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

Optimal. Leaf size=130 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}{a-x}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}{a-x}\right )-\frac {4 \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)} \]

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Rubi [F]  time = 28.98, 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 {(-2 a b+(3 a-b) x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{x (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(a - x)*(1 - x/b)^(1/4)*AppellF1[-1/2, -11/4, 5/4, 1/2, x/a, x/b])/(((a - x)*(b - x)*x^2)^(1/4)*(1 - x/a)^(
3/4)) + (2*(3*a - 7*b)*Sqrt[x]*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][(-a + x^2)^(11/4)/((-b +
x^2)^(5/4)*(a^3 - 3*a^2*x^2 + 3*a*(1 - (b*d)/(3*a))*x^4 - (1 - d)*x^6)), x], x, Sqrt[x]])/((a - x)*(b - x)*x^2
)^(1/4) + (4*b*(3*a - b*d)*Sqrt[x]*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][(x^2*(-a + x^2)^(11/4
))/((-b + x^2)^(5/4)*(a^3 - 3*a^2*x^2 + 3*a*(1 - (b*d)/(3*a))*x^4 - (1 - d)*x^6)), x], x, Sqrt[x]])/(a^2*((a -
x)*(b - x)*x^2)^(1/4)) - (4*b*(1 - d)*Sqrt[x]*(-a + x)^(1/4)*(-b + x)^(1/4)*Defer[Subst][Defer[Int][(x^4*(-a
+ x^2)^(11/4))/((-b + x^2)^(5/4)*(a^3 - 3*a^2*x^2 + 3*a*(1 - (b*d)/(3*a))*x^4 - (1 - d)*x^6)), x], x, Sqrt[x]]
)/(a^2*((a - x)*(b - x)*x^2)^(1/4))

Rubi steps

\begin {align*} \int \frac {(-2 a b+(3 a-b) x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{x (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-2 a b+(3 a-b) x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{x^{3/2} \sqrt [4]{-a+x} (-b+x)^{5/4} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-a+x)^{3/4} (-2 a b+(3 a-b) x) \left (a^2-2 a x+x^2\right )}{x^{3/2} (-b+x)^{5/4} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \int \frac {(-a+x)^{11/4} (-2 a b+(3 a-b) x)}{x^{3/2} (-b+x)^{5/4} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^2\right )^{11/4} \left (-2 a b+(3 a-b) x^2\right )}{x^2 \left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+(3 a-b d) x^4+(-1+d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 b \left (-a+x^2\right )^{11/4}}{a^2 x^2 \left (-b+x^2\right )^{5/4}}+\frac {\left (-a+x^2\right )^{11/4} \left (a^2 (3 a-7 b)+2 b (3 a-b d) x^2-2 b (1-d) x^4\right )}{a^2 \left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^2\right )^{11/4} \left (a^2 (3 a-7 b)+2 b (3 a-b d) x^2-2 b (1-d) x^4\right )}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}-\frac {\left (4 b \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^2\right )^{11/4}}{x^2 \left (-b+x^2\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a^2 (3 a-7 b) \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )}+\frac {2 b (3 a-b d) x^2 \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )}+\frac {2 b (-1+d) x^4 \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}-\frac {\left (4 b \sqrt {x} (-a+x) \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{a}\right )^{11/4}}{x^2 \left (-b+x^2\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (1-\frac {x}{a}\right )^{3/4}}\\ &=\frac {\left (2 (3 a-7 b) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2 (-a+x) (-b+x)}}-\frac {\left (4 b (1-d) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}+\frac {\left (4 b (3 a-b d) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}+\frac {\left (4 \sqrt {x} (-a+x) \sqrt [4]{1-\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{a}\right )^{11/4}}{x^2 \left (1-\frac {x^2}{b}\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2 (-a+x) (-b+x)} \left (1-\frac {x}{a}\right )^{3/4}}\\ &=\frac {4 (a-x) \sqrt [4]{1-\frac {x}{b}} F_1\left (-\frac {1}{2};-\frac {11}{4},\frac {5}{4};\frac {1}{2};\frac {x}{a},\frac {x}{b}\right )}{\sqrt [4]{(a-x) (b-x) x^2} \left (1-\frac {x}{a}\right )^{3/4}}+\frac {\left (2 (3 a-7 b) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {\left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2 (-a+x) (-b+x)}}-\frac {\left (4 b (1-d) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}+\frac {\left (4 b (3 a-b d) \sqrt {x} \sqrt [4]{-a+x} \sqrt [4]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a+x^2\right )^{11/4}}{\left (-b+x^2\right )^{5/4} \left (a^3-3 a^2 x^2+3 a \left (1-\frac {b d}{3 a}\right ) x^4-(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt [4]{x^2 (-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 7.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-2 a b+(3 a-b) x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{x (-b+x) \sqrt [4]{x^2 (-a+x) (-b+x)} \left (a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.72, size = 130, normalized size = 1.00 \begin {gather*} -\frac {4 \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x^2 (-b+x)}+2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((-2*a*b+(3*a-b)*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/x/(-b+x)/(x^2*(-a+x)*(-b+x))^(1/4)/(a^3-3*a^2*x+(-b*d+
3*a)*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^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (2 \, a b - {\left (3 \, a - b\right )} x\right )}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{4}} {\left ({\left (d - 1\right )} x^{3} + a^{3} - 3 \, a^{2} x - {\left (b d - 3 \, a\right )} x^{2}\right )} {\left (b - x\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-2*a*b+(3*a-b)*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/x/(-b+x)/(x^2*(-a+x)*(-b+x))^(1/4)/(a^3-3*a^2*x+(-b*d+3*a)*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^{3} - 3 \, a^{2} x + 3 \, a x^{2} - x^{3}\right )} {\left (2 \, a b - {\left (3 \, a - b\right )} x\right )}}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{4}} {\left ({\left (d - 1\right )} x^{3} + a^{3} - 3 \, a^{2} x - {\left (b d - 3 \, a\right )} x^{2}\right )} {\left (b - x\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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