3.28.57 \(\int \frac {(a-5 b+4 x) (-a^3+3 a^2 x-3 a x^2+x^3)}{((-a+x) (-b+x))^{2/3} (-a^5+b d-(-5 a^4+d) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5)} \, dx\)

Optimal. Leaf size=258 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a^2-2 \sqrt {3} a x+\sqrt {3} x^2}{a^2+2 \sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a x+x^2}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^3-2 a^2 x-a \sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}+a x^2\right )}{\sqrt [3]{d}}-\frac {\log \left (a^6-4 a^5 x+6 a^4 x^2-4 a^3 x^3+a^2 d^{2/3} \left (x (-a-b)+a b+x^2\right )^{2/3}+a^2 x^4+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2\right )\right )}{2 \sqrt [3]{d}} \]

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Rubi [F]  time = 9.11, 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-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x))^(2/3)*(-a^5 + b*d - (-5*a^4 + d)*x
- 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)),x]

[Out]

(-3*(a - 5*b)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^9/((a - b + x^3)^(2/3)*(a*(1 - b/a)*d +
d*x^3 - x^15)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(2/3) + (12*a*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Su
bst][Defer[Int][x^9/((a - b + x^3)^(2/3)*(-(a*(1 - b/a)*d) - d*x^3 + x^15)), x], x, (-a + x)^(1/3)])/((a - x)*
(b - x))^(2/3) + (12*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^12/((a - b + x^3)^(2/3)*(-(a*(1 -
b/a)*d) - d*x^3 + x^15)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(2/3)

Rubi steps

\begin {align*} \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x} (a-5 b+4 x) \left (a^2-2 a x+x^2\right )}{(-b+x)^{2/3} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {(-a+x)^{7/3} (a-5 b+4 x)}{(-b+x)^{2/3} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {(-a+5 b-4 x) (-a+x)^{7/3}}{(-b+x)^{2/3} \left (a^5 \left (1-\frac {b d}{a^5}\right )+\left (-5 a^4+d\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \left (\frac {5 \left (1-\frac {a}{5 b}\right ) b (-a+x)^{7/3}}{(-b+x)^{2/3} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}+\frac {4 x (-a+x)^{7/3}}{(-b+x)^{2/3} \left (-a^5 \left (1-\frac {b d}{a^5}\right )+5 a^4 \left (1-\frac {d}{5 a^4}\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )}\right ) \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (4 (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {x (-a+x)^{7/3}}{(-b+x)^{2/3} \left (-a^5 \left (1-\frac {b d}{a^5}\right )+5 a^4 \left (1-\frac {d}{5 a^4}\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx}{((-a+x) (-b+x))^{2/3}}+\frac {\left ((-a+5 b) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {(-a+x)^{7/3}}{(-b+x)^{2/3} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (12 (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9 \left (a+x^3\right )}{\left (a-b+x^3\right )^{2/3} \left (-a d+b d-d x^3+x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}+\frac {\left (3 (-a+5 b) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (a-b+x^3\right )^{2/3} \left (a d-b d+d x^3-x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (12 (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9 \left (-a-x^3\right )}{\left (a-b+x^3\right )^{2/3} \left (a \left (1-\frac {b}{a}\right ) d+d x^3-x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}+\frac {\left (3 (-a+5 b) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (a-b+x^3\right )^{2/3} \left (a \left (1-\frac {b}{a}\right ) d+d x^3-x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (12 (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^9}{\left (a-b+x^3\right )^{2/3} \left (-a \left (1-\frac {b}{a}\right ) d-d x^3+x^{15}\right )}+\frac {x^{12}}{\left (a-b+x^3\right )^{2/3} \left (-a \left (1-\frac {b}{a}\right ) d-d x^3+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}+\frac {\left (3 (-a+5 b) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (a-b+x^3\right )^{2/3} \left (a \left (1-\frac {b}{a}\right ) d+d x^3-x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (12 (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\left (a-b+x^3\right )^{2/3} \left (-a \left (1-\frac {b}{a}\right ) d-d x^3+x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}+\frac {\left (12 a (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (a-b+x^3\right )^{2/3} \left (-a \left (1-\frac {b}{a}\right ) d-d x^3+x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}+\frac {\left (3 (-a+5 b) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (a-b+x^3\right )^{2/3} \left (a \left (1-\frac {b}{a}\right ) d+d x^3-x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{((-a+x) (-b+x))^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 1.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-5 b+4 x) \left (-a^3+3 a^2 x-3 a x^2+x^3\right )}{((-a+x) (-b+x))^{2/3} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x))^(2/3)*(-a^5 + b*d - (-5*a^4
+ d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)),x]

[Out]

Integrate[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x))^(2/3)*(-a^5 + b*d - (-5*a^4
+ d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)), x]

