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

Optimal. Leaf size=215 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+x^4 \left (a^2+6 a b+3 b^2\right )+x^3 \left (-3 a^2 b-6 a b^2-b^3\right )+x^2 \left (3 a^2 b^2+2 a b^3\right )+x^5 (-2 a-3 b)+x^6}}{b-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+x^4 \left (a^2+6 a b+3 b^2\right )+x^3 \left (-3 a^2 b-6 a b^2-b^3\right )+x^2 \left (3 a^2 b^2+2 a b^3\right )+x^5 (-2 a-3 b)+x^6}}{b-x}\right )}{d^{3/4}} \]

________________________________________________________________________________________

Rubi [F]  time = 14.67, 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 {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(12*b*x^(1/4)*Sqrt[-a + x]*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^6*Sqrt[-a + x^4])/((-b + x^4)^(3/4)*(-b +
(1 - a^2*d)*x^4 + 2*a*d*x^8 - d*x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(1/4) + (4*a*b*x^(1/4)*Sq
rt[-a + x]*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^2*Sqrt[-a + x^4])/((-b + x^4)^(3/4)*(b - (1 - a^2*d)*x^4
- 2*a*d*x^8 + d*x^12)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(1/4) + (8*x^(1/4)*Sqrt[-a + x]*(-b + x)^(
3/4)*Defer[Subst][Defer[Int][(x^10*Sqrt[-a + x^4])/((-b + x^4)^(3/4)*(b - (1 - a^2*d)*x^4 - 2*a*d*x^8 + d*x^12
)), x], x, x^(1/4)])/(-((a - x)^2*(b - x)^3*x))^(1/4)

Rubi steps

\begin {align*} \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \int \frac {\sqrt {-a+x} \left (a b-3 b x+2 x^2\right )}{\sqrt [4]{x} (-b+x)^{3/4} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^4} \left (a b-3 b x^4+2 x^8\right )}{\left (-b+x^4\right )^{3/4} \left (b+\left (-1+a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {3 b x^6 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (-b+\left (1-a^2 d\right ) x^4+2 a d x^8-d x^{12}\right )}+\frac {a b x^2 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )}+\frac {2 x^{10} \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}\\ &=\frac {\left (8 \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{10} \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (-b+\left (1-a^2 d\right ) x^4+2 a d x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}+\frac {\left (4 a b \sqrt [4]{x} \sqrt {-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-a+x^4}}{\left (-b+x^4\right )^{3/4} \left (b-\left (1-a^2 d\right ) x^4-2 a d x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x)^2 (-b+x)^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 4.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.54, size = 215, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-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}}{b-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-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}}{b-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} b - 4 \, a b x + {\left (2 \, a + 3 \, b\right )} x^{2} - 2 \, x^{3}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (2 \, a d x^{2} - d x^{3} - {\left (a^{2} d - 1\right )} x - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-a^{2} b +4 a b x -\left (2 a +3 b \right ) x^{2}+2 x^{3}}{\left (x \left (-a +x \right )^{2} \left (-b +x \right )^{3}\right )^{\frac {1}{4}} \left (b +\left (a^{2} d -1\right ) x -2 a d \,x^{2}+d \,x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} b - 4 \, a b x + {\left (2 \, a + 3 \, b\right )} x^{2} - 2 \, x^{3}}{\left (-{\left (a - x\right )}^{2} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}} {\left (2 \, a d x^{2} - d x^{3} - {\left (a^{2} d - 1\right )} x - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________