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

Optimal. Leaf size=150 \[ -2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}\right )-\frac {4 \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{x} \]

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Rubi [F]  time = 36.93, 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+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(a - x)*(b - x)^2*AppellF1[-3/4, -1/4, -3/2, 1/4, x/a, x/b])/((-((a - x)*(b - x)^2*x))^(3/4)*(1 - x/a)^(1/4
)*Sqrt[1 - x/b]) - (4*(5*a + b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][((-a + x^4)^(1/4
)*(-b + x^4)^(3/2))/(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2), x], x, x^(1/4)])/(-((a - x)*(b - x
)^2*x))^(3/4) + (12*(a + 2*b)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(x^4*(-a + x^4)^(1
/4)*(-b + x^4)^(3/2))/(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2), x], x, x^(1/4)])/(b*(-((a - x)*(
b - x)^2*x))^(3/4)) - (12*(1 - d)*x^(3/4)*(-a + x)^(3/4)*(-b + x)^(3/2)*Defer[Subst][Defer[Int][(x^8*(-a + x^4
)^(1/4)*(-b + x^4)^(3/2))/(-(b^2*x^4) + 2*b*x^8 + (-1 + d)*x^12 + a*(b - x^4)^2), x], x, x^(1/4)])/(b*(-((a -
x)*(b - x)^2*x))^(3/4))

Rubi steps

\begin {align*} \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{-a+x} (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x^{7/4} (-b+x)^{3/2} \left (a b^2-b (2 a+b) x+(a+2 b) 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 {\sqrt [4]{-a+x} (-3 a b+(a+2 b) x) \left (b^2-2 b x+x^2\right )}{x^{7/4} \sqrt {-b+x} \left (a b^2-b (2 a+b) x+(a+2 b) 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 {\sqrt [4]{-a+x} (-b+x)^{3/2} (-3 a b+(a+2 b) x)}{x^{7/4} \left (a b^2-b (2 a+b) x+(a+2 b) 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 {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2} \left (-3 a b+(a+2 b) x^4\right )}{x^4 \left (a b^2-b (2 a+b) x^4+(a+2 b) 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 {3 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{b x^4}+\frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2} \left (-b (5 a+b)+3 (a+2 b) x^4-3 (1-d) x^8\right )}{b \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\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 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2} \left (-b (5 a+b)+3 (a+2 b) x^4-3 (1-d) x^8\right )}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{b \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 {(-5 a-b) b \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}}+\frac {3 (a+2 b) x^4 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}}+\frac {3 (-1+d) x^8 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}}\right ) \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 x^{3/4} (-a+x) (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (-b+x^4\right )^{3/2} \sqrt [4]{1-\frac {x^4}{a}}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4} \sqrt [4]{1-\frac {x}{a}}}\\ &=-\frac {\left (4 (5 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (12 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 (1-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^4+a \left (1+\frac {2 b}{a}\right ) x^8-(1-d) x^{12}} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (12 x^{3/4} (-a+x) (-b+x)^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x^4}{a}} \left (1-\frac {x^4}{b}\right )^{3/2}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}}\\ &=\frac {4 (a-x) (b-x)^2 F_1\left (-\frac {3}{4};-\frac {1}{4},-\frac {3}{2};\frac {1}{4};\frac {x}{a},\frac {x}{b}\right )}{\left (-\left ((a-x) (b-x)^2 x\right )\right )^{3/4} \sqrt [4]{1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}}-\frac {\left (4 (5 a+b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2} \, dx,x,\sqrt [4]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (12 (a+2 b) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}-\frac {\left (12 (1-d) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt [4]{-a+x^4} \left (-b+x^4\right )^{3/2}}{-b^2 x^4+2 b x^8+(-1+d) x^{12}+a \left (b-x^4\right )^2} \, dx,x,\sqrt [4]{x}\right )}{b \left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 7.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-a+x) (-3 a b+(a+2 b) x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{x \left (x (-a+x) (-b+x)^2\right )^{3/4} \left (a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 4.82, size = 150, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}-2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )+2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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