Integrand size = 93, antiderivative size = 150 \[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=-\frac {4 \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}-2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )+2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right ) \]
-4*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4)/x-2*d^(1/4)*arctan(d^ (1/4)*x/(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4))+2*d^(1/4)*arcta nh(d^(1/4)*x/(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4))
\[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=\int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx \]
Integrate[((-a + x)*(-3*a*b + (a + 2*b)*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3 ))/(x*(x*(-a + x)*(-b + x)^2)^(3/4)*(a*b^2 - b*(2*a + b)*x + (a + 2*b)*x^2 + (-1 + d)*x^3)),x]
Integrate[((-a + x)*(-3*a*b + (a + 2*b)*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3 ))/(x*(x*(-a + x)*(-b + x)^2)^(3/4)*(a*b^2 - b*(2*a + b)*x + (a + 2*b)*x^2 + (-1 + d)*x^3)), x]
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.
| \(\displaystyle \int \frac {(x-a) \left (-b^3+3 b^2 x-3 b x^2+x^3\right ) (x (a+2 b)-3 a b)}{x \left (x (x-a) (x-b)^2\right )^{3/4} \left (a b^2+x^2 (a+2 b)-b x (2 a+b)+(d-1) x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {(x-a) (x-b)^3 (x (a+2 b)-3 a b)}{x \left (x (x-a) (x-b)^2\right )^{3/4} \left (a b^2+x^2 (a+2 b)-b x (2 a+b)+(d-1) x^3\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int -\frac {(a-x) (b-x)^3 (3 a b-(a+2 b) x)}{x^{7/4} \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4} \left (-\left ((1-d) x^3\right )+(a+2 b) x^2-b (2 a+b) x+a b^2\right )}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x) (b-x)^3 (3 a b-(a+2 b) x)}{x^{7/4} \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4} \left (-\left ((1-d) x^3\right )+(a+2 b) x^2-b (2 a+b) x+a b^2\right )}dx}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x) (b-x)^3 (3 a b-(a+2 b) x)}{x \left (x^3-(a+2 b) x^2+b (2 a+b) x-a b^2\right )^{3/4} \left (-\left ((1-d) x^3\right )+(a+2 b) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [4]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {4 x^{3/4} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {(a-x) (b-x)^3 (3 a b-(a+2 b) x)}{x \left (-\left ((a-x) (x-b)^2\right )\right )^{3/4} \left (-\left ((1-d) x^3\right )+(a+2 b) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [4]{x}}{\left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} (b-x)^3 (3 a b-(a+2 b) x)}{x (x-b)^{3/2} \left (-\left ((1-d) x^3\right )+(a+2 b) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \frac {\sqrt [4]{a-x} (x-b)^{3/2} (3 a b-(a+2 b) x)}{x \left (-\left ((1-d) x^3\right )+(a+2 b) x^2-b (2 a+b) x+a b^2\right )}d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \int \left (\frac {3 \sqrt [4]{a-x} (x-b)^{3/2}}{b x}+\frac {\sqrt [4]{a-x} \left (3 (1-d) x^2-3 (a+2 b) x+b (5 a+b)\right ) (x-b)^{3/2}}{b \left (-\left ((1-d) x^3\right )+a \left (\frac {2 b}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2\right )}\right )d\sqrt [4]{x}}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{3/2} \left (-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3\right )^{3/4} \left ((5 a+b) \int \frac {\sqrt [4]{a-x} (x-b)^{3/2}}{-\left ((1-d) x^3\right )+a \left (\frac {2 b}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2}d\sqrt [4]{x}-\frac {3 (a+2 b) \int \frac {\sqrt [4]{a-x} x (x-b)^{3/2}}{-\left ((1-d) x^3\right )+a \left (\frac {2 b}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2}d\sqrt [4]{x}}{b}+\frac {3 (1-d) \int \frac {\sqrt [4]{a-x} x^2 (x-b)^{3/2}}{-\left ((1-d) x^3\right )+a \left (\frac {2 b}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2}d\sqrt [4]{x}}{b}+\frac {\sqrt [4]{a-x} \sqrt {x-b} \operatorname {AppellF1}\left (-\frac {3}{4},-\frac {1}{4},-\frac {3}{2},\frac {1}{4},\frac {x}{a},\frac {x}{b}\right )}{x^{3/4} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}}\right )}{\left (-\left ((a-x) (b-x)^2\right )\right )^{3/4} \left (-\left (x (a-x) (b-x)^2\right )\right )^{3/4}}\) |
