3.11.76 \(\int \frac {-2 a b x^2+(a+b) x^3}{(x^2 (-a+x) (-b+x))^{3/4} (-a b+(a+b) x+(-1+d) x^2)} \, dx\) [1076]

3.11.76.1 Optimal result

Integrand size = 53, antiderivative size = 81 \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{3/4}} \]

output

-2*arctan(d^(1/4)*x/(a*b*x^2+(-a-b)*x^3+x^4)^(1/4))/d^(3/4)+2*arctanh(d^(1 
/4)*x/(a*b*x^2+(-a-b)*x^3+x^4)^(1/4))/d^(3/4)
 

3.11.76.2 Mathematica [F]

\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \]

input

Integrate[(-2*a*b*x^2 + (a + b)*x^3)/((x^2*(-a + x)*(-b + x))^(3/4)*(-(a*b 
) + (a + b)*x + (-1 + d)*x^2)),x]
 

output

Integrate[(-2*a*b*x^2 + (a + b)*x^3)/((x^2*(-a + x)*(-b + x))^(3/4)*(-(a*b 
) + (a + b)*x + (-1 + d)*x^2)), x]
 

3.11.76.3 Rubi [F]

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.

\(\displaystyle \int \frac {x^3 (a+b)-2 a b x^2}{\left (x^2 (x-a) (x-b)\right )^{3/4} \left (x (a+b)-a b+(d-1) x^2\right )} \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {x^2 (x (a+b)-2 a b)}{\left (x^2 (x-a) (x-b)\right )^{3/4} \left (x (a+b)-a b+(d-1) x^2\right )}dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \frac {\sqrt {x} (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{3/4} \left ((1-d) x^2-(a+b) x+a b\right )}dx}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {2 x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \frac {x (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{3/4} \left ((1-d) x^2-(a+b) x+a b\right )}d\sqrt {x}}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\)

\(\Big \downarrow \) 7279

\(\displaystyle \frac {2 x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \int \left (-\frac {a+b}{(1-d) \left (x^2-(a+b) x+a b\right )^{3/4}}-\frac {a b (a+b)-\left (a^2+2 b d a+b^2\right ) x}{(d-1) \left (x^2-(a+b) x+a b\right )^{3/4} \left ((1-d) x^2+(-a-b) x+a b\right )}\right )d\sqrt {x}}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 x^{3/2} \left (-x (a+b)+a b+x^2\right )^{3/4} \left (-\frac {\left ((a+b) \sqrt {a^2-2 a b (1-2 d)+b^2}+a^2+2 a b d+b^2\right ) \int \frac {1}{\left (-a-b+2 (1-d) x-\sqrt {a^2-2 b a+4 b d a+b^2}\right ) \left (x^2+(-a-b) x+a b\right )^{3/4}}d\sqrt {x}}{1-d}-\frac {\left (-(a+b) \sqrt {a^2-2 a b (1-2 d)+b^2}+a^2+2 a b d+b^2\right ) \int \frac {1}{\left (-a-b+2 (1-d) x+\sqrt {a^2-2 b a+4 b d a+b^2}\right ) \left (x^2+(-a-b) x+a b\right )^{3/4}}d\sqrt {x}}{1-d}-\frac {\sqrt {x} (a+b) (b-x) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) \left (-x (a+b)+a b+x^2\right )^{3/4}}\right )}{\left (x^2 (a-x) (b-x)\right )^{3/4}}\)

input

Int[(-2*a*b*x^2 + (a + b)*x^3)/((x^2*(-a + x)*(-b + x))^(3/4)*(-(a*b) + (a 
 + b)*x + (-1 + d)*x^2)),x]
 

output

$Aborted
 

3.11.76.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2035

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]
 

rule 2467

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]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

3.11.76.4 Maple [A] (verified)

Time = 3.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{-d^{\frac {1}{4}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{4}}}{x \,d^{\frac {1}{4}}}\right )}{d^{\frac {3}{4}}}\)	\(81\)

input

int((-2*a*b*x^2+(a+b)*x^3)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x+(d-1)*x 
^2),x,method=_RETURNVERBOSE)
 

output

1/d^(3/4)*(ln((d^(1/4)*x+(x^2*(a-x)*(b-x))^(1/4))/(-d^(1/4)*x+(x^2*(a-x)*( 
b-x))^(1/4)))+2*arctan((x^2*(a-x)*(b-x))^(1/4)/x/d^(1/4)))
 

3.11.76.5 Fricas [F(-1)]

Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]

input

integrate((-2*a*b*x^2+(a+b)*x^3)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x+( 
-1+d)*x^2),x, algorithm="fricas")
 

output

Timed out
 

3.11.76.6 Sympy [F(-1)]

Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]

input

integrate((-2*a*b*x**2+(a+b)*x**3)/(x**2*(-a+x)*(-b+x))**(3/4)/(-a*b+(a+b) 
*x+(-1+d)*x**2),x)
 

output

Timed out
 

3.11.76.7 Maxima [F]

\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b x^{2} - {\left (a + b\right )} x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]

input

integrate((-2*a*b*x^2+(a+b)*x^3)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x+( 
-1+d)*x^2),x, algorithm="maxima")
 

output

-integrate((2*a*b*x^2 - (a + b)*x^3)/(((a - x)*(b - x)*x^2)^(3/4)*((d - 1) 
*x^2 - a*b + (a + b)*x)), x)
 

3.11.76.8 Giac [F]

\[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int { -\frac {2 \, a b x^{2} - {\left (a + b\right )} x^{3}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {3}{4}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}} \,d x } \]

input

integrate((-2*a*b*x^2+(a+b)*x^3)/(x^2*(-a+x)*(-b+x))^(3/4)/(-a*b+(a+b)*x+( 
-1+d)*x^2),x, algorithm="giac")
 

output

integrate(-(2*a*b*x^2 - (a + b)*x^3)/(((a - x)*(b - x)*x^2)^(3/4)*((d - 1) 
*x^2 - a*b + (a + b)*x)), x)
 

3.11.76.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-2 a b x^2+(a+b) x^3}{\left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\int \frac {x^3\,\left (a+b\right )-2\,a\,b\,x^2}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{3/4}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\right )} \,d x \]

input

int((x^3*(a + b) - 2*a*b*x^2)/((x^2*(a - x)*(b - x))^(3/4)*(x*(a + b) - a* 
b + x^2*(d - 1))),x)
 

output

int((x^3*(a + b) - 2*a*b*x^2)/((x^2*(a - x)*(b - x))^(3/4)*(x*(a + b) - a* 
b + x^2*(d - 1))), x)