Integrand size = 94, antiderivative size = 141 \[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}\right )}{d^{3/4}} \]
2*arctan(d^(1/4)*x/(a*b^3*x+(-3*a*b^2-b^3)*x^2+(3*a*b+3*b^2)*x^3+(-a-3*b)* x^4+x^5)^(1/4))/d^(3/4)-2*arctanh(d^(1/4)*x/(a*b^3*x+(-3*a*b^2-b^3)*x^2+(3 *a*b+3*b^2)*x^3+(-a-3*b)*x^4+x^5)^(1/4))/d^(3/4)
\[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx \]
Integrate[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x *(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 - b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - ( a + 3*b + d)*x^3 + x^4)),x]
Integrate[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x *(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 - b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - ( a + 3*b + d)*x^3 + x^4)), 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.
| \(\displaystyle \int \frac {x^2 \left (-3 a b^3+2 b^2 x (3 a+b)-3 b x^2 (a+b)+x^4\right )}{\left (x (x-a) (x-b)^3\right )^{3/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b+d)+3 b x^2 (a+b)+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int -\frac {x^{5/4} \left (-x^4+3 b (a+b) x^2-2 b^2 (3 a+b) x+3 a b^3\right )}{\left (x^4-(a+3 b) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )^{3/4} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}dx}{\left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{3/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int \frac {x^{5/4} \left (-x^4+3 b (a+b) x^2-2 b^2 (3 a+b) x+3 a b^3\right )}{\left (x^4-(a+3 b) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )^{3/4} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}dx}{\left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 x^{3/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int \frac {x^2 \left (-x^4+3 b (a+b) x^2-2 b^2 (3 a+b) x+3 a b^3\right )}{\left (x^4-(a+3 b) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )^{3/4} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}d\sqrt [4]{x}}{\left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {4 x^{3/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int \frac {(b-x)^2 x^2 \left (-x^2-2 b x+3 a b\right )}{\left (-\left ((a-x) (x-b)^3\right )\right )^{3/4} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}d\sqrt [4]{x}}{\left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{9/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int \frac {(b-x)^2 x^2 \left (-x^2-2 b x+3 a b\right )}{(a-x)^{3/4} (x-b)^{9/4} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}d\sqrt [4]{x}}{\left ((a-x) (b-x)^3\right )^{3/4} \left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{9/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int \frac {x^2 \left (-x^2-2 b x+3 a b\right )}{(a-x)^{3/4} \sqrt [4]{x-b} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}d\sqrt [4]{x}}{\left ((a-x) (b-x)^3\right )^{3/4} \left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{9/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \int \left (\frac {a b^3-(3 a+b) x b^2+3 (2 a+b) x^2 b-(a+5 b+d) x^3}{(a-x)^{3/4} \sqrt [4]{x-b} \left (x^4-(a+3 b+d) x^3+3 b (a+b) x^2-b^2 (3 a+b) x+a b^3\right )}-\frac {1}{(a-x)^{3/4} \sqrt [4]{x-b}}\right )d\sqrt [4]{x}}{\left ((a-x) (b-x)^3\right )^{3/4} \left (x (a-x) (b-x)^3\right )^{3/4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 x^{3/4} (a-x)^{3/4} (x-b)^{9/4} \left (a b^3-b^2 x (3 a+b)-x^3 (a+3 b)+3 b x^2 (a+b)+x^4\right )^{3/4} \left (a b^3 \int \frac {1}{(a-x)^{3/4} \sqrt [4]{x-b} \left (x^4-a \left (\frac {3 b+d}{a}+1\right ) x^3+3 a b \left (\frac {b}{a}+1\right ) x^2-3 a b^2 \left (\frac {b}{3 a}+1\right ) x+a b^3\right )}d\sqrt [4]{x}-b^2 (3 a+b) \int \frac {x}{(a-x)^{3/4} \sqrt [4]{x-b} \left (x^4-a \left (\frac {3 b+d}{a}+1\right ) x^3+3 a b \left (\frac {b}{a}+1\right ) x^2-3 a b^2 \left (\frac {b}{3 a}+1\right ) x+a b^3\right )}d\sqrt [4]{x}+3 b (2 a+b) \int \frac {x^2}{(a-x)^{3/4} \sqrt [4]{x-b} \left (x^4-a \left (\frac {3 b+d}{a}+1\right ) x^3+3 a b \left (\frac {b}{a}+1\right ) x^2-3 a b^2 \left (\frac {b}{3 a}+1\right ) x+a b^3\right )}d\sqrt [4]{x}-(a+5 b+d) \int \frac {x^3}{(a-x)^{3/4} \sqrt [4]{x-b} \left (x^4-a \left (\frac {3 b+d}{a}+1\right ) x^3+3 a b \left (\frac {b}{a}+1\right ) x^2-3 a b^2 \left (\frac {b}{3 a}+1\right ) x+a b^3\right )}d\sqrt [4]{x}-\frac {\sqrt [4]{x} \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1-\frac {x}{b}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},\frac {1}{4},\frac {5}{4},\frac {x}{a},\frac {x}{b}\right )}{(a-x)^{3/4} \sqrt [4]{x-b}}\right )}{\left ((a-x) (b-x)^3\right )^{3/4} \left (x (a-x) (b-x)^3\right )^{3/4}}\) |
