\(\int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx\) [1932]

Optimal result

Integrand size = 26, antiderivative size = 134 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\frac {\sqrt [4]{b x^3+a x^4} \left (7315 b^4-4180 a b^3 x+3040 a^2 b^2 x^2-2432 a^3 b x^3+2048 a^4 x^4\right )}{10240 a^5}+\frac {4389 b^5 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}-\frac {4389 b^5 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.90, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2081, 52, 65, 338, 304, 209, 212} \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\frac {4389 b^5 \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {4389 b^5 \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {1463 b^4 \sqrt [4]{a x^4+b x^3}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{a x^4+b x^3}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{a x^4+b x^3}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{a x^4+b x^3}}{80 a^2}+\frac {x^4 \sqrt [4]{a x^4+b x^3}}{5 a} \]

[In]

Int[(x^4*(b*x^3 + a*x^4)^(1/4))/(b + a*x),x]

[Out]

(1463*b^4*(b*x^3 + a*x^4)^(1/4))/(2048*a^5) - (209*b^3*x*(b*x^3 + a*x^4)^(1/4))/(512*a^4) + (19*b^2*x^2*(b*x^3
+ a*x^4)^(1/4))/(64*a^3) - (19*b*x^3*(b*x^3 + a*x^4)^(1/4))/(80*a^2) + (x^4*(b*x^3 + a*x^4)^(1/4))/(5*a) + (4
389*b^5*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(4096*a^(23/4)*x^(3/4)*(b + a*x)^(1/4
)) - (4389*b^5*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(4096*a^(23/4)*x^(3/4)*(b + a
*x)^(1/4))

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 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 304

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

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{19/4}}{(b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (19 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{15/4}}{(b+a x)^{3/4}} \, dx}{20 a x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {\left (57 b^2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{11/4}}{(b+a x)^{3/4}} \, dx}{64 a^2 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (209 b^3 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{7/4}}{(b+a x)^{3/4}} \, dx}{256 a^3 x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {\left (1463 b^4 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{(b+a x)^{3/4}} \, dx}{2048 a^4 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{8192 a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4389 b^5 \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4389 b^5 \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {4389 b^5 \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\frac {x^{9/4} \left (2 a^{3/4} x^{3/4} \left (7315 b^5+3135 a b^4 x-1140 a^2 b^3 x^2+608 a^3 b^2 x^3-384 a^4 b x^4+2048 a^5 x^5\right )-21945 b^5 (b+a x)^{3/4} \arctan \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{a} \sqrt [4]{x}}\right )-21945 b^5 (b+a x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{a} \sqrt [4]{x}}\right )\right )}{20480 a^{23/4} \left (x^3 (b+a x)\right )^{3/4}} \]

[In]

Integrate[(x^4*(b*x^3 + a*x^4)^(1/4))/(b + a*x),x]

[Out]

(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 + 20
48*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)*
ArcTanh[(b + a*x)^(1/4)/(a^(1/4)*x^(1/4))]))/(20480*a^(23/4)*(x^3*(b + a*x))^(3/4))

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {19 \left (\frac {1155 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{5}}{512}+\frac {1155 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{5}}{256}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} \left (a^{\frac {15}{4}} b \,x^{3}-\frac {16 a^{\frac {19}{4}} x^{4}}{19}-\frac {385 a^{\frac {3}{4}} b^{4}}{128}+\frac {55 a^{\frac {7}{4}} b^{3} x}{32}-\frac {5 a^{\frac {11}{4}} b^{2} x^{2}}{4}\right )\right )}{80 a^{\frac {23}{4}}}\)	\(132\)

[In]

int(x^4*(a*x^4+b*x^3)^(1/4)/(a*x+b),x,method=_RETURNVERBOSE)

[Out]

-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*ar
ctan(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))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.05 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=-\frac {21945 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (\frac {4389 \, {\left (a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) - 21945 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) - 21945 i \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (i \, a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) + 21945 i \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (-i \, a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) - 4 \, {\left (2048 \, a^{4} x^{4} - 2432 \, a^{3} b x^{3} + 3040 \, a^{2} b^{2} x^{2} - 4180 \, a b^{3} x + 7315 \, b^{4}\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{40960 \, a^{5}} \]

[In]

integrate(x^4*(a*x^4+b*x^3)^(1/4)/(a*x+b),x, algorithm="fricas")

[Out]

-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) - 2194
5*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 +
3040*a^2*b^2*x^2 - 4180*a*b^3*x + 7315*b^4)*(a*x^4 + b*x^3)^(1/4))/a^5

Sympy [F]

\[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\int \frac {x^{4} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x + b}\, dx \]

[In]

integrate(x**4*(a*x**4+b*x**3)**(1/4)/(a*x+b),x)

[Out]

Integral(x**4*(x**3*(a*x + b))**(1/4)/(a*x + b), x)

Maxima [F]

\[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x^{4}}{a x + b} \,d x } \]

[In]

integrate(x^4*(a*x^4+b*x^3)^(1/4)/(a*x+b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b*x^3)^(1/4)*x^4/(a*x + b), x)

Giac [B] (verification not implemented)

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 \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=-\frac {\frac {43890 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{6}} + \frac {43890 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{6}} + \frac {21945 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{6}} + \frac {21945 \, \sqrt {2} b^{6} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} - \frac {8 \, {\left (7315 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} b^{6} - 33440 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a b^{6} + 59470 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{2} b^{6} - 50312 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{3} b^{6} + 19015 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{4} b^{6}\right )} x^{5}}{a^{5} b^{5}}}{81920 \, b} \]

[In]

integrate(x^4*(a*x^4+b*x^3)^(1/4)/(a*x+b),x, algorithm="giac")

[Out]

-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*arctan(-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) + sqrt(-a) + sqrt(a + b/x))/a^6 + 219
45*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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx=\int \frac {x^4\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{b+a\,x} \,d x \]

[In]

int((x^4*(a*x^4 + b*x^3)^(1/4))/(b + a*x),x)

[Out]

int((x^4*(a*x^4 + b*x^3)^(1/4))/(b + a*x), x)