3.13.83 \(\int \frac {2 a b x+(-3 a+b) x^2}{\sqrt [4]{x^2 (-a+x) (-b+x)} (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3)} \, dx\) [1283]

3.13.83.1 Optimal result

Integrand size = 65, antiderivative size = 93 \[ \int \frac {2 a b x+(-3 a+b) x^2}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )}{d^{3/4}} \]

output

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

3.13.83.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(93)=186\).

Time = 33.04 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.27 \[ \int \frac {2 a b x+(-3 a+b) x^2}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx=-\frac {x \sqrt [4]{\frac {-b+x}{a-x}} \left (\arctan \left (\frac {a-x \left (1+\sqrt {d} \sqrt {\frac {-b+x}{a-x}}\right )}{\sqrt {2} \sqrt [4]{d} \sqrt {\frac {x}{a-x}} (-a+x) \sqrt [4]{\frac {-b+x}{a-x}}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} x \sqrt [4]{\frac {-b+x}{a-x}}}{\sqrt {\frac {x}{2 a-2 x}} \left (a+x \left (-1+\sqrt {d} \sqrt {\frac {-b+x}{a-x}}\right )\right )}\right )\right )}{d^{3/4} \sqrt {\frac {x}{2 a-2 x}} \sqrt [4]{x^2 (-a+x) (-b+x)}} \]

input

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

output

-((x*((-b + x)/(a - x))^(1/4)*(ArcTan[(a - x*(1 + Sqrt[d]*Sqrt[(-b + x)/(a 
 - x)]))/(Sqrt[2]*d^(1/4)*Sqrt[x/(a - x)]*(-a + x)*((-b + x)/(a - x))^(1/4 
))] - ArcTanh[(d^(1/4)*x*((-b + x)/(a - x))^(1/4))/(Sqrt[x/(2*a - 2*x)]*(a 
 + x*(-1 + Sqrt[d]*Sqrt[(-b + x)/(a - x)])))]))/(d^(3/4)*Sqrt[x/(2*a - 2*x 
)]*(x^2*(-a + x)*(-b + x))^(1/4)))
 

3.13.83.3 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.

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.13.83.3.1 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 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.13.83.4 Maple [F]

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

input

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

output

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

3.13.83.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.13.83.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.13.83.7 Maxima [F]

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

input

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

output

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

3.13.83.8 Giac [F]

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

input

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

output

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

3.13.83.9 Mupad [F(-1)]

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

input

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

output

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