\(\int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx\) [227]

Optimal result

Integrand size = 15, antiderivative size = 23 \[ \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx=\frac {4 (-3+4 x) \left (x^3+x^4\right )^{3/4}}{21 x^4} \]

[Out]

4/21*(-3+4*x)*(x^4+x^3)^(3/4)/x^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2041, 2039} \[ \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx=\frac {16 \left (x^4+x^3\right )^{3/4}}{21 x^3}-\frac {4 \left (x^4+x^3\right )^{3/4}}{7 x^4} \]

[In]

Int[1/(x^2*(x^3 + x^4)^(1/4)),x]

[Out]

(-4*(x^3 + x^4)^(3/4))/(7*x^4) + (16*(x^3 + x^4)^(3/4))/(21*x^3)

Rule 2039

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*(n - j)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (x^3+x^4\right )^{3/4}}{7 x^4}-\frac {4}{7} \int \frac {1}{x \sqrt [4]{x^3+x^4}} \, dx \\ & = -\frac {4 \left (x^3+x^4\right )^{3/4}}{7 x^4}+\frac {16 \left (x^3+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx=\frac {4 \left (x^3 (1+x)\right )^{3/4} (-3+4 x)}{21 x^4} \]

[In]

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

[Out]

(4*(x^3*(1 + x))^(3/4)*(-3 + 4*x))/(21*x^4)

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70

method	result	size
meijerg	\(-\frac {4 \left (1-\frac {4 x}{3}\right ) \left (1+x \right )^{\frac {3}{4}}}{7 x^{\frac {7}{4}}}\)	\(16\)
trager	\(\frac {4 \left (-3+4 x \right ) \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{21 x^{4}}\)	\(20\)
pseudoelliptic	\(\frac {4 \left (-3+4 x \right ) \left (x^{3} \left (1+x \right )\right )^{\frac {3}{4}}}{21 x^{4}}\)	\(20\)
gosper	\(\frac {4 \left (1+x \right ) \left (-3+4 x \right )}{21 x \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\)	\(23\)
risch	\(\frac {-\frac {4}{7}+\frac {4}{21} x +\frac {16}{21} x^{2}}{x \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(23\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx=\frac {4 \, {\left (x^{4} + x^{3}\right )}^{\frac {3}{4}} {\left (4 \, x - 3\right )}}{21 \, x^{4}} \]

[In]

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

[Out]

4/21*(x^4 + x^3)^(3/4)*(4*x - 3)/x^4

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx=-\frac {4}{7} \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx=\frac {16\,x\,{\left (x^4+x^3\right )}^{3/4}-12\,{\left (x^4+x^3\right )}^{3/4}}{21\,x^4} \]

[In]

int(1/(x^2*(x^3 + x^4)^(1/4)),x)

[Out]

(16*x*(x^3 + x^4)^(3/4) - 12*(x^3 + x^4)^(3/4))/(21*x^4)