\(\int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx\) [1478]

Optimal result

Integrand size = 25, antiderivative size = 104 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {2}{3} a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )+\frac {2}{3} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right ) \]

[Out]

4/9*(a*x^4-b*x)^(3/4)/x^3+2/3*a^(3/4)*arctan(a^(1/4)*(a*x^4-b*x)^(3/4)/(a*x^3-b))+2/3*a^(3/4)*arctanh(a^(1/4)*
(a*x^4-b*x)^(3/4)/(a*x^3-b))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.48, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2063, 2036, 335, 281, 246, 218, 212, 209} \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx=\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a x^4-b x}}+\frac {4 \left (a x^4-b x\right )^{3/4}}{9 x^3} \]

[In]

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

[Out]

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

Rule 209

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

Rule 212

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

Rule 218

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2063

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Dist[(a*d*(m + j*p + 1
) - b*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)), Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x], x] /; F
reeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m
+ j*p, -1] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && (GtQ[e, 0] || IntegersQ[j,
n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+a \int \frac {1}{\sqrt [4]{-b x+a x^4}} \, dx \\ & = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (a \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-b+a x^3}} \, dx}{\sqrt [4]{-b x+a x^4}} \\ & = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^4}} \\ & = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ & = \frac {4 \left (-b x+a x^4\right )^{3/4}}{9 x^3}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}+\frac {2 a^{3/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.69 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.26 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx=\frac {-4 b+4 a x^3+6 a^{3/4} x^{9/4} \sqrt [4]{-b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )+6 a^{3/4} x^{9/4} \sqrt [4]{-b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{9 x^2 \sqrt [4]{-b x+a x^4}} \]

[In]

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

[Out]

(-4*b + 4*a*x^3 + 6*a^(3/4)*x^(9/4)*(-b + a*x^3)^(1/4)*ArcTan[(a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)] + 6*a^(3/4
)*x^(9/4)*(-b + a*x^3)^(1/4)*ArcTanh[(a^(1/4)*x^(3/4))/(-b + a*x^3)^(1/4)])/(9*x^2*(-(b*x) + a*x^4)^(1/4))

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {-6 \arctan \left (\frac {{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{x \,a^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} x^{3}+3 \ln \left (\frac {a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} x^{3}+4 {\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {3}{4}}}{9 x^{3}}\)	\(102\)

[In]

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

[Out]

1/9*(-6*arctan((x*(a*x^3-b))^(1/4)/x/a^(1/4))*a^(3/4)*x^3+3*ln((a^(1/4)*x+(x*(a*x^3-b))^(1/4))/(-a^(1/4)*x+(x*
(a*x^3-b))^(1/4)))*a^(3/4)*x^3+4*(x*(a*x^3-b))^(3/4))/x^3

Fricas [F(-1)]

Timed out. \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (84) = 168\).

Time = 0.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.85 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx=\frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) + \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right ) + \frac {4}{9} \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {3}{4}} \]

[In]

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

[Out]

1/3*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x^3)^(1/4))/(-a)^(1/4)) + 1/3*sqrt(2)
*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a - b/x^3)^(1/4))/(-a)^(1/4)) - 1/6*sqrt(2)*(-a)^(3/4
)*log(sqrt(2)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(-a) + sqrt(a - b/x^3)) + 1/6*sqrt(2)*(-a)^(3/4)*log(-sqrt(2)
*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(-a) + sqrt(a - b/x^3)) + 4/9*(a - b/x^3)^(3/4)

Mupad [B] (verification not implemented)

Time = 6.66 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58 \[ \int \frac {b+a x^3}{x^3 \sqrt [4]{-b x+a x^4}} \, dx=\frac {4\,{\left (a\,x^4-b\,x\right )}^{3/4}}{9\,x^3}+\frac {4\,a\,x\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {a\,x^3}{b}\right )}{3\,{\left (a\,x^4-b\,x\right )}^{1/4}} \]

[In]

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

[Out]

(4*(a*x^4 - b*x)^(3/4))/(9*x^3) + (4*a*x*(1 - (a*x^3)/b)^(1/4)*hypergeom([1/4, 1/4], 5/4, (a*x^3)/b))/(3*(a*x^
4 - b*x)^(1/4))