\(\int \sqrt [4]{b x^5+a x^8} \, dx\) [1258]

Optimal result

Integrand size = 15, antiderivative size = 91 \[ \int \sqrt [4]{b x^5+a x^8} \, dx=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {b \arctan \left (\frac {\sqrt [4]{b x^5+a x^8}}{\sqrt [4]{a} x^2}\right )}{6 a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{b x^5+a x^8}}{\sqrt [4]{a} x^2}\right )}{6 a^{3/4}} \]

[Out]

1/3*x*(a*x^8+b*x^5)^(1/4)+1/6*b*arctan((a*x^8+b*x^5)^(1/4)/a^(1/4)/x^2)/a^(3/4)+1/6*b*arctanh((a*x^8+b*x^5)^(1
/4)/a^(1/4)/x^2)/a^(3/4)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.64, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2029, 2057, 335, 281, 338, 304, 209, 212} \[ \int \sqrt [4]{b x^5+a x^8} \, dx=-\frac {b x^{15/4} \left (a x^3+b\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{6 a^{3/4} \left (a x^8+b x^5\right )^{3/4}}+\frac {b x^{15/4} \left (a x^3+b\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{6 a^{3/4} \left (a x^8+b x^5\right )^{3/4}}+\frac {1}{3} x \sqrt [4]{a x^8+b x^5} \]

[In]

Int[(b*x^5 + a*x^8)^(1/4),x]

[Out]

(x*(b*x^5 + a*x^8)^(1/4))/3 - (b*x^(15/4)*(b + a*x^3)^(3/4)*ArcTan[(a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(6*a^
(3/4)*(b*x^5 + a*x^8)^(3/4)) + (b*x^(15/4)*(b + a*x^3)^(3/4)*ArcTanh[(a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(6*
a^(3/4)*(b*x^5 + a*x^8)^(3/4))

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 2029

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {1}{4} b \int \frac {x^5}{\left (b x^5+a x^8\right )^{3/4}} \, dx \\ & = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (b+a x^3\right )^{3/4}} \, dx}{4 \left (b x^5+a x^8\right )^{3/4}} \\ & = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (b+a x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (b x^5+a x^8\right )^{3/4}} \\ & = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{3 \left (b x^5+a x^8\right )^{3/4}} \\ & = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \left (b x^5+a x^8\right )^{3/4}} \\ & = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 \sqrt {a} \left (b x^5+a x^8\right )^{3/4}}-\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 \sqrt {a} \left (b x^5+a x^8\right )^{3/4}} \\ & = \frac {1}{3} x \sqrt [4]{b x^5+a x^8}-\frac {b x^{15/4} \left (b+a x^3\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 a^{3/4} \left (b x^5+a x^8\right )^{3/4}}+\frac {b x^{15/4} \left (b+a x^3\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 a^{3/4} \left (b x^5+a x^8\right )^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.70 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26 \[ \int \sqrt [4]{b x^5+a x^8} \, dx=\frac {\sqrt [4]{x^5 \left (b+a x^3\right )} \left (2 a^{3/4} x^{9/4} \sqrt [4]{b+a x^3}-b \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )+b \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )\right )}{6 a^{3/4} x^{5/4} \sqrt [4]{b+a x^3}} \]

[In]

Integrate[(b*x^5 + a*x^8)^(1/4),x]

[Out]

((x^5*(b + a*x^3))^(1/4)*(2*a^(3/4)*x^(9/4)*(b + a*x^3)^(1/4) - b*ArcTan[(a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)]
+ b*ArcTanh[(a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)]))/(6*a^(3/4)*x^(5/4)*(b + a*x^3)^(1/4))

Maple [A] (verified)

Time = 3.52 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {4 \left (x^{5} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}} x \,a^{\frac {3}{4}}+\ln \left (\frac {-a^{\frac {1}{4}} x^{2}-\left (x^{5} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x^{2}-\left (x^{5} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (x^{5} \left (a \,x^{3}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x^{2}}\right ) b}{12 a^{\frac {3}{4}}}\)	\(103\)

[In]

int((a*x^8+b*x^5)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/12*(4*(x^5*(a*x^3+b))^(1/4)*x*a^(3/4)+ln((-a^(1/4)*x^2-(x^5*(a*x^3+b))^(1/4))/(a^(1/4)*x^2-(x^5*(a*x^3+b))^(
1/4)))*b+2*arctan((x^5*(a*x^3+b))^(1/4)/a^(1/4)/x^2)*b)/a^(3/4)

Fricas [F(-1)]

Timed out. \[ \int \sqrt [4]{b x^5+a x^8} \, dx=\text {Timed out} \]

[In]

integrate((a*x^8+b*x^5)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \sqrt [4]{b x^5+a x^8} \, dx=\int \sqrt [4]{a x^{8} + b x^{5}}\, dx \]

[In]

integrate((a*x**8+b*x**5)**(1/4),x)

[Out]

Integral((a*x**8 + b*x**5)**(1/4), x)

Maxima [F]

\[ \int \sqrt [4]{b x^5+a x^8} \, dx=\int { {\left (a x^{8} + b x^{5}\right )}^{\frac {1}{4}} \,d x } \]

[In]

integrate((a*x^8+b*x^5)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^8 + b*x^5)^(1/4), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (71) = 142\).

Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.34 \[ \int \sqrt [4]{b x^5+a x^8} \, dx=\frac {8 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} b x^{3} + \frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{a} + \frac {\sqrt {2} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{\left (-a\right )^{\frac {3}{4}}}}{24 \, b} \]

[In]

integrate((a*x^8+b*x^5)^(1/4),x, algorithm="giac")

[Out]

1/24*(8*(a + b/x^3)^(1/4)*b*x^3 + 2*sqrt(2)*(-a)^(1/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x
^3)^(1/4))/(-a)^(1/4))/a + 2*sqrt(2)*(-a)^(1/4)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x^3)^(1
/4))/(-a)^(1/4))/a + sqrt(2)*(-a)^(1/4)*b^2*log(sqrt(2)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(-a) + sqrt(a + b/x
^3))/a + sqrt(2)*b^2*log(-sqrt(2)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(-a) + sqrt(a + b/x^3))/(-a)^(3/4))/b

Mupad [B] (verification not implemented)

Time = 5.94 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.46 \[ \int \sqrt [4]{b x^5+a x^8} \, dx=\frac {4\,x\,{\left (a\,x^8+b\,x^5\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {a\,x^3}{b}\right )}{9\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}} \]

[In]

int((a*x^8 + b*x^5)^(1/4),x)

[Out]

(4*x*(a*x^8 + b*x^5)^(1/4)*hypergeom([-1/4, 3/4], 7/4, -(a*x^3)/b))/(9*((a*x^3)/b + 1)^(1/4))