\(\int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx\) [2366]

Optimal result

Integrand size = 23, antiderivative size = 188 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} x \sqrt [4]{-x^3+x^4}}{\sqrt {a-b} x^2-\sqrt {b} \sqrt {-x^3+x^4}}\right )}{\sqrt [4]{a-b} b^{3/4}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a-b} x^2}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{-x^3+x^4}}\right )}{\sqrt [4]{a-b} b^{3/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.74, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2081, 95, 218, 214, 211} \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {2 \sqrt [4]{x-1} x^{3/4} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{b^{3/4} \sqrt [4]{x^4-x^3} \sqrt [4]{b-a}}-\frac {2 \sqrt [4]{x-1} x^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{b^{3/4} \sqrt [4]{x^4-x^3} \sqrt [4]{b-a}} \]

[In]

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

[Out]

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

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {\sqrt {2} \sqrt [4]{-1+x} x^{3/4} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}{-\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}\right )-\text {arctanh}\left (\frac {\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}\right )\right )}{\sqrt [4]{a-b} b^{3/4} \sqrt [4]{(-1+x) x^3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}\right )+2 \arctan \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x +\sqrt {2}\, \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x -\sqrt {2}\, \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right )\right )}{2 \left (\frac {a -b}{b}\right )^{\frac {1}{4}} b}\)	\(217\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, {\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, {\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (147) = 294\).

Time = 0.29 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {2 \, {\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} + \frac {2 \, {\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} - \frac {{\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \log \left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} + \frac {{\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \log \left (-\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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