\(\int \frac {1}{(-b+a x^3) \sqrt [4]{b x+a x^4}} \, dx\) [1463]

Optimal result

Integrand size = 25, antiderivative size = 103 \[ \int \frac {1}{\left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a} b}-\frac {2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a} b} \]

[Out]

-1/3*2^(3/4)*arctan(2^(1/4)*a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))/a^(1/4)/b-1/3*2^(3/4)*arctanh(2^(1/4)*a^(1/4)
*(a*x^4+b*x)^(3/4)/(a*x^3+b))/a^(1/4)/b

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.45, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2081, 477, 476, 385, 218, 212, 209} \[ \int \frac {1}{\left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3+b} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3+b} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4+b x}} \]

[In]

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

[Out]

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

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 476

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

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 2081

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {4 x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {2 a x^3}{b+a x^3}\right )}{3 b \sqrt [4]{x \left (b+a x^3\right )}} \]

[In]

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

[Out]

(-4*x*Hypergeometric2F1[1/4, 1, 5/4, (2*a*x^3)/(b + a*x^3)])/(3*b*(x*(b + a*x^3))^(1/4))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right )\right )}{6 a^{\frac {1}{4}} b}\)	\(87\)

[In]

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

[Out]

-1/6*2^(3/4)*(-2*arctan(1/2*(x*(a*x^3+b))^(1/4)/x*2^(3/4)/a^(1/4))+ln((-x*2^(1/4)*a^(1/4)-(x*(a*x^3+b))^(1/4))
/(x*2^(1/4)*a^(1/4)-(x*(a*x^3+b))^(1/4))))/a^(1/4)/b

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 57.02 (sec) , antiderivative size = 516, normalized size of antiderivative = 5.01 \[ \int \frac {1}{\left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} + b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}}}{a x^{3} - b}\right ) + \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} + b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}}}{a x^{3} - b}\right ) + \frac {1}{6} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{3} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}}}{a x^{3} - b}\right ) - \frac {1}{6} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{3} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}}}{a x^{3} - b}\right ) \]

[In]

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

[Out]

-1/6*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log((4*(1/2)^(3/4)*sqrt(a*x^4 + b*x)*a*b^3*x*(1/(a*b^4))^(3/4) + 4*sqrt(1/2
)*(a*x^4 + b*x)^(1/4)*a*b^2*x^2*sqrt(1/(a*b^4)) + (1/2)^(1/4)*(3*a*b*x^3 + b^2)*(1/(a*b^4))^(1/4) + 2*(a*x^4 +
b*x)^(3/4))/(a*x^3 - b)) + 1/6*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(-(4*(1/2)^(3/4)*sqrt(a*x^4 + b*x)*a*b^3*x*(1
/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 + b*x)^(1/4)*a*b^2*x^2*sqrt(1/(a*b^4)) + (1/2)^(1/4)*(3*a*b*x^3 + b^2)*(1
/(a*b^4))^(1/4) - 2*(a*x^4 + b*x)^(3/4))/(a*x^3 - b)) + 1/6*I*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log((4*I*(1/2)^(3/
4)*sqrt(a*x^4 + b*x)*a*b^3*x*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 + b*x)^(1/4)*a*b^2*x^2*sqrt(1/(a*b^4)) - (
1/2)^(1/4)*(3*I*a*b*x^3 + I*b^2)*(1/(a*b^4))^(1/4) + 2*(a*x^4 + b*x)^(3/4))/(a*x^3 - b)) - 1/6*I*(1/2)^(1/4)*(
1/(a*b^4))^(1/4)*log((-4*I*(1/2)^(3/4)*sqrt(a*x^4 + b*x)*a*b^3*x*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 + b*x)
^(1/4)*a*b^2*x^2*sqrt(1/(a*b^4)) - (1/2)^(1/4)*(-3*I*a*b*x^3 - I*b^2)*(1/(a*b^4))^(1/4) + 2*(a*x^4 + b*x)^(3/4
))/(a*x^3 - b))

Sympy [F]

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

[In]

integrate(1/(a*x**3-b)/(a*x**4+b*x)**(1/4),x)

[Out]

Integral(1/((x*(a*x**3 + b))**(1/4)*(a*x**3 - b)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (79) = 158\).

Time = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} - \frac {2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a b} \]

[In]

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

[Out]

-1/3*2^(1/4)*(-a)^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/(a*b) - 1/3*
2^(1/4)*(-a)^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/(a*b) - 1/6*2^(1
/4)*log(2^(3/4)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/((-a)^(1/4)*b) - 1/6*2^(1/4
)*(-a)^(3/4)*log(-2^(3/4)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/(a*b)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\int \frac {1}{{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \]

[In]

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

[Out]

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