Integrand size = 26, antiderivative size = 134 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\frac {\sqrt [4]{b x^3+a x^4} \left (7315 b^4-4180 a b^3 x+3040 a^2 b^2 x^2-2432 a^3 b x^3+2048 a^4 x^4\right )}{10240 a^5}+\frac {4389 b^5 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}-\frac {4389 b^5 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}} \]
1/10240*(a*x^4+b*x^3)^(1/4)*(2048*a^4*x^4-2432*a^3*b*x^3+3040*a^2*b^2*x^2- 4180*a*b^3*x+7315*b^4)/a^5+4389/4096*b^5*arctan(a^(1/4)*x/(a*x^4+b*x^3)^(1 /4))/a^(23/4)-4389/4096*b^5*arctanh(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4 )
Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\frac {x^{9/4} \left (2 a^{3/4} x^{3/4} \left (7315 b^5+3135 a b^4 x-1140 a^2 b^3 x^2+608 a^3 b^2 x^3-384 a^4 b x^4+2048 a^5 x^5\right )-21945 b^5 (b+a x)^{3/4} \arctan \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{a} \sqrt [4]{x}}\right )-21945 b^5 (b+a x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{a} \sqrt [4]{x}}\right )\right )}{20480 a^{23/4} \left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(2*a^(3/4)*x^(3/4)*(7315*b^5 + 3135*a*b^4*x - 1140*a^2*b^3*x^2 + 608*a^3*b^2*x^3 - 384*a^4*b*x^4 + 2048*a^5*x^5) - 21945*b^5*(b + a*x)^(3/4 )*ArcTan[(b + a*x)^(1/4)/(a^(1/4)*x^(1/4))] - 21945*b^5*(b + a*x)^(3/4)*Ar cTanh[(b + a*x)^(1/4)/(a^(1/4)*x^(1/4))]))/(20480*a^(23/4)*(x^3*(b + a*x)) ^(3/4))
Time = 0.42 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.76, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2467, 60, 60, 60, 60, 60, 73, 854, 827, 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.
| \(\displaystyle \int \frac {x^4 \sqrt [4]{a x^4+b x^3}}{a x+b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int \frac {x^{19/4}}{(b+a x)^{3/4}}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \int \frac {x^{15/4}}{(b+a x)^{3/4}}dx}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \int \frac {x^{11/4}}{(b+a x)^{3/4}}dx}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \int \frac {x^{7/4}}{(b+a x)^{3/4}}dx}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \int \frac {x^{3/4}}{(b+a x)^{3/4}}dx}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}}dx}{4 a}\right )}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {\sqrt {x}}{(b+a x)^{3/4}}d\sqrt [4]{x}}{a}\right )}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{a}\right )}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}\right )}{a}\right )}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \left (\frac {x^{19/4} \sqrt [4]{a x+b}}{5 a}-\frac {19 b \left (\frac {x^{15/4} \sqrt [4]{a x+b}}{4 a}-\frac {15 b \left (\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}-\frac {11 b \left (\frac {x^{7/4} \sqrt [4]{a x+b}}{2 a}-\frac {7 b \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )}{8 a}\right )}{12 a}\right )}{16 a}\right )}{20 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
((b*x^3 + a*x^4)^(1/4)*((x^(19/4)*(b + a*x)^(1/4))/(5*a) - (19*b*((x^(15/4 )*(b + a*x)^(1/4))/(4*a) - (15*b*((x^(11/4)*(b + a*x)^(1/4))/(3*a) - (11*b *((x^(7/4)*(b + a*x)^(1/4))/(2*a) - (7*b*((x^(3/4)*(b + a*x)^(1/4))/a - (3 *b*(-1/2*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/a^(3/4) + ArcTanh[(a^(1 /4)*x^(1/4))/(b + a*x)^(1/4)]/(2*a^(3/4))))/a))/(8*a)))/(12*a)))/(16*a)))/ (20*a)))/(x^(3/4)*(b + a*x)^(1/4))
3.20.32.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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])
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])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(-\frac {19 \left (\frac {1155 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{5}}{512}+\frac {1155 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{5}}{256}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} \left (a^{\frac {15}{4}} b \,x^{3}-\frac {16 a^{\frac {19}{4}} x^{4}}{19}-\frac {385 a^{\frac {3}{4}} b^{4}}{128}+\frac {55 a^{\frac {7}{4}} b^{3} x}{32}-\frac {5 a^{\frac {11}{4}} b^{2} x^{2}}{4}\right )\right )}{80 a^{\frac {23}{4}}}\) | \(132\) |
