\(\int \frac {x^4 (c+d x^4)}{\sqrt [4]{a+b x^4}} \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=\frac {(8 b c-5 a d) x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}-\frac {a (8 b c-5 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}-\frac {a (8 b c-5 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}} \] Output:

1/32*(-5*a*d+8*b*c)*x*(b*x^4+a)^(3/4)/b^2+1/8*d*x^5*(b*x^4+a)^(3/4)/b-1/64 
*a*(-5*a*d+8*b*c)*arctan(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(9/4)-1/64*a*(-5*a*d 
+8*b*c)*arctanh(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(9/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=\frac {2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4} \left (8 b c-5 a d+4 b d x^4\right )+a (-8 b c+5 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+a (-8 b c+5 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}} \] Input:

Integrate[(x^4*(c + d*x^4))/(a + b*x^4)^(1/4),x]
 

Output:

(2*b^(1/4)*x*(a + b*x^4)^(3/4)*(8*b*c - 5*a*d + 4*b*d*x^4) + a*(-8*b*c + 5 
*a*d)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)] + a*(-8*b*c + 5*a*d)*ArcTanh[( 
b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(9/4))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 843, 770, 756, 216, 219}

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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {(8 b c-5 a d) \int \frac {x^4}{\sqrt [4]{b x^4+a}}dx}{8 b}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {(8 b c-5 a d) \left (\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \int \frac {1}{\sqrt [4]{b x^4+a}}dx}{4 b}\right )}{8 b}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {(8 b c-5 a d) \left (\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \int \frac {1}{1-\frac {b x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}}{4 b}\right )}{8 b}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {(8 b c-5 a d) \left (\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}+1}d\frac {x}{\sqrt [4]{b x^4+a}}\right )}{4 b}\right )}{8 b}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {(8 b c-5 a d) \left (\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}+\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{4 b}\right )}{8 b}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(8 b c-5 a d) \left (\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \left (\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}\right )}{4 b}\right )}{8 b}+\frac {d x^5 \left (a+b x^4\right )^{3/4}}{8 b}\)

Input:

Int[(x^4*(c + d*x^4))/(a + b*x^4)^(1/4),x]
 

Output:

(d*x^5*(a + b*x^4)^(3/4))/(8*b) + ((8*b*c - 5*a*d)*((x*(a + b*x^4)^(3/4))/ 
(4*b) - (a*(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))))/(4*b)))/(8*b)
 

Defintions of rubi rules used

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 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 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 843

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

rule 959

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(\frac {\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} x \left (\frac {d \,x^{4}}{2}+c \right ) b^{\frac {5}{4}}}{4}+\frac {5 \left (-4 \left (b \,x^{4}+a \right )^{\frac {3}{4}} x d \,b^{\frac {1}{4}}+\left (\ln \left (\frac {x \,b^{\frac {1}{4}}+\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{-x \,b^{\frac {1}{4}}+\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x \,b^{\frac {1}{4}}}\right )\right ) \left (a d -\frac {8 c b}{5}\right )\right ) a}{128}}{b^{\frac {9}{4}}}\)	\(113\)

Input:

int(x^4*(d*x^4+c)/(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

5/128*(32/5*(b*x^4+a)^(3/4)*x*(1/2*d*x^4+c)*b^(5/4)+(-4*(b*x^4+a)^(3/4)*x* 
d*b^(1/4)+(ln((x*b^(1/4)+(b*x^4+a)^(1/4))/(-x*b^(1/4)+(b*x^4+a)^(1/4)))-2* 
arctan((b*x^4+a)^(1/4)/x/b^(1/4)))*(a*d-8/5*c*b))*a)/b^(9/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 809, normalized size of antiderivative = 6.37 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx =\text {Too large to display} \] Input:

integrate(x^4*(d*x^4+c)/(b*x^4+a)^(1/4),x, algorithm="fricas")
 

Output:

