Integrand size = 35, antiderivative size = 126 \[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (b x+a x^4\right )^{3/4}}{9 b x^3}-\frac {2\ 2^{3/4} a^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b}-\frac {2\ 2^{3/4} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b} \]
4/9*(a*x^4+b*x)^(3/4)/b/x^3-2/3*2^(3/4)*a^(3/4)*arctan(2^(1/4)*a^(1/4)*(a* x^4+b*x)^(3/4)/(a*x^3+b))/b-2/3*2^(3/4)*a^(3/4)*arctanh(2^(1/4)*a^(1/4)*(a *x^4+b*x)^(3/4)/(a*x^3+b))/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.37 \[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (x \left (b+a x^3\right )\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},\frac {2 a x^3}{b+a x^3}\right )}{9 b x^3} \]
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.51, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2467, 25, 966, 965, 1013, 1012}
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.
| \(\displaystyle \int \frac {a x^3+b}{x^3 \left (a x^3-b\right ) \sqrt [4]{a x^4+b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int -\frac {\left (a x^3+b\right )^{3/4}}{x^{13/4} \left (b-a x^3\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {\left (a x^3+b\right )^{3/4}}{x^{13/4} \left (b-a x^3\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {\left (a x^3+b\right )^{3/4}}{x^{5/2} \left (b-a x^3\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {(b+a x)^{3/4}}{x (b-a x)}dx^{3/4}}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} (a x+b)^{3/4} \sqrt [4]{a x^3+b} \int \frac {\left (\frac {a x}{b}+1\right )^{3/4}}{x (b-a x)}dx^{3/4}}{3 \left (\frac {a x}{b}+1\right )^{3/4} \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle \frac {4 (a x+b)^{3/4} \sqrt [4]{a x^3+b} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},\frac {2 a x}{b+a x}\right )}{9 b \sqrt {x} \sqrt [4]{a x^4+b x}}\) |
(4*(b + a*x)^(3/4)*(b + a*x^3)^(1/4)*Hypergeometric2F1[-3/4, 1, 1/4, (2*a* x)/(b + a*x)])/(9*b*Sqrt[x]*(b*x + a*x^4)^(1/4))
3.19.48.3.1 Defintions of rubi rules used
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[((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] /; 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]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[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])
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 = 0.50 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {\left (6 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) a \,x^{3}-3 \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right ) a \,x^{3}+2 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}} 2^{\frac {1}{4}} a^{\frac {1}{4}}\right ) 2^{\frac {3}{4}}}{9 x^{3} a^{\frac {1}{4}} b}\) | \(119\) |
1/9*(6*arctan(1/2*(x*(a*x^3+b))^(1/4)/x*2^(3/4)/a^(1/4))*a*x^3-3*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*x^3+2*(x*(a*x^3+b))^(3/4)*2^(1/4)*a^(1/4))/x^3*2^(3/4)/a^(1/4)/b
Result contains complex when optimal does not.
Time = 48.54 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.56 \[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {3 \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} + 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} + 8^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) + 3 i \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} + i \cdot 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} - 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} + 8^{\frac {1}{4}} {\left (-3 i \, a^{2} b x^{3} - i \, a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 3 i \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} - i \cdot 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} - 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} + 8^{\frac {1}{4}} {\left (3 i \, a^{2} b x^{3} + i \, a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 3 \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} - 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} - 8^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 8 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}}}{18 \, b x^{3}} \]
-1/18*(3*8^(1/4)*b*x^3*(a^3/b^4)^(1/4)*log((4*sqrt(2)*(a*x^4 + b*x)^(1/4)* a*b^2*x^2*sqrt(a^3/b^4) + 8^(3/4)*sqrt(a*x^4 + b*x)*b^3*x*(a^3/b^4)^(3/4) + 4*(a*x^4 + b*x)^(3/4)*a^2 + 8^(1/4)*(3*a^2*b*x^3 + a*b^2)*(a^3/b^4)^(1/4 ))/(a*x^3 - b)) + 3*I*8^(1/4)*b*x^3*(a^3/b^4)^(1/4)*log(-(4*sqrt(2)*(a*x^4 + b*x)^(1/4)*a*b^2*x^2*sqrt(a^3/b^4) + I*8^(3/4)*sqrt(a*x^4 + b*x)*b^3*x* (a^3/b^4)^(3/4) - 4*(a*x^4 + b*x)^(3/4)*a^2 + 8^(1/4)*(-3*I*a^2*b*x^3 - I* a*b^2)*(a^3/b^4)^(1/4))/(a*x^3 - b)) - 3*I*8^(1/4)*b*x^3*(a^3/b^4)^(1/4)*l og(-(4*sqrt(2)*(a*x^4 + b*x)^(1/4)*a*b^2*x^2*sqrt(a^3/b^4) - I*8^(3/4)*sqr t(a*x^4 + b*x)*b^3*x*(a^3/b^4)^(3/4) - 4*(a*x^4 + b*x)^(3/4)*a^2 + 8^(1/4) *(3*I*a^2*b*x^3 + I*a*b^2)*(a^3/b^4)^(1/4))/(a*x^3 - b)) - 3*8^(1/4)*b*x^3 *(a^3/b^4)^(1/4)*log((4*sqrt(2)*(a*x^4 + b*x)^(1/4)*a*b^2*x^2*sqrt(a^3/b^4 ) - 8^(3/4)*sqrt(a*x^4 + b*x)*b^3*x*(a^3/b^4)^(3/4) + 4*(a*x^4 + b*x)^(3/4 )*a^2 - 8^(1/4)*(3*a^2*b*x^3 + a*b^2)*(a^3/b^4)^(1/4))/(a*x^3 - b)) - 8*(a *x^4 + b*x)^(3/4))/(b*x^3)
\[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{3} + b}{x^{3} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \]
\[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{3} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (98) = 196\).
Time = 0.32 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.65 \[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, b} - \frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}}}{9 \, b} \]
-2/3*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))/b - 2/3*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))/b + 1/3*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))/b - 1/3*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))/b + 4/9*(a + b/x^3)^(3/4)/b
Timed out. \[ \int \frac {b+a x^3}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\int \frac {a\,x^3+b}{x^3\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \]