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 d}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \int \frac {1}{1-\frac {b x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}}{d}-\frac {4 (b c-a d)^2 \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 d}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}+1}d\frac {x}{\sqrt [4]{b x^4+a}}\right )}{d}-\frac {4 (b c-a d)^2 \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {4 (b c-a d)^2 \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {4 (b c-a d)^2 \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 d}\)

\(\Big \downarrow \) 902

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {4 (b c-a d)^2 \int \frac {1}{c-\frac {(b c-a d) x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}}{d}}{4 d}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {4 (b c-a d)^2 \left (\frac {\int \frac {1}{\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\int \frac {1}{\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}\right )}{d}}{4 d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {4 (b c-a d)^2 \left (\frac {\int \frac {1}{\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}\right )}{d}}{4 d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {\frac {b (4 b c-7 a d) \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{d}-\frac {4 (b c-a d)^2 \left (\frac {\arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}\right )}{d}}{4 d}\)

Input:

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

Output:

(b*x*(a + b*x^4)^(3/4))/(4*d) - ((b*(4*b*c - 7*a*d)*(ArcTan[(b^(1/4)*x)/(a 
 + b*x^4)^(1/4)]/(2*b^(1/4)) + ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b 
^(1/4))))/d - (4*(b*c - a*d)^2*(ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + 
 b*x^4)^(1/4))]/(2*c^(3/4)*(b*c - a*d)^(1/4)) + ArcTanh[((b*c - a*d)^(1/4) 
*x)/(c^(1/4)*(a + b*x^4)^(1/4))]/(2*c^(3/4)*(b*c - a*d)^(1/4))))/d)/(4*d)
 

Defintions of rubi rules used

rule 25

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

rule 216

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

rule 218

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

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

rule 756

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

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 902

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su 
bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b 
, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
 

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 1026

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(399\) vs. \(2(167)=334\).

Time = 10.43 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.90

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {2}\, \left (a d -b c \right )^{2} \ln \left (\frac {-\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}\right )}{2}+\sqrt {2}\, \left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )+\sqrt {2}\, \left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}+\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )+\frac {7 c \left (\left (-\frac {b^{\frac {3}{4}} a d}{2}+\frac {2 b^{\frac {7}{4}} c}{7}\right ) \ln \left (\frac {-b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )+\left (b^{\frac {3}{4}} a d -\frac {4 b^{\frac {7}{4}} c}{7}\right ) \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )-\frac {2 \left (b \,x^{4}+a \right )^{\frac {3}{4}} x b d}{7}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}}}{2}}{4 \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} d^{2} c}\)	\(400\)

Input:

int((b*x^4+a)^(7/4)/(d*x^4+c),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-1/4/((a*d-b*c)/c)^(1/4)*(1/2*2^(1/2)*(a*d-b*c)^2*ln((-((a*d-b*c)/c)^(1/4) 
*(b*x^4+a)^(1/4)*2^(1/2)*x+((a*d-b*c)/c)^(1/2)*x^2+(b*x^4+a)^(1/2))/(((a*d 
-b*c)/c)^(1/4)*(b*x^4+a)^(1/4)*2^(1/2)*x+((a*d-b*c)/c)^(1/2)*x^2+(b*x^4+a) 
^(1/2)))+2^(1/2)*(a*d-b*c)^2*arctan((2^(1/2)*(b*x^4+a)^(1/4)-((a*d-b*c)/c) 
^(1/4)*x)/((a*d-b*c)/c)^(1/4)/x)+2^(1/2)*(a*d-b*c)^2*arctan((2^(1/2)*(b*x^ 
4+a)^(1/4)+((a*d-b*c)/c)^(1/4)*x)/((a*d-b*c)/c)^(1/4)/x)+7/2*c*((-1/2*b^(3 
/4)*a*d+2/7*b^(7/4)*c)*ln((-b^(1/4)*x-(b*x^4+a)^(1/4))/(b^(1/4)*x-(b*x^4+a 
)^(1/4)))+(b^(3/4)*a*d-4/7*b^(7/4)*c)*arctan(1/b^(1/4)/x*(b*x^4+a)^(1/4))- 
2/7*(b*x^4+a)^(3/4)*x*b*d)*((a*d-b*c)/c)^(1/4))/d^2/c
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.47 (sec) , antiderivative size = 1962, normalized size of antiderivative = 9.30 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{c+d x^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/16*(4*(b*x^4 + a)^(3/4)*b*x + 4*d*((b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5 
*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 
7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(1/4)*log(-(c^2*d^6*x*((b^7*c^7 - 7*a* 
b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 
 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(3/4) + (b^5*c^5 
 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 
 - a^5*d^5)*(b*x^4 + a)^(1/4))/x) - 4*d*((b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2 
*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^ 
5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(1/4)*log((c^2*d^6*x*((b^7*c^7 - 7 
*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^ 
4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(3/4) - (b^5* 
c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c* 
d^4 - a^5*d^5)*(b*x^4 + a)^(1/4))/x) + 4*I*d*((b^7*c^7 - 7*a*b^6*c^6*d + 2 
1*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c 
^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(1/4)*log((I*c^2*d^6*x*((b^7* 
c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3 
*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)/(c^3*d^8))^(3/4) 
- (b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a 
^4*b*c*d^4 - a^5*d^5)*(b*x^4 + a)^(1/4))/x) - 4*I*d*((b^7*c^7 - 7*a*b^6*c^ 
6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((a + b*x**4)**(3/4)*b*x + 4*int((a + b*x**4)**(3/4)/(a*c + a*d*x**4 + b*c 
*x**4 + b*d*x**8),x)*a**2*d - int((a + b*x**4)**(3/4)/(a*c + a*d*x**4 + b* 
c*x**4 + b*d*x**8),x)*a*b*c + 7*int(((a + b*x**4)**(3/4)*x**4)/(a*c + a*d* 
x**4 + b*c*x**4 + b*d*x**8),x)*a*b*d - 4*int(((a + b*x**4)**(3/4)*x**4)/(a 
*c + a*d*x**4 + b*c*x**4 + b*d*x**8),x)*b**2*c)/(4*d)