Int[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x*(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 - b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - (a + 3* b + d)*x^3 + x^4)),x]
3.20.97.3.1 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]
Time = 2.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(-\frac {\ln \left (\frac {d^{\frac {1}{4}} x +\left (x \left (a -x \right ) \left (b -x \right )^{3}\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (x \left (a -x \right ) \left (b -x \right )^{3}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x \left (a -x \right ) \left (b -x \right )^{3}\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )}{d^{\frac {3}{4}}}\) | \(82\) |
int(x^2*(-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^3)^( 3/4)/(a*b^3-b^2*(3*a+b)*x+3*b*(a+b)*x^2-(a+3*b+d)*x^3+x^4),x,method=_RETUR NVERBOSE)
-1/d^(3/4)*(ln((d^(1/4)*x+(x*(a-x)*(b-x)^3)^(1/4))/(-d^(1/4)*x+(x*(a-x)*(b -x)^3)^(1/4)))+2*arctan((x*(a-x)*(b-x)^3)^(1/4)/x/d^(1/4)))
Timed out. \[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate(x^2*(-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x )^3)^(3/4)/(a*b^3-b^2*(3*a+b)*x+3*b*(a+b)*x^2-(a+3*b+d)*x^3+x^4),x, algori thm="fricas")
Timed out. \[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate(x**2*(-3*a*b**3+2*b**2*(3*a+b)*x-3*b*(a+b)*x**2+x**4)/(x*(-a+x)* (-b+x)**3)**(3/4)/(a*b**3-b**2*(3*a+b)*x+3*b*(a+b)*x**2-(a+3*b+d)*x**3+x** 4),x)
\[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\int { -\frac {{\left (3 \, a b^{3} - 2 \, {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - x^{4}\right )} x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (a b^{3} - {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - {\left (a + 3 \, b + d\right )} x^{3} + x^{4}\right )}} \,d x } \]
integrate(x^2*(-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x )^3)^(3/4)/(a*b^3-b^2*(3*a+b)*x+3*b*(a+b)*x^2-(a+3*b+d)*x^3+x^4),x, algori thm="maxima")
-integrate((3*a*b^3 - 2*(3*a + b)*b^2*x + 3*(a + b)*b*x^2 - x^4)*x^2/(((a - x)*(b - x)^3*x)^(3/4)*(a*b^3 - (3*a + b)*b^2*x + 3*(a + b)*b*x^2 - (a + 3*b + d)*x^3 + x^4)), x)
\[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\int { -\frac {{\left (3 \, a b^{3} - 2 \, {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - x^{4}\right )} x^{2}}{\left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (a b^{3} - {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - {\left (a + 3 \, b + d\right )} x^{3} + x^{4}\right )}} \,d x } \]
integrate(x^2*(-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x )^3)^(3/4)/(a*b^3-b^2*(3*a+b)*x+3*b*(a+b)*x^2-(a+3*b+d)*x^3+x^4),x, algori thm="giac")
integrate(-(3*a*b^3 - 2*(3*a + b)*b^2*x + 3*(a + b)*b*x^2 - x^4)*x^2/(((a - x)*(b - x)^3*x)^(3/4)*(a*b^3 - (3*a + b)*b^2*x + 3*(a + b)*b*x^2 - (a + 3*b + d)*x^3 + x^4)), x)
Timed out. \[ \int \frac {x^2 \left (-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right )}{\left (x (-a+x) (-b+x)^3\right )^{3/4} \left (a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right )} \, dx=\int -\frac {x^2\,\left (3\,a\,b^3-x^4-2\,b^2\,x\,\left (3\,a+b\right )+3\,b\,x^2\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,{\left (b-x\right )}^3\right )}^{3/4}\,\left (a\,b^3-x^3\,\left (a+3\,b+d\right )+x^4-b^2\,x\,\left (3\,a+b\right )+3\,b\,x^2\,\left (a+b\right )\right )} \,d x \]
int(-(x^2*(3*a*b^3 - x^4 - 2*b^2*x*(3*a + b) + 3*b*x^2*(a + b)))/((x*(a - x)*(b - x)^3)^(3/4)*(a*b^3 - x^3*(a + 3*b + d) + x^4 - b^2*x*(3*a + b) + 3 *b*x^2*(a + b))),x)