-1/128*(b^2*((4096*a^4*b^4*c^4 - 10240*a^5*b^3*c^3*d + 9600*a^6*b^2*c^2*d^ 
2 - 4000*a^7*b*c*d^3 + 625*a^8*d^4)/b^9)^(1/4)*log(-(b^7*x*((4096*a^4*b^4* 
c^4 - 10240*a^5*b^3*c^3*d + 9600*a^6*b^2*c^2*d^2 - 4000*a^7*b*c*d^3 + 625* 
a^8*d^4)/b^9)^(3/4) + (512*a^3*b^3*c^3 - 960*a^4*b^2*c^2*d + 600*a^5*b*c*d 
^2 - 125*a^6*d^3)*(b*x^4 + a)^(1/4))/x) - b^2*((4096*a^4*b^4*c^4 - 10240*a 
^5*b^3*c^3*d + 9600*a^6*b^2*c^2*d^2 - 4000*a^7*b*c*d^3 + 625*a^8*d^4)/b^9) 
^(1/4)*log((b^7*x*((4096*a^4*b^4*c^4 - 10240*a^5*b^3*c^3*d + 9600*a^6*b^2* 
c^2*d^2 - 4000*a^7*b*c*d^3 + 625*a^8*d^4)/b^9)^(3/4) - (512*a^3*b^3*c^3 - 
960*a^4*b^2*c^2*d + 600*a^5*b*c*d^2 - 125*a^6*d^3)*(b*x^4 + a)^(1/4))/x) + 
 I*b^2*((4096*a^4*b^4*c^4 - 10240*a^5*b^3*c^3*d + 9600*a^6*b^2*c^2*d^2 - 4 
000*a^7*b*c*d^3 + 625*a^8*d^4)/b^9)^(1/4)*log((I*b^7*x*((4096*a^4*b^4*c^4 
- 10240*a^5*b^3*c^3*d + 9600*a^6*b^2*c^2*d^2 - 4000*a^7*b*c*d^3 + 625*a^8* 
d^4)/b^9)^(3/4) - (512*a^3*b^3*c^3 - 960*a^4*b^2*c^2*d + 600*a^5*b*c*d^2 - 
 125*a^6*d^3)*(b*x^4 + a)^(1/4))/x) - I*b^2*((4096*a^4*b^4*c^4 - 10240*a^5 
*b^3*c^3*d + 9600*a^6*b^2*c^2*d^2 - 4000*a^7*b*c*d^3 + 625*a^8*d^4)/b^9)^( 
1/4)*log((-I*b^7*x*((4096*a^4*b^4*c^4 - 10240*a^5*b^3*c^3*d + 9600*a^6*b^2 
*c^2*d^2 - 4000*a^7*b*c*d^3 + 625*a^8*d^4)/b^9)^(3/4) - (512*a^3*b^3*c^3 - 
 960*a^4*b^2*c^2*d + 600*a^5*b*c*d^2 - 125*a^6*d^3)*(b*x^4 + a)^(1/4))/x) 
- 4*(4*b*d*x^5 + (8*b*c - 5*a*d)*x)*(b*x^4 + a)^(3/4))/b^2
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.63 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=\frac {c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {9}{4}\right )} + \frac {d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {13}{4}\right )} \] Input:

integrate(x**4*(d*x**4+c)/(b*x**4+a)**(1/4),x)
 

Output:

c*x**5*gamma(5/4)*hyper((1/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a 
**(1/4)*gamma(9/4)) + d*x**9*gamma(9/4)*hyper((1/4, 9/4), (13/4,), b*x**4* 
exp_polar(I*pi)/a)/(4*a**(1/4)*gamma(13/4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (103) = 206\).

Time = 0.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.09 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=-\frac {1}{128} \, d {\left (\frac {5 \, a^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {1}{4}}}\right )}}{b^{2}} - \frac {4 \, {\left (\frac {9 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{2} b}{x^{3}} - \frac {5 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{x^{7}}\right )}}{b^{4} - \frac {2 \, {\left (b x^{4} + a\right )} b^{3}}{x^{4}} + \frac {{\left (b x^{4} + a\right )}^{2} b^{2}}{x^{8}}}\right )} + \frac {1}{16} \, c {\left (\frac {a {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {1}{4}}}\right )}}{b} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a}{{\left (b^{2} - \frac {{\left (b x^{4} + a\right )} b}{x^{4}}\right )} x^{3}}\right )} \] Input:

integrate(x^4*(d*x^4+c)/(b*x^4+a)^(1/4),x, algorithm="maxima")
 

Output:

-1/128*d*(5*a^2*(2*arctan((b*x^4 + a)^(1/4)/(b^(1/4)*x))/b^(1/4) + log(-(b 
^(1/4) - (b*x^4 + a)^(1/4)/x)/(b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(1/4))/b^ 
2 - 4*(9*(b*x^4 + a)^(3/4)*a^2*b/x^3 - 5*(b*x^4 + a)^(7/4)*a^2/x^7)/(b^4 - 
 2*(b*x^4 + a)*b^3/x^4 + (b*x^4 + a)^2*b^2/x^8)) + 1/16*c*(a*(2*arctan((b* 
x^4 + a)^(1/4)/(b^(1/4)*x))/b^(1/4) + log(-(b^(1/4) - (b*x^4 + a)^(1/4)/x) 
/(b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(1/4))/b - 4*(b*x^4 + a)^(3/4)*a/((b^2 
 - (b*x^4 + a)*b/x^4)*x^3))
 

Giac [F]

\[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{4}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x^4*(d*x^4+c)/(b*x^4+a)^(1/4),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)*x^4/(b*x^4 + a)^(1/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=\int \frac {x^4\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{1/4}} \,d x \] Input:

int((x^4*(c + d*x^4))/(a + b*x^4)^(1/4),x)
 

Output:

int((x^4*(c + d*x^4))/(a + b*x^4)^(1/4), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt [4]{a+b x^4}} \, dx=\left (\int \frac {x^{8}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x \right ) d +\left (\int \frac {x^{4}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x \right ) c \] Input:

int(x^4*(d*x^4+c)/(b*x^4+a)^(1/4),x)
 

Output:

int(x**8/(a + b*x**4)**(1/4),x)*d + int(x**4/(a + b*x**4)**(1/4),x)*c