Int[((-a + x)*(-3*a*b + (a + 2*b)*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(x* (x*(-a + x)*(-b + x)^2)^(3/4)*(a*b^2 - b*(2*a + b)*x + (a + 2*b)*x^2 + (-1 + d)*x^3)),x]
3.21.75.3.1 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[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
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]
Time = 1.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.73
| method | result | size |
| pseudoelliptic | \(\frac {x \left (\ln \left (\frac {d^{\frac {1}{4}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )\right ) d^{\frac {1}{4}}-4 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x}\) | \(109\) |
int((-a+x)*(-3*a*b+(a+2*b)*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/x/(x*(-a+x)*(-b+x )^2)^(3/4)/(a*b^2-b*(2*a+b)*x+(a+2*b)*x^2+(-1+d)*x^3),x,method=_RETURNVERB OSE)
(x*(ln((d^(1/4)*x+(-x*(a-x)*(b-x)^2)^(1/4))/(-d^(1/4)*x+(-x*(a-x)*(b-x)^2) ^(1/4)))+2*arctan((-x*(a-x)*(b-x)^2)^(1/4)/x/d^(1/4)))*d^(1/4)-4*(-x*(a-x) *(b-x)^2)^(1/4))/x
Timed out. \[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-3*a*b+(a+2*b)*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/x/(x*(-a+x) *(-b+x)^2)^(3/4)/(a*b^2-b*(2*a+b)*x+(a+2*b)*x^2+(-1+d)*x^3),x, algorithm=" fricas")
Timed out. \[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-3*a*b+(a+2*b)*x)*(-b**3+3*b**2*x-3*b*x**2+x**3)/x/(x*(- a+x)*(-b+x)**2)**(3/4)/(a*b**2-b*(2*a+b)*x+(a+2*b)*x**2+(-1+d)*x**3),x)
\[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=\int { -\frac {{\left (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (3 \, a b - {\left (a + 2 \, b\right )} x\right )} {\left (a - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{3} + a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b\right )} x^{2}\right )} x} \,d x } \]
integrate((-a+x)*(-3*a*b+(a+2*b)*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/x/(x*(-a+x) *(-b+x)^2)^(3/4)/(a*b^2-b*(2*a+b)*x+(a+2*b)*x^2+(-1+d)*x^3),x, algorithm=" maxima")
-integrate((b^3 - 3*b^2*x + 3*b*x^2 - x^3)*(3*a*b - (a + 2*b)*x)*(a - x)/( (-(a - x)*(b - x)^2*x)^(3/4)*((d - 1)*x^3 + a*b^2 - (2*a + b)*b*x + (a + 2 *b)*x^2)*x), x)
\[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=\int { -\frac {{\left (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (3 \, a b - {\left (a + 2 \, b\right )} x\right )} {\left (a - x\right )}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{3} + a b^{2} - {\left (2 \, a + b\right )} b x + {\left (a + 2 \, b\right )} x^{2}\right )} x} \,d x } \]
integrate((-a+x)*(-3*a*b+(a+2*b)*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/x/(x*(-a+x) *(-b+x)^2)^(3/4)/(a*b^2-b*(2*a+b)*x+(a+2*b)*x^2+(-1+d)*x^3),x, algorithm=" giac")
integrate(-(b^3 - 3*b^2*x + 3*b*x^2 - x^3)*(3*a*b - (a + 2*b)*x)*(a - x)/( (-(a - x)*(b - x)^2*x)^(3/4)*((d - 1)*x^3 + a*b^2 - (2*a + b)*b*x + (a + 2 *b)*x^2)*x), x)
Timed out. \[ \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx=\int -\frac {\left (a-x\right )\,\left (3\,a\,b-x\,\left (a+2\,b\right )\right )\,\left (b^3-3\,b^2\,x+3\,b\,x^2-x^3\right )}{x\,{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a\,b^2+x^2\,\left (a+2\,b\right )+x^3\,\left (d-1\right )-b\,x\,\left (2\,a+b\right )\right )} \,d x \]
int(-((a - x)*(3*a*b - x*(a + 2*b))*(3*b*x^2 - 3*b^2*x + b^3 - x^3))/(x*(- x*(a - x)*(b - x)^2)^(3/4)*(a*b^2 + x^2*(a + 2*b) + x^3*(d - 1) - b*x*(2*a + b))),x)