\(\int \frac {(3+x^2) (1+x^2+x^3)}{x^6 \sqrt [4]{x+x^3}} \, dx\) [314]

Optimal result

Integrand size = 26, antiderivative size = 28 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \left (x+x^3\right )^{3/4} \left (3+3 x^2+7 x^3\right )}{21 x^6} \]

[Out]

-4/21*(x^3+x)^(3/4)*(7*x^3+3*x^2+3)/x^6

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 21, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2077, 2050, 2036, 371, 2057} \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \left (x^3+x\right )^{3/4}}{3 x^3}-\frac {4 \left (x^3+x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^3+x\right )^{3/4}}{7 x^4} \]

[In]

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

[Out]

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

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 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

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^3}}+\frac {4}{x^4 \sqrt [4]{x+x^3}}+\frac {3}{x^3 \sqrt [4]{x+x^3}}+\frac {1}{x^2 \sqrt [4]{x+x^3}}+\frac {1}{x \sqrt [4]{x+x^3}}\right ) \, dx \\ & = 3 \int \frac {1}{x^6 \sqrt [4]{x+x^3}} \, dx+3 \int \frac {1}{x^3 \sqrt [4]{x+x^3}} \, dx+4 \int \frac {1}{x^4 \sqrt [4]{x+x^3}} \, dx+\int \frac {1}{x^2 \sqrt [4]{x+x^3}} \, dx+\int \frac {1}{x \sqrt [4]{x+x^3}} \, dx \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {16 \left (x+x^3\right )^{3/4}}{13 x^4}-\frac {4 \left (x+x^3\right )^{3/4}}{3 x^3}-\frac {4 \left (x+x^3\right )^{3/4}}{5 x^2}-\frac {4 \left (x+x^3\right )^{3/4}}{x}+\frac {1}{5} \int \frac {1}{\sqrt [4]{x+x^3}} \, dx-\frac {15}{7} \int \frac {1}{x^4 \sqrt [4]{x+x^3}} \, dx-\frac {28}{13} \int \frac {1}{x^2 \sqrt [4]{x+x^3}} \, dx+5 \int \frac {x}{\sqrt [4]{x+x^3}} \, dx-\int \frac {1}{x \sqrt [4]{x+x^3}} \, dx \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}-\frac {4 \left (x+x^3\right )^{3/4}}{3 x^3}+\frac {12 \left (x+x^3\right )^{3/4}}{13 x^2}-\frac {28}{65} \int \frac {1}{\sqrt [4]{x+x^3}} \, dx+\frac {15}{13} \int \frac {1}{x^2 \sqrt [4]{x+x^3}} \, dx-5 \int \frac {x}{\sqrt [4]{x+x^3}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1+x^2}} \, dx}{5 \sqrt [4]{x+x^3}}+\frac {\left (5 \sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{x+x^3}} \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}-\frac {4 \left (x+x^3\right )^{3/4}}{3 x^3}+\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{8},\frac {11}{8},-x^2\right )}{15 \sqrt [4]{x+x^3}}+\frac {20 x^2 \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{8},\frac {15}{8},-x^2\right )}{7 \sqrt [4]{x+x^3}}+\frac {3}{13} \int \frac {1}{\sqrt [4]{x+x^3}} \, dx-\frac {\left (28 \sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1+x^2}} \, dx}{65 \sqrt [4]{x+x^3}}-\frac {\left (5 \sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{x+x^3}} \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}-\frac {4 \left (x+x^3\right )^{3/4}}{3 x^3}-\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{8},\frac {11}{8},-x^2\right )}{13 \sqrt [4]{x+x^3}}+\frac {\left (3 \sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1+x^2}} \, dx}{13 \sqrt [4]{x+x^3}} \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}-\frac {4 \left (x+x^3\right )^{3/4}}{3 x^3} \\ \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 = 124, normalized size of antiderivative = 4.43 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \sqrt [4]{1+x^2} \left (195 \operatorname {Hypergeometric2F1}\left (-\frac {21}{8},\frac {1}{4},-\frac {13}{8},-x^2\right )+7 x^2 \left (60 \operatorname {Hypergeometric2F1}\left (-\frac {13}{8},\frac {1}{4},-\frac {5}{8},-x^2\right )+13 x \left (5 \operatorname {Hypergeometric2F1}\left (-\frac {9}{8},\frac {1}{4},-\frac {1}{8},-x^2\right )+3 x \left (\operatorname {Hypergeometric2F1}\left (-\frac {5}{8},\frac {1}{4},\frac {3}{8},-x^2\right )+5 x \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {1}{4},\frac {7}{8},-x^2\right )\right )\right )\right )\right )}{1365 x^5 \sqrt [4]{x+x^3}} \]

