Integrand size = 108, antiderivative size = 215 \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right )}{d^{3/4}} \]
-2*arctan(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^ 3)*x^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(1/4)/(a-x))/d^(3/4)+2*ar ctanh(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^3)*x ^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(1/4)/(a-x))/d^(3/4)
Time = 15.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x)^2 (-b+x)^3}}{a-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(a-x)^2 x (-b+x)^3}}{-a+x}\right )\right )}{d^{3/4}} \]
Integrate[(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/ ((x*(-a + x)^2*(-b + x)^3)^(1/4)*(-a^2 + (2*a - b^3*d)*x + (-1 + 3*b^2*d)* x^2 - 3*b*d*x^3 + d*x^4)),x]
(-2*(ArcTan[(d^(1/4)*(x*(-a + x)^2*(-b + x)^3)^(1/4))/(a - x)] + ArcTanh[( d^(1/4)*((a - x)^2*x*(-b + x)^3)^(1/4))/(-a + x)]))/d^(3/4)
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.
| \(\displaystyle \int \frac {a b^3-b^2 x (6 a-b)-x^3 (4 a+3 b)+9 a b x^2+2 x^4}{\sqrt [4]{x (x-a)^2 (x-b)^3} \left (-a^2+x \left (2 a-b^3 d\right )+x^2 \left (3 b^2 d-1\right )-3 b d x^3+d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int -\frac {2 x^4-(4 a+3 b) x^3+9 a b x^2-(6 a-b) b^2 x+a b^3}{\sqrt [4]{x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-\left (2 a-b^3 d\right ) x+a^2\right ) \sqrt [4]{x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3}}dx}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {2 x^4-(4 a+3 b) x^3+9 a b x^2-(6 a-b) b^2 x+a b^3}{\sqrt [4]{x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-\left (2 a-b^3 d\right ) x+a^2\right ) \sqrt [4]{x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3}}dx}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {\sqrt {x} \left (2 x^4-(4 a+3 b) x^3+9 a b x^2-(6 a-b) b^2 x+a b^3\right )}{\left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-\left (2 a-b^3 d\right ) x+a^2\right ) \sqrt [4]{x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3}}d\sqrt [4]{x}}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {(b-x)^2 \sqrt {x} \left (2 x^2-(4 a-b) x+a b\right )}{\sqrt [4]{(a-x)^2 (x-b)^3} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-\left (2 a-b^3 d\right ) x+a^2\right )}d\sqrt [4]{x}}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {(b-x)^2 \sqrt {x} \left (2 x^2-(4 a-b) x+a b\right )}{\sqrt {a-x} (x-b)^{3/4} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-\left (2 a-b^3 d\right ) x+a^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {\sqrt {x} (x-b)^{5/4} \left (2 x^2-(4 a-b) x+a b\right )}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-\left (2 a-b^3 d\right ) x+a^2\right )}d\sqrt [4]{x}}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \left (\frac {2 (x-b)^{5/4} x^{5/2}}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x+a^2\right )}+\frac {(b-4 a) (x-b)^{5/4} x^{3/2}}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x+a^2\right )}+\frac {a b (x-b)^{5/4} \sqrt {x}}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x+a^2\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \left (a b \int \frac {\sqrt {x} (x-b)^{5/4}}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x+a^2\right )}d\sqrt [4]{x}-(4 a-b) \int \frac {x^{3/2} (x-b)^{5/4}}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x+a^2\right )}d\sqrt [4]{x}+2 \int \frac {x^{5/2} (x-b)^{5/4}}{\sqrt {a-x} \left (-d x^4+3 b d x^3+\left (1-3 b^2 d\right ) x^2-2 a \left (1-\frac {b^3 d}{2 a}\right ) x+a^2\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\) |
