Integrand size = 35, antiderivative size = 136 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (3 b+10 a x^3\right ) \left (b x+a x^4\right )^{3/4}}{63 b^2 x^6}-\frac {2\ 2^{3/4} a^{7/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2}-\frac {2\ 2^{3/4} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b^2} \]
4/63*(10*a*x^3+3*b)*(a*x^4+b*x)^(3/4)/b^2/x^6-2/3*2^(3/4)*a^(7/4)*arctan(2 ^(1/4)*a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))/b^2-2/3*2^(3/4)*a^(7/4)*arctan h(2^(1/4)*a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))/b^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 14.67 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {\left (x \left (b+a x^3\right )\right )^{7/4} \operatorname {Gamma}\left (-\frac {3}{4}\right ) \left (\left (3 b^2-7 a b x^3+4 a^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (1,1,\frac {1}{4},-\frac {2 a x^3}{b-a x^3}\right )+32 a x^3 \left (b+a x^3\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{4},-\frac {2 a x^3}{b-a x^3}\right )\right )}{21 b^2 x^7 \left (b-a x^3\right )^2 \operatorname {Gamma}\left (\frac {1}{4}\right )} \]
-1/21*((x*(b + a*x^3))^(7/4)*Gamma[-3/4]*((3*b^2 - 7*a*b*x^3 + 4*a^2*x^6)* Hypergeometric2F1[1, 1, 1/4, (-2*a*x^3)/(b - a*x^3)] + 32*a*x^3*(b + a*x^3 )*Hypergeometric2F1[2, 2, 5/4, (-2*a*x^3)/(b - a*x^3)]))/(b^2*x^7*(b - a*x ^3)^2*Gamma[1/4])
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.86 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \frac {a x^3+b}{x^6 \left (a x^3-b\right ) \sqrt [4]{a x^4+b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int -\frac {\left (a x^3+b\right )^{3/4}}{x^{25/4} \left (b-a x^3\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {\left (a x^3+b\right )^{3/4}}{x^{25/4} \left (b-a x^3\right )}dx}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {\left (a x^3+b\right )^{3/4}}{x^{11/2} \left (b-a x^3\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 965 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {(b+a x)^{3/4}}{x^2 (b-a x)}dx^{3/4}}{3 \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} (a x+b)^{3/4} \sqrt [4]{a x^3+b} \int \frac {\left (\frac {a x}{b}+1\right )^{3/4}}{x^2 (b-a x)}dx^{3/4}}{3 \left (\frac {a x}{b}+1\right )^{3/4} \sqrt [4]{a x^4+b x}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle -\frac {\operatorname {Gamma}\left (-\frac {3}{4}\right ) (a x+b)^{3/4} \left (\frac {a x}{b}+1\right ) \sqrt [4]{a x^3+b} \left (\left (4 a^2 x^2-7 a b x+3 b^2\right ) \operatorname {Hypergeometric2F1}\left (1,1,\frac {1}{4},-\frac {2 a x}{b-a x}\right )+32 a x (a x+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{4},-\frac {2 a x}{b-a x}\right )\right )}{21 b x^{3/2} \operatorname {Gamma}\left (\frac {1}{4}\right ) (b-a x)^2 \sqrt [4]{a x^4+b x}}\) |
-1/21*((b + a*x)^(3/4)*(1 + (a*x)/b)*(b + a*x^3)^(1/4)*Gamma[-3/4]*((3*b^2 - 7*a*b*x + 4*a^2*x^2)*Hypergeometric2F1[1, 1, 1/4, (-2*a*x)/(b - a*x)] + 32*a*x*(b + a*x)*Hypergeometric2F1[2, 2, 5/4, (-2*a*x)/(b - a*x)]))/(b*x^ (3/2)*(b - a*x)^2*(b*x + a*x^4)^(1/4)*Gamma[1/4])
3.20.43.3.1 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 = 0.56 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04
| method | result | size |
| pseudoelliptic | \(\frac {\left (42 \arctan \left (\frac {{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) a^{2} x^{6}-21 \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+{\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {1}{4}}}\right ) a^{2} x^{6}+20 {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}} a^{\frac {5}{4}} x^{3} 2^{\frac {1}{4}}+6 b 2^{\frac {1}{4}} a^{\frac {1}{4}} {\left (x \left (a \,x^{3}+b \right )\right )}^{\frac {3}{4}}\right ) 2^{\frac {3}{4}}}{63 x^{6} a^{\frac {1}{4}} b^{2}}\) | \(142\) |
1/63/x^6*(42*arctan(1/2*(x*(a*x^3+b))^(1/4)/x*2^(3/4)/a^(1/4))*a^2*x^6-21* 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^2*x^6+20*(x*(a*x^3+b))^(3/4)*a^(5/4)*x^3*2^(1/4)+6*b*2^(1/4)* a^(1/4)*(x*(a*x^3+b))^(3/4))*2^(3/4)/a^(1/4)/b^2
Result contains complex when optimal does not.
