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

Optimal. Leaf size=95 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 (-a-b)+a b x+x^3\right )^{3/4}}{x (x-b)}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 (-a-b)+a b x+x^3\right )^{3/4}}{x (x-b)}\right )}{d^{3/4}} \]

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Rubi [F]  time = 11.60, 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 (a-b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+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*(a - b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + (b + 3*a^2*d)*
x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 1.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2-2 a x+x^2\right ) \left (-a b+2 (a-b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+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*(a - b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + (b + 3*a
^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 3.37, size = 95, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (a b x+(-a-b) x^2+x^3\right )^{3/4}}{x (-b+x)}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (a b x+(-a-b) x^2+x^3\right )^{3/4}}{x (-b+x)}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(3/4))/(x*(-b + x))])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b*x + (
-a - b)*x^2 + x^3)^(3/4))/(x*(-b + x))])/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*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+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^{2} - 2 \, a x + x^{2}\right )} {\left (a b - 2 \, {\left (a - b\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} 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*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2-2*a*x+x^2)*(-a*b+2*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+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^{2} - 2 \, a x + x^{2}\right )} {\left (a b - 2 \, {\left (a - b\right )} x - x^{2}\right )}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} 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*(a-b)*x+x^2)/(x*(-a+x)*(-b+x))^(3/4)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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