-19/80/a^(23/4)*(1155/512*ln((a^(1/4)*x+(x^3*(a*x+b))^(1/4))/(-a^(1/4)*x+( x^3*(a*x+b))^(1/4)))*b^5+1155/256*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))* b^5+(x^3*(a*x+b))^(1/4)*(a^(15/4)*b*x^3-16/19*a^(19/4)*x^4-385/128*a^(3/4) *b^4+55/32*a^(7/4)*b^3*x-5/4*a^(11/4)*b^2*x^2))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.05 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=-\frac {21945 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (\frac {4389 \, {\left (a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) - 21945 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) - 21945 i \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (i \, a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) + 21945 i \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (-i \, a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) - 4 \, {\left (2048 \, a^{4} x^{4} - 2432 \, a^{3} b x^{3} + 3040 \, a^{2} b^{2} x^{2} - 4180 \, a b^{3} x + 7315 \, b^{4}\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{40960 \, a^{5}} \]
-1/40960*(21945*a^5*(b^20/a^23)^(1/4)*log(4389*(a^6*x*(b^20/a^23)^(1/4) + (a*x^4 + b*x^3)^(1/4)*b^5)/x) - 21945*a^5*(b^20/a^23)^(1/4)*log(-4389*(a^6 *x*(b^20/a^23)^(1/4) - (a*x^4 + b*x^3)^(1/4)*b^5)/x) - 21945*I*a^5*(b^20/a ^23)^(1/4)*log(-4389*(I*a^6*x*(b^20/a^23)^(1/4) - (a*x^4 + b*x^3)^(1/4)*b^ 5)/x) + 21945*I*a^5*(b^20/a^23)^(1/4)*log(-4389*(-I*a^6*x*(b^20/a^23)^(1/4 ) - (a*x^4 + b*x^3)^(1/4)*b^5)/x) - 4*(2048*a^4*x^4 - 2432*a^3*b*x^3 + 304 0*a^2*b^2*x^2 - 4180*a*b^3*x + 7315*b^4)*(a*x^4 + b*x^3)^(1/4))/a^5
\[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\int \frac {x^{4} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x + b}\, dx \]
\[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x^{4}}{a x + b} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (114) = 228\).
Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.20 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=-\frac {\frac {43890 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{6}} + \frac {43890 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{6}} + \frac {21945 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{6}} + \frac {21945 \, \sqrt {2} b^{6} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} - \frac {8 \, {\left (7315 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} b^{6} - 33440 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a b^{6} + 59470 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{2} b^{6} - 50312 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{3} b^{6} + 19015 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{4} b^{6}\right )} x^{5}}{a^{5} b^{5}}}{81920 \, b} \]
-1/81920*(43890*sqrt(2)*(-a)^(1/4)*b^6*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1 /4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^6 + 43890*sqrt(2)*(-a)^(1/4)*b^6*ar ctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^6 + 21945*sqrt(2)*(-a)^(1/4)*b^6*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + s qrt(-a) + sqrt(a + b/x))/a^6 + 21945*sqrt(2)*b^6*log(-sqrt(2)*(-a)^(1/4)*( a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) - 8*(7315*(a + b/x)^(17/4)*b^6 - 33440*(a + b/x)^(13/4)*a*b^6 + 59470*(a + b/x)^(9/4)*a^ 2*b^6 - 50312*(a + b/x)^(5/4)*a^3*b^6 + 19015*(a + b/x)^(1/4)*a^4*b^6)*x^5 /(a^5*b^5))/b
Timed out. \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\int \frac {x^4\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{b+a\,x} \,d x \]