\(\int \frac {(a+b x^4)^{7/4}}{c+d x^4} \, dx\) [193]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {427, 544, 246, 218, 212, 209, 385, 214, 211} \[ \int \frac {\left (a+b x^4\right )^{7/4}}{c+d x^4} \, dx=-\frac {b^{3/4} \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ) (4 b c-7 a d)}{8 d^2}+\frac {(b c-a d)^{7/4} \arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}-\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ) (4 b c-7 a d)}{8 d^2}+\frac {(b c-a d)^{7/4} \text {arctanh}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d^2}+\frac {b x \left (a+b x^4\right )^{3/4}}{4 d} \]

[In]

Int[(a + b*x^4)^(7/4)/(c + d*x^4),x]

[Out]

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

Rule 209

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

Rule 211

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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 218

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[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 427

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] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[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] /; F
reeQ[{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] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}+\frac {\int \frac {-a (b c-4 a d)-b (4 b c-7 a d) x^4}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 d} \\ & = \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {(b (4 b c-7 a d)) \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{4 d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{d^2} \\ & = \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {(b (4 b c-7 a d)) \text {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 d^2}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{d^2} \\ & = \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {(b (4 b c-7 a d)) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}-\frac {(b (4 b c-7 a d)) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\sqrt {c}-\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt {c} d^2}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\sqrt {c}+\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt {c} d^2} \\ & = \frac {b x \left (a+b x^4\right )^{3/4}}{4 d}-\frac {b^{3/4} (4 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac {(b c-a d)^{7/4} \tan ^{-1}\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) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 d^2}+\frac {(b c-a d)^{7/4} \tanh ^{-1}\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} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.80 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{c+d x^4} \, dx=\frac {2 b d x \left (a+b x^4\right )^{3/4}-b^{3/4} (4 b c-7 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {(2+2 i) (b c-a d)^{7/4} \arctan \left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}-\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{c^{3/4}}-b^{3/4} (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {(2+2 i) (b c-a d)^{7/4} \text {arctanh}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}+\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{c^{3/4}}}{8 d^2} \]

[In]

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

[Out]

(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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(167)=334\).

Time = 8.93 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.90

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {2}\, \left (a d -b c \right )^{2} \ln \left (\frac {-\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}\right )}{2}+\sqrt {2}\, \left (a d -b c \right )^{2} \arctan \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x -\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )-\sqrt {2}\, \left (a d -b c \right )^{2} \arctan \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x +\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )-\frac {7 \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} c \left (\left (-\frac {a d \,b^{\frac {3}{4}}}{2}+\frac {2 b^{\frac {7}{4}} c}{7}\right ) \ln \left (\frac {-b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )+\left (a d \,b^{\frac {3}{4}}-\frac {4 b^{\frac {7}{4}} c}{7}\right ) \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )-\frac {2 \left (b \,x^{4}+a \right )^{\frac {3}{4}} x b d}{7}\right )}{2}}{4 \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} d^{2} c}\)	\(401\)

[In]

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

[Out]

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((((a*d-b*c)/c)^(1/4)*x-2^(1/2)*(b*x^4+a)^(1/4))/((a*d-b*c)/c)^(1/4)/x)-2^(1/2)*(a*d-b*c)^2*a
rctan((((a*d-b*c)/c)^(1/4)*x+2^(1/2)*(b*x^4+a)^(1/4))/((a*d-b*c)/c)^(1/4)/x)-7/2*((a*d-b*c)/c)^(1/4)*c*((-1/2*
a*d*b^(3/4)+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)))+(a*d*b^(3/4)-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 = 1.71 (sec) , antiderivative size = 1962, normalized size of antiderivative = 9.30 \[ \int \frac {\left (a+b x^4\right )^{7/4}}{c+d x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 + 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((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*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)*lo
g((-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) - d*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 470
4*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(1/4)*log(-(d^6*x*((256*b^7*c^4 - 1792*a*b^6*c
^3*d + 4704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4) + (64*b^5*c^3 - 336*a*b^4*c^2*
d + 588*a^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4 + a)^(1/4))/x) + d*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2
*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(1/4)*log((d^6*x*((256*b^7*c^4 - 1792*a*b^6*c^3*d +
4704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4) - (64*b^5*c^3 - 336*a*b^4*c^2*d + 58
8*a^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4 + a)^(1/4))/x) - I*d*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5
*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(1/4)*log((I*d^6*x*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4
704*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4) - (64*b^5*c^3 - 336*a*b^4*c^2*d + 588*
a^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4 + a)^(1/4))/x) + I*d*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 4704*a^2*b^5*c
^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(1/4)*log((-I*d^6*x*((256*b^7*c^4 - 1792*a*b^6*c^3*d + 47
04*a^2*b^5*c^2*d^2 - 5488*a^3*b^4*c*d^3 + 2401*a^4*b^3*d^4)/d^8)^(3/4) - (64*b^5*c^3 - 336*a*b^4*c^2*d + 588*a
^2*b^3*c*d^2 - 343*a^3*b^2*d^3)*(b*x^4 + a)^(1/4))/x))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x^4)^(7/4)/(c + d*x^4),x)

[Out]

int((a + b*x^4)^(7/4)/(c + d*x^4), x)