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

Optimal. Leaf size=125 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{3/4}}{x (x-b)^2}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{3/4}}{x (x-b)^2}\right )}{d^{3/4}} \]

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Rubi [F]  time = 8.44, 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 (a^2-2 a x+x^2\right ) \left (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*a*b*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(-a + x^4)^(5/4)/(Sqrt[-b + x^4]*(-(a^3*d
) - b^2*(1 - (3*a^2*d)/b^2)*x^4 + 2*b*(1 - (3*a*d)/(2*b))*x^8 - (1 - d)*x^12)), x], x, x^(1/4)])/(-((a - x)*(b
- x)^2*x))^(3/4) + (4*(3*a - 2*b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(x^4*(-a + x^
4)^(5/4))/(Sqrt[-b + x^4]*(a^3*d + b^2*(1 - (3*a^2*d)/b^2)*x^4 - 2*b*(1 - (3*a*d)/(2*b))*x^8 + (1 - d)*x^12)),
x], x, x^(1/4)])/(-((a - x)*(b - x)^2*x))^(3/4)

Rubi steps

\begin {align*} \int \frac {\left (a^2-2 a x+x^2\right ) \left (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx &=\int \frac {(-a+x)^2 \left (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(-a+x)^{5/4} \left (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right )}{x^{3/4} (-b+x)^{3/2} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {(-a+x)^{5/4} (-a b+(3 a-2 b) x)}{x^{3/4} \sqrt {-b+x} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^4\right )^{5/4} \left (-a b+(3 a-2 b) x^4\right )}{\sqrt {-b+x^4} \left (a^3 d+\left (b^2-3 a^2 d\right ) x^4+(-2 b+3 a d) x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (4 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \left (\frac {a b \left (-a+x^4\right )^{5/4}}{\sqrt {-b+x^4} \left (-a^3 d-b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x^4+2 b \left (1-\frac {3 a d}{2 b}\right ) x^8-(1-d) x^{12}\right )}+\frac {(3 a-2 b) x^4 \left (-a+x^4\right )^{5/4}}{\sqrt {-b+x^4} \left (a^3 d+b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x^4-2 b \left (1-\frac {3 a d}{2 b}\right ) x^8+(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (4 (3 a-2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-a+x^4\right )^{5/4}}{\sqrt {-b+x^4} \left (a^3 d+b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x^4-2 b \left (1-\frac {3 a d}{2 b}\right ) x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (4 a b x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^4\right )^{5/4}}{\sqrt {-b+x^4} \left (-a^3 d-b^2 \left (1-\frac {3 a^2 d}{b^2}\right ) x^4+2 b \left (1-\frac {3 a d}{2 b}\right ) x^8-(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 5.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2-2 a x+x^2\right ) \left (a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a^3 d+\left (b^2-3 a^2 d\right ) x+(-2 b+3 a d) x^2+(1-d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.67, size = 125, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{x (-b+x)^2}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{x (-b+x)^2}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x^2\,\left (3\,a-2\,b\right )+a\,b^2-2\,b\,x\,\left (2\,a-b\right )\right )\,\left (a^2-2\,a\,x+x^2\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (x^2\,\left (2\,b-3\,a\,d\right )-a^3\,d+x\,\left (3\,a^2\,d-b^2\right )+x^3\,\left (d-1\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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