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

Optimal. Leaf size=291 \[ -\frac {2 \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \text {ArcTan}\left (\frac {(1+i) \sqrt [4]{d} \sqrt [4]{-b c+a d} x \sqrt [4]{-b x^3+a x^4}}{\sqrt {-b c+a d} x^2-i \sqrt {d} \sqrt {-b x^3+a x^4}}\right )}{c \sqrt [4]{d}}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{-b c+a d} x^2}{\sqrt [4]{d}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{d} \sqrt {-b x^3+a x^4}}{\sqrt [4]{-b c+a d}}}{x \sqrt [4]{-b x^3+a x^4}}\right )}{c \sqrt [4]{d}} \]

[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)

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.19, antiderivative size = 59, normalized size of antiderivative = 0.20, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2067, 129, 525, 524} \begin {gather*} -\frac {4 \sqrt [4]{a x^4-b x^3} F_1\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 {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 1.03, size = 303, normalized size = 1.04 \begin {gather*} -\frac {x^{9/4} (-b+a x)^{3/4} \left (2 \sqrt [4]{a} \sqrt [4]{d} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \text {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} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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)))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{x \left (c x -d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.39, size = 441, normalized size = 1.52 \begin {gather*} 4 \, \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \arctan \left (-\frac {c^{3} d x \sqrt {\frac {c^{2} x^{2} \sqrt {-\frac {b c - a d}{c^{4} d}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {3}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} c^{3} d \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {3}{4}}}{{\left (b c - a d\right )} x}\right ) - 4 \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {c^{3} x \sqrt {\frac {c^{2} x^{2} \sqrt {\frac {a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} c^{3} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}}}{a 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 ) - \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 {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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{x \left (c x - d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (228) = 456\).
time = 0.45, size = 505, normalized size = 1.74 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (d-c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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