Int[(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/((x*(- a + x)^2*(-b + x)^3)^(1/4)*(-a^2 + (2*a - b^3*d)*x + (-1 + 3*b^2*d)*x^2 - 3*b*d*x^3 + d*x^4)),x]
3.26.53.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
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]
\[\int \frac {a \,b^{3}-\left (6 a -b \right ) b^{2} x +9 a b \,x^{2}-\left (4 a +3 b \right ) x^{3}+2 x^{4}}{\left (x \left (-a +x \right )^{2} \left (-b +x \right )^{3}\right )^{\frac {1}{4}} \left (-a^{2}+\left (-b^{3} d +2 a \right ) x +\left (3 b^{2} d -1\right ) x^{2}-3 b d \,x^{3}+d \,x^{4}\right )}d x\]
int((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x) ^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x)
int((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2*(-b+x) ^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x)
Timed out. \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]
integrate((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2* (-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x, a lgorithm="fricas")
Timed out. \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]
integrate((a*b**3-(6*a-b)*b**2*x+9*a*b*x**2-(4*a+3*b)*x**3+2*x**4)/(x*(-a+ x)**2*(-b+x)**3)**(1/4)/(-a**2+(-b**3*d+2*a)*x+(3*b**2*d-1)*x**2-3*b*d*x** 3+d*x**4),x)
\[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int { -\frac {a b^{3} - {\left (6 \, a - b\right )} b^{2} x + 9 \, a b x^{2} - {\left (4 \, a + 3 \, b\right )} x^{3} + 2 \, x^{4}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (3 \, b d x^{3} - d x^{4} - {\left (3 \, b^{2} d - 1\right )} x^{2} + a^{2} + {\left (b^{3} d - 2 \, a\right )} x\right )}} \,d x } \]
integrate((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2* (-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x, a lgorithm="maxima")
-integrate((a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4) /((-(a - x)^2*(b - x)^3*x)^(1/4)*(3*b*d*x^3 - d*x^4 - (3*b^2*d - 1)*x^2 + a^2 + (b^3*d - 2*a)*x)), x)
\[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int { -\frac {a b^{3} - {\left (6 \, a - b\right )} b^{2} x + 9 \, a b x^{2} - {\left (4 \, a + 3 \, b\right )} x^{3} + 2 \, x^{4}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (3 \, b d x^{3} - d x^{4} - {\left (3 \, b^{2} d - 1\right )} x^{2} + a^{2} + {\left (b^{3} d - 2 \, a\right )} x\right )}} \,d x } \]
integrate((a*b^3-(6*a-b)*b^2*x+9*a*b*x^2-(4*a+3*b)*x^3+2*x^4)/(x*(-a+x)^2* (-b+x)^3)^(1/4)/(-a^2+(-b^3*d+2*a)*x+(3*b^2*d-1)*x^2-3*b*d*x^3+d*x^4),x, a lgorithm="giac")
integrate(-(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4) /((-(a - x)^2*(b - x)^3*x)^(1/4)*(3*b*d*x^3 - d*x^4 - (3*b^2*d - 1)*x^2 + a^2 + (b^3*d - 2*a)*x)), x)
Timed out. \[ \int \frac {a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (-a^2+\left (2 a-b^3 d\right ) x+\left (-1+3 b^2 d\right ) x^2-3 b d x^3+d x^4\right )} \, dx=\int \frac {a\,b^3-x^3\,\left (4\,a+3\,b\right )+2\,x^4-b^2\,x\,\left (6\,a-b\right )+9\,a\,b\,x^2}{{\left (-x\,{\left (a-x\right )}^2\,{\left (b-x\right )}^3\right )}^{1/4}\,\left (x^2\,\left (3\,b^2\,d-1\right )+x\,\left (2\,a-b^3\,d\right )+d\,x^4-a^2-3\,b\,d\,x^3\right )} \,d x \]
int((a*b^3 - x^3*(4*a + 3*b) + 2*x^4 - b^2*x*(6*a - b) + 9*a*b*x^2)/((-x*( a - x)^2*(b - x)^3)^(1/4)*(x^2*(3*b^2*d - 1) + x*(2*a - b^3*d) + d*x^4 - a ^2 - 3*b*d*x^3)),x)