\(\int \frac {(1-x^3+x^5) (-3+2 x^5)}{x^6 \sqrt [4]{x+x^6}} \, dx\) [335]

Optimal result

Integrand size = 30, antiderivative size = 28 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (3-7 x^3+3 x^5\right ) \left (x+x^6\right )^{3/4}}{21 x^6} \]

[Out]

4/21*(3*x^5-7*x^3+3)*(x^6+x)^(3/4)/x^6

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 16, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2077, 2050, 2057, 371} \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (x^6+x\right )^{3/4}}{7 x}+\frac {4 \left (x^6+x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^6+x\right )^{3/4}}{3 x^3} \]

[In]

Int[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^6*(x + x^6)^(1/4)),x]

[Out]

(4*(x + x^6)^(3/4))/(7*x^6) - (4*(x + x^6)^(3/4))/(3*x^3) + (4*(x + x^6)^(3/4))/(7*x)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2050

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(j - 1)*(c*x)^(m - j +
1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 2077

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3}{x^6 \sqrt [4]{x+x^6}}+\frac {3}{x^3 \sqrt [4]{x+x^6}}-\frac {1}{x \sqrt [4]{x+x^6}}-\frac {2 x^2}{\sqrt [4]{x+x^6}}+\frac {2 x^4}{\sqrt [4]{x+x^6}}\right ) \, dx \\ & = -\left (2 \int \frac {x^2}{\sqrt [4]{x+x^6}} \, dx\right )+2 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx-3 \int \frac {1}{x^6 \sqrt [4]{x+x^6}} \, dx+3 \int \frac {1}{x^3 \sqrt [4]{x+x^6}} \, dx-\int \frac {1}{x \sqrt [4]{x+x^6}} \, dx \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{x}+\frac {6}{7} \int \frac {1}{x \sqrt [4]{x+x^6}} \, dx+2 \int \frac {x^2}{\sqrt [4]{x+x^6}} \, dx-14 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}-\frac {8 x^3 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {8 x^5 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )}{19 \sqrt [4]{x+x^6}}+12 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}-\frac {48 x^5 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )}{19 \sqrt [4]{x+x^6}}+\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.

Time = 10.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 4.50 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \sqrt [4]{1+x^5} \left (627 \operatorname {Hypergeometric2F1}\left (-\frac {21}{20},\frac {1}{4},-\frac {1}{20},-x^5\right )+7 x^3 \left (-209 \operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )+627 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{20},\frac {1}{4},\frac {19}{20},-x^5\right )-114 x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )+66 x^7 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )\right )\right )}{4389 x^5 \sqrt [4]{x+x^6}} \]

[In]

Integrate[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^6*(x + x^6)^(1/4)),x]

[Out]

(4*(1 + x^5)^(1/4)*(627*Hypergeometric2F1[-21/20, 1/4, -1/20, -x^5] + 7*x^3*(-209*Hypergeometric2F1[-9/20, 1/4
, 11/20, -x^5] + 627*x^2*Hypergeometric2F1[-1/20, 1/4, 19/20, -x^5] - 114*x^5*Hypergeometric2F1[1/4, 11/20, 31
/20, -x^5] + 66*x^7*Hypergeometric2F1[1/4, 19/20, 39/20, -x^5])))/(4389*x^5*(x + x^6)^(1/4))

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
trager	\(\frac {4 \left (3 x^{5}-7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}\)	\(25\)
pseudoelliptic	\(\frac {4 \left (3 x^{5}-7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}\)	\(25\)
risch	\(\frac {\frac {4}{7} x^{10}+\frac {8}{7} x^{5}+\frac {4}{7}-\frac {4}{3} x^{8}-\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}\)	\(37\)
gosper	\(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (3 x^{5}-7 x^{3}+3\right )}{21 x^{5} \left (x^{6}+x \right )^{\frac {1}{4}}}\)	\(44\)
meijerg	\(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {21}{20}, \frac {1}{4}\right ], \left [-\frac {1}{20}\right ], -x^{5}\right )}{7 x^{\frac {21}{4}}}+\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{20}, \frac {1}{4}\right ], \left [\frac {19}{20}\right ], -x^{5}\right )}{x^{\frac {1}{4}}}+\frac {8 x^{\frac {19}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {19}{20}\right ], \left [\frac {39}{20}\right ], -x^{5}\right )}{19}-\frac {8 x^{\frac {11}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {11}{20}\right ], \left [\frac {31}{20}\right ], -x^{5}\right )}{11}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {9}{20}, \frac {1}{4}\right ], \left [\frac {11}{20}\right ], -x^{5}\right )}{3 x^{\frac {9}{4}}}\)	\(82\)

[In]

int((x^5-x^3+1)*(2*x^5-3)/x^6/(x^6+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/21*(3*x^5-7*x^3+3)*(x^6+x)^(3/4)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} - 7 \, x^{3} + 3\right )}}{21 \, x^{6}} \]

[In]

integrate((x^5-x^3+1)*(2*x^5-3)/x^6/(x^6+x)^(1/4),x, algorithm="fricas")

[Out]

4/21*(x^6 + x)^(3/4)*(3*x^5 - 7*x^3 + 3)/x^6

Sympy [F]

\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2 x^{5} - 3\right ) \left (x^{5} - x^{3} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

[In]

integrate((x**5-x**3+1)*(2*x**5-3)/x**6/(x**6+x)**(1/4),x)

[Out]

Integral((2*x**5 - 3)*(x**5 - x**3 + 1)/(x**6*(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))**(1/4)), x)

Maxima [F]

\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]

[In]

integrate((x^5-x^3+1)*(2*x^5-3)/x^6/(x^6+x)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)*(x^5 - x^3 + 1)/((x^6 + x)^(1/4)*x^6), x)

Giac [F]

\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]

[In]

integrate((x^5-x^3+1)*(2*x^5-3)/x^6/(x^6+x)^(1/4),x, algorithm="giac")

[Out]

integrate((2*x^5 - 3)*(x^5 - x^3 + 1)/((x^6 + x)^(1/4)*x^6), x)

Mupad [B] (verification not implemented)

Time = 5.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {12\,{\left (x^6+x\right )}^{3/4}-28\,x^3\,{\left (x^6+x\right )}^{3/4}+12\,x^5\,{\left (x^6+x\right )}^{3/4}}{21\,x^6} \]

[In]

int(((2*x^5 - 3)*(x^5 - x^3 + 1))/(x^6*(x + x^6)^(1/4)),x)

[Out]

(12*(x + x^6)^(3/4) - 28*x^3*(x + x^6)^(3/4) + 12*x^5*(x + x^6)^(3/4))/(21*x^6)