3.28.100 \(\int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} (b-a^5 d-(1-5 a^4 d) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5)} \, dx\)

Optimal. Leaf size=272 \[ \frac {\log \left (a^2 \sqrt {d}-\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a \sqrt {d} x+\sqrt {d} x^2\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x (-a-b)+a b+x^2}}{2 a^2 \sqrt [3]{d}+\sqrt [3]{x (-a-b)+a b+x^2}-4 a \sqrt [3]{d} x+2 \sqrt [3]{d} x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^4 d-4 a^3 d x+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )+6 a^2 d x^2+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-4 a d x^3+d x^4\right )}{2 \sqrt [3]{d}} \]

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Rubi [F]  time = 7.48, 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 (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 10*a^3*
d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)),x]

[Out]

(-3*(a - 5*b)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^4/((a - b + x^3)^(1/3)*(a*(1 - b/a) + x^
3 - d*x^15)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(1/3) + (12*a*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subs
t][Defer[Int][x^4/((a - b + x^3)^(1/3)*(-(a*(1 - b/a)) - x^3 + d*x^15)), x], x, (-a + x)^(1/3)])/((a - x)*(b -
x))^(1/3) + (12*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^7/((a - b + x^3)^(1/3)*(-(a*(1 - b/a)
) - x^3 + d*x^15)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(1/3)

Rubi steps

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

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Mathematica [F]  time = 4.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 1
0*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)),x]

[Out]

Integrate[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 1
0*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)), x]

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IntegrateAlgebraic [A]  time = 0.71, size = 272, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a^2 \sqrt [3]{d}-4 a \sqrt [3]{d} x+2 \sqrt [3]{d} x^2+\sqrt [3]{a b+(-a-b) x+x^2}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^2 \sqrt {d}-2 a \sqrt {d} x+\sqrt {d} x^2-\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^4 d-4 a^3 d x+6 a^2 d x^2-4 a d x^3+d x^4+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(b - a^5*d - (1 - 5*a^4
*d)*x - 10*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b + (-a - b)*x + x^2)^(1/3))/(2*a^2*d^(1/3) - 4*a*d^(1/3)*x + 2*d^(1/3)*x^2 + (a*b
+ (-a - b)*x + x^2)^(1/3))])/d^(1/3) + Log[a^2*Sqrt[d] - 2*a*Sqrt[d]*x + Sqrt[d]*x^2 - d^(1/6)*(a*b + (-a - b
)*x + x^2)^(1/3)]/d^(1/3) - Log[a^4*d - 4*a^3*d*x + 6*a^2*d*x^2 - 4*a*d*x^3 + d*x^4 + d^(1/3)*(a*b + (-a - b)*
x + x^2)^(2/3) + (a*b + (-a - b)*x + x^2)^(1/3)*(a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*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*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(b-a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x
^3-5*a*d*x^4+d*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 - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(b-a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x
^3-5*a*d*x^4+d*x^5),x, algorithm="giac")

[Out]

integrate(((a - 5*b)*a + (3*a + 5*b)*x - 4*x^2)/((a^5*d + 10*a^3*d*x^2 - 10*a^2*d*x^3 + 5*a*d*x^4 - d*x^5 - (5
*a^4*d - 1)*x - b)*((a - x)*(b - x))^(1/3)), x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {-a \left (a -5 b \right )-\left (3 a +5 b \right ) x +4 x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (b -a^{5} d -\left (-5 a^{4} d +1\right ) x -10 a^{3} d \,x^{2}+10 a^{2} d \,x^{3}-5 a d \,x^{4}+d \,x^{5}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(b-a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a
*d*x^4+d*x^5),x)

[Out]

int((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(b-a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x^3-5*a
*d*x^4+d*x^5),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(b-a^5*d-(-5*a^4*d+1)*x-10*a^3*d*x^2+10*a^2*d*x
^3-5*a*d*x^4+d*x^5),x, algorithm="maxima")

[Out]

integrate(((a - 5*b)*a + (3*a + 5*b)*x - 4*x^2)/((a^5*d + 10*a^3*d*x^2 - 10*a^2*d*x^3 + 5*a*d*x^4 - d*x^5 - (5
*a^4*d - 1)*x - b)*((a - x)*(b - x))^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*(a - 5*b) + x*(3*a + 5*b) - 4*x^2)/(((a - x)*(b - x))^(1/3)*(b - a^5*d + d*x^5 + x*(5*a^4*d - 1) + 10*
a^2*d*x^3 - 10*a^3*d*x^2 - 5*a*d*x^4)),x)

[Out]

-int((a*(a - 5*b) + x*(3*a + 5*b) - 4*x^2)/(((a - x)*(b - x))^(1/3)*(b - a^5*d + d*x^5 + x*(5*a^4*d - 1) + 10*
a^2*d*x^3 - 10*a^3*d*x^2 - 5*a*d*x^4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x**2)/((-a+x)*(-b+x))**(1/3)/(b-a**5*d-(-5*a**4*d+1)*x-10*a**3*d*x**2+10*a
**2*d*x**3-5*a*d*x**4+d*x**5),x)

[Out]

Integral((-a + x)*(a - 5*b + 4*x)/(((-a + x)*(-b + x))**(1/3)*(-a**5*d + 5*a**4*d*x - 10*a**3*d*x**2 + 10*a**2
*d*x**3 - 5*a*d*x**4 + b + d*x**5 - x)), x)

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