3.10.49 \(\int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx\)

Optimal. Leaf size=72 \[ \frac {4389 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )}{4096}-\frac {4389 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )}{4096}+\frac {\sqrt [4]{x^4+x^3} \left (2048 x^4-2432 x^3+3040 x^2-4180 x+7315\right )}{10240} \]

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Rubi [B]  time = 0.15, antiderivative size = 168, normalized size of antiderivative = 2.33, number of steps used = 11, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2056, 50, 63, 331, 298, 203, 206} \begin {gather*} \frac {1}{5} \sqrt [4]{x^4+x^3} x^4-\frac {19}{80} \sqrt [4]{x^4+x^3} x^3-\frac {209}{512} \sqrt [4]{x^4+x^3} x+\frac {1463 \sqrt [4]{x^4+x^3}}{2048}+\frac {4389 \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4096 \sqrt [4]{x+1} x^{3/4}}-\frac {4389 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4096 \sqrt [4]{x+1} x^{3/4}}+\frac {19}{64} \sqrt [4]{x^4+x^3} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 50

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 63

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 203

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

Rule 206

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

Rule 298

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 331

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 2056

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*} \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{19/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (19 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{15/4}}{(1+x)^{3/4}} \, dx}{20 x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {\left (57 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{11/4}}{(1+x)^{3/4}} \, dx}{64 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (209 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{256 x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {\left (1463 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{2048 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{8192 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2048 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {4389 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}-\frac {4389 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 38, normalized size = 0.53 \begin {gather*} \frac {4 x^8 (x+1)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {23}{4};\frac {27}{4};-x\right )}{23 \left (x^3 (x+1)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x^8*(1 + x)^(3/4)*Hypergeometric2F1[3/4, 23/4, 27/4, -x])/(23*(x^3*(1 + x))^(3/4))

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IntegrateAlgebraic [A]  time = 0.41, size = 72, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{x^3+x^4} \left (7315-4180 x+3040 x^2-2432 x^3+2048 x^4\right )}{10240}+\frac {4389 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096}-\frac {4389 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(x^3 + x^4)^(1/4))/(1 + x),x]

[Out]

((x^3 + x^4)^(1/4)*(7315 - 4180*x + 3040*x^2 - 2432*x^3 + 2048*x^4))/10240 + (4389*ArcTan[x/(x^3 + x^4)^(1/4)]
)/4096 - (4389*ArcTanh[x/(x^3 + x^4)^(1/4)])/4096

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fricas [A]  time = 0.44, size = 87, normalized size = 1.21 \begin {gather*} \frac {1}{10240} \, {\left (2048 \, x^{4} - 2432 \, x^{3} + 3040 \, x^{2} - 4180 \, x + 7315\right )} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} - \frac {4389}{4096} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {4389}{8192} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {4389}{8192} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10240*(2048*x^4 - 2432*x^3 + 3040*x^2 - 4180*x + 7315)*(x^4 + x^3)^(1/4) - 4389/4096*arctan((x^4 + x^3)^(1/4
)/x) - 4389/8192*log((x + (x^4 + x^3)^(1/4))/x) + 4389/8192*log(-(x - (x^4 + x^3)^(1/4))/x)

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giac [A]  time = 0.32, size = 87, normalized size = 1.21 \begin {gather*} \frac {1}{10240} \, {\left (7315 \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{4}} - 33440 \, {\left (\frac {1}{x} + 1\right )}^{\frac {13}{4}} + 59470 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 50312 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 19015 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} - \frac {4389}{4096} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {4389}{8192} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4389}{8192} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10240*(7315*(1/x + 1)^(17/4) - 33440*(1/x + 1)^(13/4) + 59470*(1/x + 1)^(9/4) - 50312*(1/x + 1)^(5/4) + 1901
5*(1/x + 1)^(1/4))*x^5 - 4389/4096*arctan((1/x + 1)^(1/4)) - 4389/8192*log((1/x + 1)^(1/4) + 1) + 4389/8192*lo
g(abs((1/x + 1)^(1/4) - 1))

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maple [C]  time = 0.69, size = 15, normalized size = 0.21

method	result	size
meijerg	\(\frac {4 x^{\frac {23}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], -x \right )}{23}\)	\(15\)
trager	\(\left (\frac {1}{5} x^{4}-\frac {19}{80} x^{3}+\frac {19}{64} x^{2}-\frac {209}{512} x +\frac {1463}{2048}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-\frac {4389 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}+x^{3}}\, x +2 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}+2 x^{3}+x^{2}}{x^{2}}\right )}{8192}+\frac {4389 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{8192}\)	\(163\)
risch	\(\frac {\left (2048 x^{4}-2432 x^{3}+3040 x^{2}-4180 x +7315\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{10240}+\frac {\left (\frac {4389 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{8192}+\frac {4389 \ln \left (\frac {2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, x +2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-2 x^{3}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}-4 x -1}{\left (1+x \right )^{2}}\right )}{8192}\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (\left (1+x \right )^{3} x \right )^{\frac {1}{4}}}{x \left (1+x \right )}\)	\(390\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/23*x^(23/4)*hypergeom([3/4,23/4],[27/4],-x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{4}}{x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,{\left (x^4+x^3\right )}^{1/4}}{x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt [4]{x^{3} \left (x + 1\right )}}{x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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