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

Optimal. Leaf size=77 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 (-a-b)+a b x+x^3}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 (-a-b)+a b x+x^3}}\right )}{d^{3/4}} \]

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Rubi [F]  time = 13.65, 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 {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(1 - 2*a*d - 2*b*d)*x*(1 - x/a)^(3/4)*(1 - x/b)^(3/4)*AppellF1[1/4, 3/4, 3/4, 5/4, x/a, x/b])/(d^2*((a - x)
*(b - x)*x)^(3/4)) + (4*x^2*(1 - x/a)^(3/4)*(1 - x/b)^(3/4)*AppellF1[5/4, 3/4, 3/4, 9/4, x/a, x/b])/(5*d*((a -
x)*(b - x)*x)^(3/4)) - (4*a*b*(1 - 2*a*d - 2*b*d)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[In
t][1/((-a + x^4)^(3/4)*(-b + x^4)^(3/4)*(a*b - a*(1 + b/a)*x^4 + x^8 - d*x^12)), x], x, x^(1/4)])/(d^2*((a - x
)*(b - x)*x)^(3/4)) + (4*(a + b - 2*a^2*d - 5*a*b*d - 2*b^2*d)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Sub
st][Defer[Int][x^4/((-a + x^4)^(3/4)*(-b + x^4)^(3/4)*(a*b - a*(1 + b/a)*x^4 + x^8 - d*x^12)), x], x, x^(1/4)]
)/(d^2*((a - x)*(b - x)*x)^(3/4)) - (4*(1 - 3*b*d - 3*a*d*(1 - b*d))*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/4)*Def
er[Subst][Defer[Int][x^8/((-a + x^4)^(3/4)*(-b + x^4)^(3/4)*(a*b - a*(1 + b/a)*x^4 + x^8 - d*x^12)), x], x, x^
(1/4)])/(d^2*((a - x)*(b - x)*x)^(3/4))

Rubi steps

\begin {align*} \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {x^{5/4} \left (3 a b-2 (a+b) x+x^2\right )}{(-a+x)^{3/4} (-b+x)^{3/4} \left (-a b+(a+b) x-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 {x^8 \left (3 a b-2 (a+b) x^4+x^8\right )}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (-a b+(a+b) x^4-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 {1-2 a d-2 b d}{d^2 \left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4}}+\frac {x^4}{d \left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4}}+\frac {a b (1-2 a d-2 b d)+\left (2 a^2 d-a (1-5 b d)-b (1-2 b d)\right ) x^4+(1-3 b d-3 a d (1-b d)) x^8}{d^2 \left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (-a b+(a+b) x^4-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 {a b (1-2 a d-2 b d)+\left (2 a^2 d-a (1-5 b d)-b (1-2 b d)\right ) x^4+(1-3 b d-3 a d (1-b d)) x^8}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (-a b+(a+b) x^4-x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (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^4}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{d (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 (1-2 a d-2 b d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{d^2 (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 (-1+2 a d+2 b d)}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )}+\frac {\left (-2 a^2 d+a (1-5 b d)+b (1-2 b d)\right ) x^4}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )}+\frac {(-1+3 b d+3 a d (1-b d)) x^8}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 x^{3/4} (-b+x)^{3/4} \left (1-\frac {x}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-b+x^4\right )^{3/4} \left (1-\frac {x^4}{a}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{d (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 (1-2 a d-2 b d) x^{3/4} (-b+x)^{3/4} \left (1-\frac {x}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-b+x^4\right )^{3/4} \left (1-\frac {x^4}{a}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}\\ &=-\frac {\left (4 a b (1-2 a d-2 b d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 \left (a+b-2 a^2 d-5 a b d-2 b^2 d\right ) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-1+3 b d+3 a d (1-b d)) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 x^{3/4} \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-\frac {x^4}{a}\right )^{3/4} \left (1-\frac {x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{d (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 (1-2 a d-2 b d) x^{3/4} \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^4}{a}\right )^{3/4} \left (1-\frac {x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}\\ &=\frac {4 (1-2 a d-2 b d) x \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4} F_1\left (\frac {1}{4};\frac {3}{4},\frac {3}{4};\frac {5}{4};\frac {x}{a},\frac {x}{b}\right )}{d^2 ((a-x) (b-x) x)^{3/4}}+\frac {4 x^2 \left (1-\frac {x}{a}\right )^{3/4} \left (1-\frac {x}{b}\right )^{3/4} F_1\left (\frac {5}{4};\frac {3}{4},\frac {3}{4};\frac {9}{4};\frac {x}{a},\frac {x}{b}\right )}{5 d ((a-x) (b-x) x)^{3/4}}-\frac {\left (4 a b (1-2 a d-2 b d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 \left (a+b-2 a^2 d-5 a b d-2 b^2 d\right ) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}+\frac {\left (4 (-1+3 b d+3 a d (1-b d)) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-a+x^4\right )^{3/4} \left (-b+x^4\right )^{3/4} \left (a b-a \left (1+\frac {b}{a}\right ) x^4+x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{d^2 (x (-a+x) (-b+x))^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 1.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (3 a b-2 (a+b) x+x^2\right )}{(x (-a+x) (-b+x))^{3/4} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.46, size = 77, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((3*a*b - 2*(a + b)*x + x^2)*x^2/((d*x^3 - a*b + (a + b)*x - x^2)*((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 {x^2\,\left (3\,a\,b+x^2-2\,x\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{3/4}\,\left (-d\,x^3+x^2+\left (-a-b\right )\,x+a\,b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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