\(\int \frac {\sqrt [4]{-b x^3+a x^4}}{x (b+a x^3)} \, dx\) [1100]

Optimal result

Integrand size = 29, antiderivative size = 82 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\frac {1}{3} \text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^4+3 a \text {$\#$1}^8-\text {$\#$1}^{12}\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(82)=164\).

Time = 1.00 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.54, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2081, 6857, 129, 525, 524} \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\frac {4 \sqrt [4]{a x^4-b x^3} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{9 b \sqrt [4]{1-\frac {a x}{b}}}+\frac {4 \sqrt [4]{a x^4-b x^3} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt [3]{-1} \sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{9 b \sqrt [4]{1-\frac {a x}{b}}}+\frac {4 \sqrt [4]{a x^4-b x^3} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {(-1)^{2/3} \sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{9 b \sqrt [4]{1-\frac {a x}{b}}} \]

[In]

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

[Out]

(4*(-(b*x^3) + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, -((a^(1/3)*x)/b^(1/3)), (a*x)/b])/(9*b*(1 - (a*x)/b)^(
1/4)) + (4*(-(b*x^3) + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, ((-1)^(1/3)*a^(1/3)*x)/b^(1/3), (a*x)/b])/(9*b
*(1 - (a*x)/b)^(1/4)) + (4*(-(b*x^3) + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, -(((-1)^(2/3)*a^(1/3)*x)/b^(1/
3)), (a*x)/b])/(9*b*(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 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}}{\sqrt [4]{x} \left (b+a x^3\right )} \, dx}{x^{3/4} \sqrt [4]{-b+a x}} \\ & = \frac {\sqrt [4]{-b x^3+a x^4} \int \left (-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )}-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{-b+a x}} \\ & = -\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} 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}}{-\sqrt [3]{b}-\sqrt [3]{a} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 b^{2/3} 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}}{-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 b^{2/3} 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}}{-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 b^{2/3} 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}}}{-\sqrt [3]{b}-\sqrt [3]{a} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 b^{2/3} x^{3/4} \sqrt [4]{1-\frac {a x}{b}}}-\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}}}{-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 b^{2/3} x^{3/4} \sqrt [4]{1-\frac {a x}{b}}}-\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}}}{-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 b^{2/3} 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},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{9 b \sqrt [4]{1-\frac {a x}{b}}}+\frac {4 \sqrt [4]{-b x^3+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt [3]{-1} \sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{9 b \sqrt [4]{1-\frac {a x}{b}}}+\frac {4 \sqrt [4]{-b x^3+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {(-1)^{2/3} \sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{9 b \sqrt [4]{1-\frac {a x}{b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\frac {x^{9/4} (-b+a x)^{3/4} \text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^4+3 a \text {$\#$1}^8-\text {$\#$1}^{12}\&,\frac {-\log \left (\sqrt [4]{x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]}{3 \left (x^3 (-b+a x)\right )^{3/4}} \]

[In]

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

[Out]

(x^(9/4)*(-b + a*x)^(3/4)*RootSum[a^3 + a*b^2 - 3*a^2*#1^4 + 3*a*#1^8 - #1^12 & , (-(Log[x^(1/4)]*#1) + Log[(-
b + a*x)^(1/4) - x^(1/4)*#1]*#1)/(-a + #1^4) & ])/(3*(x^3*(-b + a*x))^(3/4))

Maple [N/A] (verified)

Time = 2.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}-3 a \,\textit {\_Z}^{8}+3 a^{2} \textit {\_Z}^{4}-a^{3}-a \,b^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right ) \textit {\_R}}{\textit {\_R}^{4}-a}\right )}{3}\)	\(70\)

[In]

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

[Out]

1/3*sum(ln((-_R*x+(x^3*(a*x-b))^(1/4))/x)*_R/(_R^4-a),_R=RootOf(_Z^12-3*_Z^8*a+3*_Z^4*a^2-a^3-a*b^2))

Fricas [C] (verification not implemented)

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

Time = 0.98 (sec) , antiderivative size = 7487, normalized size of antiderivative = 91.30 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 2.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{x \left (a x^{3} + b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (a x^{3} + b\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (a x^{3} + b\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx=\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (a\,x^3+b\right )} \,d x \]

[In]

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

[Out]

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