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

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

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Rubi [A]  time = 0.40, antiderivative size = 309, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2042, 105, 63, 331, 298, 203, 206, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a d-b c} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a d-b c} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [4]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{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]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

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

Rule 298

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 2042

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [C]  time = 0.07, size = 105, normalized size = 0.36 \begin {gather*} -\frac {4 \sqrt [4]{x^3 (a x-b)} \left ((b c-a d) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {b c x-a d x}{b d-a d x}\right )+a d \left (1-\frac {a x}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {a x}{b}\right )\right )}{3 c d (b-a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(x^3*(-b + a*x))^(1/4)*(a*d*(1 - (a*x)/b)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (a*x)/b] + (b*c - a*d)*Hy
pergeometric2F1[3/4, 1, 7/4, (b*c*x - a*d*x)/(b*d - a*d*x)]))/(3*c*d*(b - a*x))

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IntegrateAlgebraic [A]  time = 1.38, size = 293, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {\sqrt {2} \sqrt [4]{b c-a d} \tan ^{-1}\left (\frac {\sqrt {2} \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-\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 {\sqrt {2} \sqrt [4]{b c-a d} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b c-a d} x^2}{\sqrt {2} \sqrt [4]{d}}+\frac {\sqrt [4]{d} \sqrt {-b x^3+a x^4}}{\sqrt {2} \sqrt [4]{b c-a d}}}{x \sqrt [4]{-b x^3+a x^4}}\right )}{c \sqrt [4]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, 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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giac [B]  time = 0.51, 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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maple [F]  time = 0.08, 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*} \int \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (c x - d\right )} x}\,{d x} \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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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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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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