[In]

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

[Out]

(-4*(1 + x^2)^(1/4)*(195*Hypergeometric2F1[-21/8, 1/4, -13/8, -x^2] + 7*x^2*(60*Hypergeometric2F1[-13/8, 1/4,
-5/8, -x^2] + 13*x*(5*Hypergeometric2F1[-9/8, 1/4, -1/8, -x^2] + 3*x*(Hypergeometric2F1[-5/8, 1/4, 3/8, -x^2]
+ 5*x*Hypergeometric2F1[-1/8, 1/4, 7/8, -x^2])))))/(1365*x^5*(x + x^3)^(1/4))

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
trager	\(-\frac {4 \left (x^{3}+x \right )^{\frac {3}{4}} \left (7 x^{3}+3 x^{2}+3\right )}{21 x^{6}}\)	\(25\)
gosper	\(-\frac {4 \left (x^{2}+1\right ) \left (7 x^{3}+3 x^{2}+3\right )}{21 x^{5} \left (x^{3}+x \right )^{\frac {1}{4}}}\)	\(30\)
risch	\(-\frac {4 \left (7 x^{5}+3 x^{4}+7 x^{3}+6 x^{2}+3\right )}{21 x^{5} {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{4}}}\)	\(37\)
meijerg	\(-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {21}{8}, \frac {1}{4}\right ], \left [-\frac {13}{8}\right ], -x^{2}\right )}{7 x^{\frac {21}{4}}}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {9}{8}, \frac {1}{4}\right ], \left [-\frac {1}{8}\right ], -x^{2}\right )}{3 x^{\frac {9}{4}}}-\frac {16 \operatorname {hypergeom}\left (\left [-\frac {13}{8}, \frac {1}{4}\right ], \left [-\frac {5}{8}\right ], -x^{2}\right )}{13 x^{\frac {13}{4}}}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{8}, \frac {1}{4}\right ], \left [\frac {7}{8}\right ], -x^{2}\right )}{x^{\frac {1}{4}}}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {5}{8}, \frac {1}{4}\right ], \left [\frac {3}{8}\right ], -x^{2}\right )}{5 x^{\frac {5}{4}}}\)	\(82\)

[In]

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

[Out]

-4/21*(x^3+x)^(3/4)*(7*x^3+3*x^2+3)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \, {\left (7 \, x^{3} + 3 \, x^{2} + 3\right )} {\left (x^{3} + x\right )}^{\frac {3}{4}}}{21 \, x^{6}} \]

[In]

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

[Out]

-4/21*(7*x^3 + 3*x^2 + 3)*(x^3 + x)^(3/4)/x^6

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4}{7} \, {\left (\frac {1}{x} + \frac {1}{x^{3}}\right )}^{\frac {7}{4}} - \frac {4}{3} \, {\left (\frac {1}{x} + \frac {1}{x^{3}}\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

-4/7*(1/x + 1/x^3)^(7/4) - 4/3*(1/x + 1/x^3)^(3/4)

Mupad [B] (verification not implemented)

Time = 5.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {12\,{\left (x^3+x\right )}^{3/4}+12\,x^2\,{\left (x^3+x\right )}^{3/4}+28\,x^3\,{\left (x^3+x\right )}^{3/4}}{21\,x^6} \]

[In]

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

[Out]

-(12*(x + x^3)^(3/4) + 12*x^2*(x + x^3)^(3/4) + 28*x^3*(x + x^3)^(3/4))/(21*x^6)