\(\int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx\) [2845]

Optimal result

Integrand size = 29, antiderivative size = 291 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \arctan \left (\frac {(1+i) \sqrt [4]{d} \sqrt [4]{-b c+a d} x \sqrt [4]{-b x^3+a x^4}}{\sqrt {-b c+a d} x^2-i \sqrt {d} \sqrt {-b x^3+a x^4}}\right )}{c \sqrt [4]{d}}+\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \text {arctanh}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{-b c+a d} x^2}{\sqrt [4]{d}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{d} \sqrt {-b x^3+a x^4}}{\sqrt [4]{-b c+a d}}}{x \sqrt [4]{-b x^3+a x^4}}\right )}{c \sqrt [4]{d}} \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2067, 129, 525, 524} \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=-\frac {4 \sqrt [4]{a x^4-b x^3} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},\frac {a x}{b},\frac {c x}{d}\right )}{3 d \sqrt [4]{1-\frac {a x}{b}}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=-\frac {x^{9/4} (-b+a x)^{3/4} \left (2 \sqrt [4]{a} \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{x} \sqrt [4]{-b+a x}}{\sqrt {b c-a d} \sqrt {x}-\sqrt {d} \sqrt {-b+a x}}\right )-2 \sqrt [4]{a} \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}+\sqrt {d} \sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{x} \sqrt [4]{-b+a x}}\right )\right )}{c \sqrt [4]{d} \left (x^3 (-b+a x)\right )^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(\frac {a^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right )+2 a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\ln \left (\frac {\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}-2 \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}{c}\)	\(208\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {i \, c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {-i \, c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (228) = 456\).

Time = 0.32 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{c d} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{c d} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {a - \frac {b}{x}} + \sqrt {\frac {b c - a d}{d}}\right )}{2 \, c d} + \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {a - \frac {b}{x}} + \sqrt {\frac {b c - a d}{d}}\right )}{2 \, c d} \]

[In]

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

[Out]

sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x)^(1/4))/(-a)^(1/4))/c + sqrt(2)*(-a)^(1
/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a - b/x)^(1/4))/(-a)^(1/4))/c + 1/2*sqrt(2)*(-a)^(1/4)*log(sq
rt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/c - 1/2*sqrt(2)*(-a)^(1/4)*log(-sqrt(2)*(-a)^(1/4
)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/c - sqrt(2)*(b*c*d^3 - a*d^4)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*
((b*c - a*d)/d)^(1/4) + 2*(a - b/x)^(1/4))/((b*c - a*d)/d)^(1/4))/(c*d) - sqrt(2)*(b*c*d^3 - a*d^4)^(1/4)*arct
an(-1/2*sqrt(2)*(sqrt(2)*((b*c - a*d)/d)^(1/4) - 2*(a - b/x)^(1/4))/((b*c - a*d)/d)^(1/4))/(c*d) - 1/2*sqrt(2)
*(b*c*d^3 - a*d^4)^(1/4)*log(sqrt(2)*(a - b/x)^(1/4)*((b*c - a*d)/d)^(1/4) + sqrt(a - b/x) + sqrt((b*c - a*d)/
d))/(c*d) + 1/2*sqrt(2)*(b*c*d^3 - a*d^4)^(1/4)*log(-sqrt(2)*(a - b/x)^(1/4)*((b*c - a*d)/d)^(1/4) + sqrt(a -
b/x) + sqrt((b*c - a*d)/d))/(c*d)

Mupad [F(-1)]

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

[In]

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

[Out]

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