Integrand size = 24, antiderivative size = 109 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 \sqrt [4]{a} b}+\frac {2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 \sqrt [4]{a} b} \]
1/3*2^(3/4)*arctan(2^(1/4)*a^(1/4)*(a*x^4-b*x)^(3/4)/(a*x^3-b))/a^(1/4)/b+ 1/3*2^(3/4)*arctanh(2^(1/4)*a^(1/4)*(a*x^4-b*x)^(3/4)/(a*x^3-b))/a^(1/4)/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {4 x \sqrt [4]{1-\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {2 a x^3}{b+a x^3}\right )}{3 b \sqrt [4]{1+\frac {a x^3}{b}} \sqrt [4]{-b x+a x^4}} \]
(4*x*(1 - (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (2*a*x^3)/(b + a*x^3)])/(3*b*(1 + (a*x^3)/b)^(1/4)*(-(b*x) + a*x^4)^(1/4))
Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2467, 966, 965, 902, 756, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (a x^3+b\right ) \sqrt [4]{a x^4-b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a x^3-b} \left (a x^3+b\right )}dx}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {\sqrt {x}}{\sqrt [4]{a x^3-b} \left (a x^3+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {1}{\sqrt [4]{a x-b} (b+a x)}dx^{3/4}}{3 \sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \int \frac {1}{b-2 a b x}d\frac {x^{3/4}}{\sqrt [4]{a x-b}}}{3 \sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \left (\frac {\int \frac {1}{1-\sqrt {2} \sqrt {a} \sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{a x-b}}}{2 b}+\frac {\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {x}+1}d\frac {x^{3/4}}{\sqrt [4]{a x-b}}}{2 b}\right )}{3 \sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \left (\frac {\int \frac {1}{1-\sqrt {2} \sqrt {a} \sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{a x-b}}}{2 b}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}\right )}{3 \sqrt [4]{a x^4-b x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^3-b} \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}\right )}{3 \sqrt [4]{a x^4-b x}}\) |
(4*x^(1/4)*(-b + a*x^3)^(1/4)*(ArcTan[(2^(1/4)*a^(1/4)*x^(3/4))/(-b + a*x) ^(1/4)]/(2*2^(1/4)*a^(1/4)*b) + ArcTanh[(2^(1/4)*a^(1/4)*x^(3/4))/(-b + a* x)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)))/(3*(-(b*x) + a*x^4)^(1/4))
3.16.90.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )+\ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}-b \right )\right )}^{\frac {1}{4}}}\right )\right )}{6 a^{\frac {1}{4}} b}\) | \(93\) |
1/6*2^(3/4)*(-2*arctan(1/2*(x*(a*x^3-b))^(1/4)/x*2^(3/4)/a^(1/4))+ln((-x*2 ^(1/4)*a^(1/4)-(x*(a*x^3-b))^(1/4))/(x*2^(1/4)*a^(1/4)-(x*(a*x^3-b))^(1/4) )))/a^(1/4)/b
Result contains complex when optimal does not.
Time = 57.53 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.81 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) - \frac {1}{6} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{3} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) + \frac {1}{6} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{3} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) \]
1/6*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log((4*(1/2)^(3/4)*sqrt(a*x^4 - b*x)*a*b ^3*x*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 - b*x)^(1/4)*a*b^2*x^2*sqrt(1/ (a*b^4)) + (1/2)^(1/4)*(3*a*b*x^3 - b^2)*(1/(a*b^4))^(1/4) + 2*(a*x^4 - b* x)^(3/4))/(a*x^3 + b)) - 1/6*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(-(4*(1/2)^( 3/4)*sqrt(a*x^4 - b*x)*a*b^3*x*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 - b* x)^(1/4)*a*b^2*x^2*sqrt(1/(a*b^4)) + (1/2)^(1/4)*(3*a*b*x^3 - b^2)*(1/(a*b ^4))^(1/4) - 2*(a*x^4 - b*x)^(3/4))/(a*x^3 + b)) - 1/6*I*(1/2)^(1/4)*(1/(a *b^4))^(1/4)*log((4*I*(1/2)^(3/4)*sqrt(a*x^4 - b*x)*a*b^3*x*(1/(a*b^4))^(3 /4) - 4*sqrt(1/2)*(a*x^4 - b*x)^(1/4)*a*b^2*x^2*sqrt(1/(a*b^4)) - (1/2)^(1 /4)*(3*I*a*b*x^3 - I*b^2)*(1/(a*b^4))^(1/4) + 2*(a*x^4 - b*x)^(3/4))/(a*x^ 3 + b)) + 1/6*I*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log((-4*I*(1/2)^(3/4)*sqrt(a *x^4 - b*x)*a*b^3*x*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 - b*x)^(1/4)*a* b^2*x^2*sqrt(1/(a*b^4)) - (1/2)^(1/4)*(-3*I*a*b*x^3 + I*b^2)*(1/(a*b^4))^( 1/4) + 2*(a*x^4 - b*x)^(3/4))/(a*x^3 + b))
\[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x \left (a x^{3} - b\right )} \left (a x^{3} + b\right )}\, dx \]
\[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (85) = 170\).
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} + \frac {2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{6 \, a b} \]
1/3*2^(1/4)*(-a)^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a - b/x ^3)^(1/4))/(-a)^(1/4))/(a*b) + 1/3*2^(1/4)*(-a)^(3/4)*arctan(-1/2*2^(1/4)* (2^(3/4)*(-a)^(1/4) - 2*(a - b/x^3)^(1/4))/(-a)^(1/4))/(a*b) + 1/6*2^(1/4) *log(2^(3/4)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a - b/ x^3))/((-a)^(1/4)*b) + 1/6*2^(1/4)*(-a)^(3/4)*log(-2^(3/4)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a - b/x^3))/(a*b)
Timed out. \[ \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx=\int \frac {1}{{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (a\,x^3+b\right )} \,d x \]