\(\int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 (-d+c x^2)} \, dx\) [2384]

Optimal result

Integrand size = 31, antiderivative size = 191 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{2 d^2} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(191)=382\).

Time = 0.90 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.24, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2081, 922, 37, 6857, 95, 304, 211, 214} \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}+\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x-b}}+\frac {4 \sqrt [4]{a x^4-b x^3}}{d x} \]

[In]

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

[Out]

(4*(-(b*x^3) + a*x^4)^(1/4))/(d*x) + ((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[((-(b*S
qrt[c]) + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(d^(9/8)*x^(3/4)*(-b + a*x)^(1/4)) + ((b*Sqrt
[c] + a*Sqrt[d])^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b +
a*x)^(1/4))])/(d^(9/8)*x^(3/4)*(-b + a*x)^(1/4)) - ((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*
ArcTanh[((-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(-b + a*x)^(1/4))])/(d^(9/8)*x^(3/4)*(-b + a*x)^(1
/4)) - ((b*Sqrt[c] + a*Sqrt[d])^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))
/(d^(1/8)*(-b + a*x)^(1/4))])/(d^(9/8)*x^(3/4)*(-b + a*x)^(1/4))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 304

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 922

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(-g)*((e*f
- d*g)/(c*f^2 + a*g^2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n, x], x] + Dist[1/(c*f^2 + a*g^2), Int[Simp[c*d*f +
a*e*g + c*(e*f - d*g)*x, x]*(d + e*x)^(m - 1)*((f + g*x)^(n + 1)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f,
g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] && LtQ[n, -1]

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]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{x^{5/4} \left (-d+c x^2\right )} \, dx}{x^{3/4} \sqrt [4]{-b+a x}} \\ & = -\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {-a d+b c x}{\sqrt [4]{x} (-b+a x)^{3/4} \left (-d+c x^2\right )} \, dx}{d x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (b \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (-b+a x)^{3/4}} \, dx}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}-\frac {\sqrt [4]{-b x^3+a x^4} \int \left (-\frac {b \sqrt {c} d-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {c} x\right )}-\frac {-b \sqrt {c} d-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {c} x\right )}\right ) \, dx}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\left (\left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {c} x\right )} \, dx}{2 d x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (b \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {c} x\right )} \, dx}{2 d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\left (2 \left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (-b \sqrt {c}+a \sqrt {d}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{d x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \left (b \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (b \sqrt {c}+a \sqrt {d}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\left (\left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {-b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{\sqrt {-b \sqrt {c}+a \sqrt {d}} d x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {-b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{\sqrt {-b \sqrt {c}+a \sqrt {d}} d x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\sqrt {b \sqrt {c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{d x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\sqrt {b \sqrt {c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {b \sqrt {c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{d x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {4 \sqrt [4]{-b x^3+a x^4}}{d x}+\frac {\sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{-b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{b \sqrt {c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{-b+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\frac {32 d x^2 (-b+a x)-x^{9/4} (-b+a x)^{3/4} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 d^2 \left (x^3 (-b+a x)\right )^{3/4}} \]

[In]

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

[Out]

(32*d*x^2*(-b + a*x) - x^(9/4)*(-b + a*x)^(3/4)*RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (b^2*c*Log[x]
- a^2*d*Log[x] - 4*b^2*c*Log[(-b + a*x)^(1/4) - x^(1/4)*#1] + 4*a^2*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#1] + a*d
*Log[x]*#1^4 - 4*a*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(8*d^2*(x^3*(-b + a*x))^
(3/4))

Maple [N/A]

Time = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {8 \left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}} d +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} d -2 \textit {\_Z}^{4} a d +a^{2} d -b^{2} c \right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d +b^{2} c \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x}{2 d^{2} x}\)	\(110\)

[In]

int((a*x^4-b*x^3)^(1/4)/x^2/(c*x^2-d),x,method=_RETURNVERBOSE)

[Out]

1/2*(8*(x^3*(a*x-b))^(1/4)*d+sum((_R^4*a*d-a^2*d+b^2*c)*ln((-_R*x+(x^3*(a*x-b))^(1/4))/x)/_R^3/(_R^4-a),_R=Roo
tOf(_Z^8*d-2*_Z^4*a*d+a^2*d-b^2*c))*x)/d^2/x

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.28 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} \log \left (\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} \log \left (-\frac {d x \sqrt {-\sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} \log \left (\frac {d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} \log \left (-\frac {d x \sqrt {-\sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 8 \, {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, d x} \]

[In]

integrate((a*x^4-b*x^3)^(1/4)/x^2/(c*x^2-d),x, algorithm="fricas")

[Out]

-1/2*(d*x*sqrt(-sqrt((d^4*sqrt(b^2*c/d^9) + a)/d^4))*log((d*x*sqrt(-sqrt((d^4*sqrt(b^2*c/d^9) + a)/d^4)) + (a*
x^4 - b*x^3)^(1/4))/x) - d*x*sqrt(-sqrt((d^4*sqrt(b^2*c/d^9) + a)/d^4))*log(-(d*x*sqrt(-sqrt((d^4*sqrt(b^2*c/d
^9) + a)/d^4)) - (a*x^4 - b*x^3)^(1/4))/x) + d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c/d^9) - a)/d^4))*log((d*x*sqrt(-sq
rt(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)) + (a*x^4 - b*x^3)^(1/4))/x) - d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c/d^9) - a)/d^
4))*log(-(d*x*sqrt(-sqrt(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)) - (a*x^4 - b*x^3)^(1/4))/x) + d*x*((d^4*sqrt(b^2*c/d
^9) + a)/d^4)^(1/4)*log((d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - d*x*((d^4*sqr
t(b^2*c/d^9) + a)/d^4)^(1/4)*log(-(d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4) - (a*x^4 - b*x^3)^(1/4))/x) + d*x
*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*log((d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4) + (a*x^4 - b*x^3)^(1/4
))/x) - d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*log(-(d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4) - (a*x^4 -
b*x^3)^(1/4))/x) - 8*(a*x^4 - b*x^3)^(1/4))/(d*x)

Sympy [N/A]

Not integrable

Time = 2.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{x^{2} \left (c x^{2} - d\right )}\, dx \]

[In]

integrate((a*x**4-b*x**3)**(1/4)/x**2/(c*x**2-d),x)

[Out]

Integral((x**3*(a*x - b))**(1/4)/(x**2*(c*x**2 - d)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (c x^{2} - d\right )} x^{2}} \,d x } \]

[In]

integrate((a*x^4-b*x^3)^(1/4)/x^2/(c*x^2-d),x, algorithm="maxima")

[Out]

integrate((a*x^4 - b*x^3)^(1/4)/((c*x^2 - d)*x^2), x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 117.64 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\frac {2 \, \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + 2 \, \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) + \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, d} + \frac {4 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}}{d} \]

[In]

integrate((a*x^4-b*x^3)^(1/4)/x^2/(c*x^2-d),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2 \left (-d+c x^2\right )} \, dx=-\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x^2\,\left (d-c\,x^2\right )} \,d x \]

[In]

int(-(a*x^4 - b*x^3)^(1/4)/(x^2*(d - c*x^2)),x)

[Out]

-int((a*x^4 - b*x^3)^(1/4)/(x^2*(d - c*x^2)), x)