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

Optimal. Leaf size=235 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-a^2 b^3 x+x^4 \left (a^2+6 a b+3 b^2\right )+x^3 \left (-3 a^2 b-6 a b^2-b^3\right )+x^2 \left (3 a^2 b^2+2 a b^3\right )+x^5 (-2 a-3 b)+x^6\right )^{3/4}}{x (x-a)^2 (b-x)^2}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-a^2 b^3 x+x^4 \left (a^2+6 a b+3 b^2\right )+x^3 \left (-3 a^2 b-6 a b^2-b^3\right )+x^2 \left (3 a^2 b^2+2 a b^3\right )+x^5 (-2 a-3 b)+x^6\right )^{3/4}}{x (x-a)^2 (b-x)^2}\right )}{d^{3/4}} \]

________________________________________________________________________________________

Rubi [F]  time = 10.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(12*b*x^(3/4)*(-a + x)^(3/2)*(-b + x)^(9/4)*Defer[Subst][Defer[Int][x^4/(Sqrt[-a + x^4]*(-b + x^4)^(1/4)*(-(b*
d) - a^2*(1 - d/a^2)*x^4 + 2*a*x^8 - x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(3/4) + (4*a*b*x^(3/4
)*(-a + x)^(3/2)*(-b + x)^(9/4)*Defer[Subst][Defer[Int][1/(Sqrt[-a + x^4]*(-b + x^4)^(1/4)*(b*d + a^2*(1 - d/a
^2)*x^4 - 2*a*x^8 + x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(3/4) + (8*x^(3/4)*(-a + x)^(3/2)*(-b
+ x)^(9/4)*Defer[Subst][Defer[Int][x^8/(Sqrt[-a + x^4]*(-b + x^4)^(1/4)*(b*d + a^2*(1 - d/a^2)*x^4 - 2*a*x^8 +
x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(3/4)

Rubi steps

\begin {align*} \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx &=\int \frac {(-b+x)^2 \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx\\ &=\frac {\left (x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{x^{3/4} (-a+x)^{3/2} \sqrt [4]{-b+x} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \int \frac {a b-3 b x+2 x^2}{x^{3/4} \sqrt {-a+x} \sqrt [4]{-b+x} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {a b-3 b x^4+2 x^8}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+\left (a^2-d\right ) x^4-2 a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \left (\frac {3 b x^4}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (-b d-a^2 \left (1-\frac {d}{a^2}\right ) x^4+2 a x^8-x^{12}\right )}+\frac {a b}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )}+\frac {2 x^8}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}\\ &=\frac {\left (8 x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}+\frac {\left (12 b x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (-b d-a^2 \left (1-\frac {d}{a^2}\right ) x^4+2 a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}+\frac {\left (4 a b x^{3/4} (-a+x)^{3/2} (-b+x)^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^4} \sqrt [4]{-b+x^4} \left (b d+a^2 \left (1-\frac {d}{a^2}\right ) x^4-2 a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 4.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b^2-2 b x+x^2\right ) \left (-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right )}{\left (x (-a+x)^2 (-b+x)^3\right )^{3/4} \left (b d+\left (a^2-d\right ) x-2 a x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.33, size = 235, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-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\right )^{3/4}}{(b-x)^2 x (-a+x)^2}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-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\right )^{3/4}}{(b-x)^2 x (-a+x)^2}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} b - 4 \, a b x + {\left (2 \, a + 3 \, b\right )} x^{2} - 2 \, x^{3}\right )} {\left (b^{2} - 2 \, b x + x^{2}\right )}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (2 \, a x^{2} - x^{3} - b d - {\left (a^{2} - d\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b^{2}-2 b x +x^{2}\right ) \left (-a^{2} b +4 a b x -\left (2 a +3 b \right ) x^{2}+2 x^{3}\right )}{\left (x \left (-a +x \right )^{2} \left (-b +x \right )^{3}\right )^{\frac {3}{4}} \left (b d +\left (a^{2}-d \right ) x -2 a \,x^{2}+x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} b - 4 \, a b x + {\left (2 \, a + 3 \, b\right )} x^{2} - 2 \, x^{3}\right )} {\left (b^{2} - 2 \, b x + x^{2}\right )}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {3}{4}} {\left (2 \, a x^{2} - x^{3} - b d - {\left (a^{2} - d\right )} x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (b^2-2\,b\,x+x^2\right )\,\left (x^2\,\left (2\,a+3\,b\right )+a^2\,b-2\,x^3-4\,a\,b\,x\right )}{{\left (-x\,{\left (a-x\right )}^2\,{\left (b-x\right )}^3\right )}^{3/4}\,\left (x^3-2\,a\,x^2+\left (a^2-d\right )\,x+b\,d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________