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

Optimal. Leaf size=133 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^3 (-a-b)+a b x^2+x^4\right )^{3/4}}{x^2 (x-b)}\right )-\frac {4 \sqrt [4]{x^3 (-a-b)+a b x^2+x^4}}{a-x} \]

________________________________________________________________________________________

Rubi [F]  time = 26.53, 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^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-4*(3*a - b)*(b - x)*x^2)/(3*a*(1 - d)*((a - x)*(b - x)*x^2)^(3/4)) - (2*(3*a - b)*((b*(a - x))/(a*(b - x)))^
(3/4)*(b - x)*x^2*Hypergeometric2F1[1/2, 3/4, 3/2, -(((a - b)*x)/(a*(b - x)))])/(3*a*(1 - d)*((a - x)*(b - x)*
x^2)^(3/4)) + (2*a^3*(3*a - b)*d*x^(3/2)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(-b + x^2)^(1/4
)/((-a + x^2)^(7/4)*(a^3*d - 3*a^2*d*x^2 - b*(1 - (3*a*d)/b)*x^4 + (1 - d)*x^6)), x], x, Sqrt[x]])/((1 - d)*((
a - x)*(b - x)*x^2)^(3/4)) - (6*a^2*(3*a - b)*d*x^(3/2)*(-a + x)^(3/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][
(x^2*(-b + x^2)^(1/4))/((-a + x^2)^(7/4)*(a^3*d - 3*a^2*d*x^2 - b*(1 - (3*a*d)/b)*x^4 + (1 - d)*x^6)), x], x,
Sqrt[x]])/((1 - d)*((a - x)*(b - x)*x^2)^(3/4)) + (2*(b^2 + 9*a^2*d - a*(b + 5*b*d))*x^(3/2)*(-a + x)^(3/4)*(-
b + x)^(3/4)*Defer[Subst][Defer[Int][(x^4*(-b + x^2)^(1/4))/((-a + x^2)^(7/4)*(a^3*d - 3*a^2*d*x^2 - b*(1 - (3
*a*d)/b)*x^4 + (1 - d)*x^6)), x], x, Sqrt[x]])/((1 - d)*((a - x)*(b - x)*x^2)^(3/4))

Rubi steps

\begin {align*} \int \frac {x^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx &=\frac {\left (x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \int \frac {x^{3/2} \sqrt [4]{-b+x} (-2 a b+(3 a-b) x)}{(-a+x)^{7/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{-b+x^2} \left (-2 a b+(3 a-b) x^2\right )}{\left (-a+x^2\right )^{7/4} \left (-a^3 d+3 a^2 d x^2+(b-3 a d) x^4+(-1+d) x^6\right )} \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (-\frac {(3 a-b) \sqrt [4]{-b+x^2}}{(1-d) \left (-a+x^2\right )^{7/4}}+\frac {\sqrt [4]{-b+x^2} \left (a^3 (3 a-b) d-3 a^2 (3 a-b) d x^2+\left (b^2+9 a^2 d-a (b+5 b d)\right ) x^4\right )}{(-1+d) \left (-a+x^2\right )^{7/4} \left (-a^3 d+3 a^2 d x^2+(b-3 a d) x^4+(-1+d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=-\frac {\left (2 (3 a-b) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4}} \, dx,x,\sqrt {x}\right )}{(1-d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+x^2} \left (a^3 (3 a-b) d-3 a^2 (3 a-b) d x^2+\left (b^2+9 a^2 d-a (b+5 b d)\right ) x^4\right )}{\left (-a+x^2\right )^{7/4} \left (-a^3 d+3 a^2 d x^2+(b-3 a d) x^4+(-1+d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=-\frac {4 (3 a-b) (b-x) x^2}{3 a (1-d) \left ((a-x) (b-x) x^2\right )^{3/4}}-\frac {\left (2 (3 a-b) b x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^2\right )^{3/4} \left (-b+x^2\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{3 a (1-d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (2 x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {a^3 (-3 a+b) d \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}+\frac {3 a^2 (3 a-b) d x^2 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}+\frac {\left (-b^2-9 a^2 d+a (b+5 b d)\right ) x^4 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ &=-\frac {4 (3 a-b) (b-x) x^2}{3 a (1-d) \left ((a-x) (b-x) x^2\right )^{3/4}}-\frac {2 (3 a-b) \left (\frac {b (a-x)}{a (b-x)}\right )^{3/4} (b-x) x^2 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {(a-b) x}{a (b-x)}\right )}{3 a (1-d) \left ((a-x) (b-x) x^2\right )^{3/4}}+\frac {\left (6 a^2 (3 a-b) d x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}-\frac {\left (2 a^3 (3 a-b) d x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}+\frac {\left (2 \left (-b^2-9 a^2 d+a (b+5 b d)\right ) x^{3/2} (-a+x)^{3/4} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [4]{-b+x^2}}{\left (-a+x^2\right )^{7/4} \left (a^3 d-3 a^2 d x^2-b \left (1-\frac {3 a d}{b}\right ) x^4+(1-d) x^6\right )} \, dx,x,\sqrt {x}\right )}{(-1+d) \left (x^2 (-a+x) (-b+x)\right )^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 5.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left (x^2 (-a+x) (-b+x)\right )^{3/4} \left (-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 6.68, size = 133, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{a b x^2+(-a-b) x^3+x^4}}{a-x}+2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x^2 (-b+x)}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (a b x^2+(-a-b) x^3+x^4\right )^{3/4}}{x^2 (-b+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (-b +x \right ) \left (-2 a b +\left (3 a -b \right ) x \right )}{\left (-a +x \right ) \left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {3}{4}} \left (-a^{3} d +3 a^{2} d x +\left (-3 a d +b \right ) x^{2}+\left (-1+d \right ) x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________