\(\int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx\) [2656]

Optimal result

Integrand size = 26, antiderivative size = 237 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {2 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{a}-\frac {\sqrt {2} \sqrt [4]{a-b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} x \sqrt [4]{-x^3+x^4}}{\sqrt {a-b} x^2-\sqrt {b} \sqrt {-x^3+x^4}}\right )}{a \sqrt [4]{b}}+\frac {2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{a}-\frac {\sqrt {2} \sqrt [4]{a-b} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a-b} x^2}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{-x^3+x^4}}\right )}{a \sqrt [4]{b}} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2067, 129, 525, 524} \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {4 \sqrt [4]{x^4-x^3} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,\frac {a x}{b}\right )}{3 b \sqrt [4]{1-x}} \]

[In]

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

[Out]

(-4*(-x^3 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x, (a*x)/b])/(3*b*(1 - x)^(1/4))

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 2067

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]
:> Dist[e^IntPart[m]*(e*x)^FracPart[m]*((a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a
+ b*x^n)^FracPart[p])), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,
p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} (-b+a x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{1-x} x^{3/4}} \\ & = -\frac {4 \sqrt [4]{-x^3+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,\frac {a x}{b}\right )}{3 b \sqrt [4]{1-x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {(-1+x)^{3/4} x^{9/4} \left (2 \sqrt [4]{b} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\sqrt {2} \sqrt [4]{a-b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}{-\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}\right )-2 \sqrt [4]{b} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\sqrt {2} \sqrt [4]{a-b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}\right )\right )}{a \sqrt [4]{b} \left ((-1+x) x^3\right )^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {-\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{\sqrt {x^{3} \left (-1+x \right )}-\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{4}} \sqrt {2}-2 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}+\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{4}} \sqrt {2}-2 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}-\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{4}} \sqrt {2}+2 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )-2 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )+4 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2 a}\)	\(301\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (-\frac {a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {i \, a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 2 \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right )}{a} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=\frac {2 \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}{a} + \frac {\log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right )}{a} - \frac {\log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right )}{a} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \log \left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{2 \, a b} + \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{2 \, a b} \]

[In]

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

[Out]

2*arctan((-1/x + 1)^(1/4))/a + log((-1/x + 1)^(1/4) + 1)/a - log(abs((-1/x + 1)^(1/4) - 1))/a - sqrt(2)*(a*b^3
- b^4)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*((a - b)/b)^(1/4) + 2*(-1/x + 1)^(1/4))/((a - b)/b)^(1/4))/(a*b) - s
qrt(2)*(a*b^3 - b^4)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*((a - b)/b)^(1/4) - 2*(-1/x + 1)^(1/4))/((a - b)/b)^(1
/4))/(a*b) - 1/2*sqrt(2)*(a*b^3 - b^4)^(1/4)*log(sqrt(2)*((a - b)/b)^(1/4)*(-1/x + 1)^(1/4) + sqrt((a - b)/b)
+ sqrt(-1/x + 1))/(a*b) + 1/2*sqrt(2)*(a*b^3 - b^4)^(1/4)*log(-sqrt(2)*((a - b)/b)^(1/4)*(-1/x + 1)^(1/4) + sq
rt((a - b)/b) + sqrt(-1/x + 1))/(a*b)

Mupad [F(-1)]

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

[In]

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

[Out]

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