\(\int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx\) [501]

Optimal result

Integrand size = 22, antiderivative size = 368 \[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}-\frac {\left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{1536 a^3 c^4 x}+\frac {(b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}+\frac {(b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}} \] Output:

-1/4*(b*x+a)^(1/4)*(d*x+c)^(3/4)/c/x^4-1/48*(-13*a*d+b*c)*(b*x+a)^(1/4)*(d 
*x+c)^(3/4)/a/c^2/x^3+1/384*(-117*a^2*d^2+10*a*b*c*d+11*b^2*c^2)*(b*x+a)^( 
1/4)*(d*x+c)^(3/4)/a^2/c^3/x^2-1/1536*(-585*a^3*d^3+63*a^2*b*c*d^2+61*a*b^ 
2*c^2*d+77*b^3*c^3)*(b*x+a)^(1/4)*(d*x+c)^(3/4)/a^3/c^4/x+1/1024*(-a*d+b*c 
)*(195*a^3*d^3+135*a^2*b*c*d^2+105*a*b^2*c^2*d+77*b^3*c^3)*arctan(c^(1/4)* 
(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(15/4)/c^(17/4)+1/1024*(-a*d+b*c)*( 
195*a^3*d^3+135*a^2*b*c*d^2+105*a*b^2*c^2*d+77*b^3*c^3)*arctanh(c^(1/4)*(b 
*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(15/4)/c^(17/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\frac {\sqrt [4]{a+b x} \left (-a (c+d x) \left (77 b^3 c^3 x^3+a b^2 c^2 x^2 (-44 c+61 d x)+a^2 b c x \left (32 c^2-40 c d x+63 d^2 x^2\right )+a^3 \left (384 c^3-416 c^2 d x+468 c d^2 x^2-585 d^3 x^3\right )\right )+3 \left (77 b^4 c^4+28 a b^3 c^3 d+30 a^2 b^2 c^2 d^2+60 a^3 b c d^3-195 a^4 d^4\right ) x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {c (a+b x)}{a (c+d x)}\right )\right )}{1536 a^4 c^4 x^4 \sqrt [4]{c+d x}} \] Input:

Integrate[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
 

Output:

((a + b*x)^(1/4)*(-(a*(c + d*x)*(77*b^3*c^3*x^3 + a*b^2*c^2*x^2*(-44*c + 6 
1*d*x) + a^2*b*c*x*(32*c^2 - 40*c*d*x + 63*d^2*x^2) + a^3*(384*c^3 - 416*c 
^2*d*x + 468*c*d^2*x^2 - 585*d^3*x^3))) + 3*(77*b^4*c^4 + 28*a*b^3*c^3*d + 
 30*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 195*a^4*d^4)*x^4*Hypergeometric2F1[ 
1/4, 1, 5/4, (c*(a + b*x))/(a*(c + d*x))]))/(1536*a^4*c^4*x^4*(c + d*x)^(1 
/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {110, 27, 168, 27, 168, 27, 168, 27, 104, 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.

\(\displaystyle \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {\int \frac {b c-13 a d-12 b d x}{4 x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {b c-13 a d-12 b d x}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {\int \frac {11 b^2 c^2+10 a b d c-117 a^2 d^2+8 b d (b c-13 a d) x}{4 x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{3 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {11 b^2 c^2+10 a b d c-117 a^2 d^2+8 b d (b c-13 a d) x}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {\int \frac {77 b^3 c^3+61 a b^2 d c^2+63 a^2 b d^2 c-585 a^3 d^3+4 b d \left (11 b^2 c^2+10 a b d c-117 a^2 d^2\right ) x}{4 x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{2 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {77 b^3 c^3+61 a b^2 d c^2+63 a^2 b d^2 c-585 a^3 d^3+4 b d \left (11 b^2 c^2+10 a b d c-117 a^2 d^2\right ) x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {3 (b c-a d) \left (77 b^3 c^3+105 a b^2 d c^2+135 a^2 b d^2 c+195 a^3 d^3\right )}{4 x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 (b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 (b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 (b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a}+\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 (b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {-\frac {-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{a c x}-\frac {3 (b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {11 b^2 c}{a}-\frac {117 a d^2}{c}+10 b d\right )}{2 x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{3 a c x^3}}{16 c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}\)

Input:

Int[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
 

Output:

-1/4*((a + b*x)^(1/4)*(c + d*x)^(3/4))/(c*x^4) + (-1/3*((b*c - 13*a*d)*(a 
+ b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x^3) - (-1/2*(((11*b^2*c)/a + 10*b*d - 
(117*a*d^2)/c)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/x^2 - (-(((77*b^3*c^3 + 61 
*a*b^2*c^2*d + 63*a^2*b*c*d^2 - 585*a^3*d^3)*(a + b*x)^(1/4)*(c + d*x)^(3/ 
4))/(a*c*x)) - (3*(b*c - a*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c* 
d^2 + 195*a^3*d^3)*(-1/2*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d* 
x)^(1/4))]/(a^(3/4)*c^(1/4)) - ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)* 
(c + d*x)^(1/4))]/(2*a^(3/4)*c^(1/4))))/(a*c))/(8*a*c))/(12*a*c))/(16*c)
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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 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]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{5} \left (x d +c \right )^{\frac {1}{4}}}d x\]

Input:

int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
 

Output:

int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 2386, normalized size of antiderivative = 6.48 \[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x, algorithm="fricas")
 

Output:

1/6144*(3*a^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 8267 
3976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12* 
c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989 
262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7 
*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 155162 
2500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c 
^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1 
/4)*log(-((77*b^4*c^4 + 28*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d 
^3 - 195*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^4*c^4*d*x + a^4*c^5 
)*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d 
^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680 
*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d 
^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^1 
0*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^ 
12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000 
*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4))/(d*x + c)) - 3* 
a^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b 
^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 
 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7 
*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + ...
 

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\int \frac {\sqrt [4]{a + b x}}{x^{5} \sqrt [4]{c + d x}}\, dx \] Input:

integrate((b*x+a)**(1/4)/x**5/(d*x+c)**(1/4),x)
 

Output:

Integral((a + b*x)**(1/4)/(x**5*(c + d*x)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{5}} \,d x } \] Input:

integrate((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5), x)
 

Giac [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{5}} \,d x } \] Input:

integrate((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/4}}{x^5\,{\left (c+d\,x\right )}^{1/4}} \,d x \] Input:

int((a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x)
 

Output:

int((a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}} x^{5}}d x \] Input:

int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
 

Output:

int((a + b*x)**(1/4)/((c + d*x)**(1/4)*x**5),x)