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IntegrateAlgebraic [A]  time = 5.27, size = 258, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a^2-2 \sqrt {3} a x+\sqrt {3} x^2}{a^2-2 a x+x^2+2 \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^3-2 a^2 x+a x^2-a \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^6-4 a^5 x+6 a^4 x^2-4 a^3 x^3+a^2 x^4+a^2 d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2\right )\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x))^(2/3)*(-a^5 + b*d -
(-5*a^4 + d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*a^2 - 2*Sqrt[3]*a*x + Sqrt[3]*x^2)/(a^2 - 2*a*x + x^2 + 2*d^(1/3)*(a*b + (-a - b)*x +
x^2)^(1/3))])/d^(1/3) + Log[a^3 - 2*a^2*x + a*x^2 - a*d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3)]/d^(1/3) - Log[a
^6 - 4*a^5*x + 6*a^4*x^2 - 4*a^3*x^3 + a^2*x^4 + a^2*d^(2/3)*(a*b + (-a - b)*x + x^2)^(2/3) + (a*b + (-a - b)*
x + x^2)^(1/3)*(a^4*d^(1/3) - 2*a^3*d^(1/3)*x + a^2*d^(1/3)*x^2)]/(2*d^(1/3))

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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-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(-a^5+b*d-(-5*a^4+d)*x-10*a^3*x^2+10*a^
2*x^3-5*a*x^4+x^5),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 (a - 5 \, b + 4 \, x\right )}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(-a^5+b*d-(-5*a^4+d)*x-10*a^3*x^2+10*a^
2*x^3-5*a*x^4+x^5),x, algorithm="giac")

[Out]

integrate((a^3 - 3*a^2*x + 3*a*x^2 - x^3)*(a - 5*b + 4*x)/((a^5 + 10*a^3*x^2 - 10*a^2*x^3 + 5*a*x^4 - x^5 - b*
d - (5*a^4 - d)*x)*((a - x)*(b - x))^(2/3)), x)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a -5 b +4 x \right ) \left (-a^{3}+3 a^{2} x -3 a \,x^{2}+x^{3}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (-a^{5}+b d -\left (-5 a^{4}+d \right ) x -10 a^{3} x^{2}+10 a^{2} x^{3}-5 a \,x^{4}+x^{5}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(-a^5+b*d-(-5*a^4+d)*x-10*a^3*x^2+10*a^2*x^3-
5*a*x^4+x^5),x)

[Out]

int((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(-a^5+b*d-(-5*a^4+d)*x-10*a^3*x^2+10*a^2*x^3-
5*a*x^4+x^5),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 (a - 5 \, b + 4 \, x\right )}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-5*b+4*x)*(-a^3+3*a^2*x-3*a*x^2+x^3)/((-a+x)*(-b+x))^(2/3)/(-a^5+b*d-(-5*a^4+d)*x-10*a^3*x^2+10*a^
2*x^3-5*a*x^4+x^5),x, algorithm="maxima")

[Out]

integrate((a^3 - 3*a^2*x + 3*a*x^2 - x^3)*(a - 5*b + 4*x)/((a^5 + 10*a^3*x^2 - 10*a^2*x^3 + 5*a*x^4 - x^5 - b*
d - (5*a^4 - d)*x)*((a - x)*(b - x))^(2/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int -\frac {\left (a-5\,b+4\,x\right )\,\left (a^3-3\,a^2\,x+3\,a\,x^2-x^3\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (5\,a\,x^4-b\,d+x\,\left (d-5\,a^4\right )+a^5-x^5-10\,a^2\,x^3+10\,a^3\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a - 5*b + 4*x)*(3*a*x^2 - 3*a^2*x + a^3 - x^3))/(((a - x)*(b - x))^(2/3)*(5*a*x^4 - b*d + x*(d - 5*a^4)
+ a^5 - x^5 - 10*a^2*x^3 + 10*a^3*x^2)),x)

[Out]

-int(-((a - 5*b + 4*x)*(3*a*x^2 - 3*a^2*x + a^3 - x^3))/(((a - x)*(b - x))^(2/3)*(5*a*x^4 - b*d + x*(d - 5*a^4
) + a^5 - x^5 - 10*a^2*x^3 + 10*a^3*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- a + x\right )^{3} \left (a - 5 b + 4 x\right )}{\left (\left (- a + x\right ) \left (- b + x\right )\right )^{\frac {2}{3}} \left (- a^{5} + 5 a^{4} x - 10 a^{3} x^{2} + 10 a^{2} x^{3} - 5 a x^{4} + b d - d x + x^{5}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-5*b+4*x)*(-a**3+3*a**2*x-3*a*x**2+x**3)/((-a+x)*(-b+x))**(2/3)/(-a**5+b*d-(-5*a**4+d)*x-10*a**3*x
**2+10*a**2*x**3-5*a*x**4+x**5),x)

[Out]

Integral((-a + x)**3*(a - 5*b + 4*x)/(((-a + x)*(-b + x))**(2/3)*(-a**5 + 5*a**4*x - 10*a**3*x**2 + 10*a**2*x*
*3 - 5*a*x**4 + b*d - d*x + x**5)), x)

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