Time = 82.93 (sec) , antiderivative size = 616, normalized size of antiderivative = 4.53 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {21 \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} + 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} + 8^{\frac {1}{4}} {\left (3 \, a^{4} b^{2} x^{3} + a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) + 21 i \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} + i \cdot 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} - 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} + 8^{\frac {1}{4}} {\left (-3 i \, a^{4} b^{2} x^{3} - i \, a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 21 i \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} - i \cdot 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} - 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} + 8^{\frac {1}{4}} {\left (3 i \, a^{4} b^{2} x^{3} + i \, a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 21 \cdot 8^{\frac {1}{4}} b^{2} x^{6} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{2} b^{4} x^{2} \sqrt {\frac {a^{7}}{b^{8}}} - 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{6} x \left (\frac {a^{7}}{b^{8}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{5} - 8^{\frac {1}{4}} {\left (3 \, a^{4} b^{2} x^{3} + a^{3} b^{3}\right )} \left (\frac {a^{7}}{b^{8}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) - 8 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} {\left (10 \, a x^{3} + 3 \, b\right )}}{126 \, b^{2} x^{6}} \]
-1/126*(21*8^(1/4)*b^2*x^6*(a^7/b^8)^(1/4)*log((4*sqrt(2)*(a*x^4 + b*x)^(1 /4)*a^2*b^4*x^2*sqrt(a^7/b^8) + 8^(3/4)*sqrt(a*x^4 + b*x)*b^6*x*(a^7/b^8)^ (3/4) + 4*(a*x^4 + b*x)^(3/4)*a^5 + 8^(1/4)*(3*a^4*b^2*x^3 + a^3*b^3)*(a^7 /b^8)^(1/4))/(a*x^3 - b)) + 21*I*8^(1/4)*b^2*x^6*(a^7/b^8)^(1/4)*log(-(4*s qrt(2)*(a*x^4 + b*x)^(1/4)*a^2*b^4*x^2*sqrt(a^7/b^8) + I*8^(3/4)*sqrt(a*x^ 4 + b*x)*b^6*x*(a^7/b^8)^(3/4) - 4*(a*x^4 + b*x)^(3/4)*a^5 + 8^(1/4)*(-3*I *a^4*b^2*x^3 - I*a^3*b^3)*(a^7/b^8)^(1/4))/(a*x^3 - b)) - 21*I*8^(1/4)*b^2 *x^6*(a^7/b^8)^(1/4)*log(-(4*sqrt(2)*(a*x^4 + b*x)^(1/4)*a^2*b^4*x^2*sqrt( a^7/b^8) - I*8^(3/4)*sqrt(a*x^4 + b*x)*b^6*x*(a^7/b^8)^(3/4) - 4*(a*x^4 + b*x)^(3/4)*a^5 + 8^(1/4)*(3*I*a^4*b^2*x^3 + I*a^3*b^3)*(a^7/b^8)^(1/4))/(a *x^3 - b)) - 21*8^(1/4)*b^2*x^6*(a^7/b^8)^(1/4)*log((4*sqrt(2)*(a*x^4 + b* x)^(1/4)*a^2*b^4*x^2*sqrt(a^7/b^8) - 8^(3/4)*sqrt(a*x^4 + b*x)*b^6*x*(a^7/ b^8)^(3/4) + 4*(a*x^4 + b*x)^(3/4)*a^5 - 8^(1/4)*(3*a^4*b^2*x^3 + a^3*b^3) *(a^7/b^8)^(1/4))/(a*x^3 - b)) - 8*(a*x^4 + b*x)^(3/4)*(10*a*x^3 + 3*b))/( b^2*x^6)
\[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int \frac {a x^{3} + b}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \]
\[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=\int { \frac {a x^{3} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{6}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (108) = 216\).
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.71 \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, b^{2}} - \frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, b^{2}} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b^{2}} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} a \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b^{2}} + \frac {4 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}} b^{12} + 7 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a b^{12}\right )}}{63 \, b^{14}} \]
-2/3*2^(1/4)*(-a)^(3/4)*a*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/b^2 - 2/3*2^(1/4)*(-a)^(3/4)*a*arctan(-1/2*2^(1/ 4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/b^2 + 1/3*2^(1/4 )*(-a)^(3/4)*a*log(2^(3/4)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/b^2 - 1/3*2^(1/4)*(-a)^(3/4)*a*log(-2^(3/4)*(-a)^(1/4) *(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/b^2 + 4/63*(3*(a + b/x^3)^(7/4)*b^12 + 7*(a + b/x^3)^(3/4)*a*b^12)/b^14
Timed out. \[ \int \frac {b+a x^3}{x^6 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx=-\int \frac {a\,x^3+b}{x^6\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \]