3.25.97 \(\int \frac {x^7 (-4 a+3 x)}{(x^2 (-a+x))^{2/3} (-a^2+2 a x-x^2+d x^8)} \, dx\)

Optimal. Leaf size=207 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \sqrt [3]{x^3-a x^2}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{2 \sqrt [3]{x^3-a x^2}+\sqrt [6]{d} x^2}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \left (x^3-a x^2\right )^{2/3}}{a-x}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\frac {\left (x^3-a x^2\right )^{2/3}}{\sqrt [6]{d}}+\sqrt [6]{d} x^4}{x^2 \sqrt [3]{x^3-a x^2}}\right )}{2 d^{5/6}} \]

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Rubi [F]  time = 3.38, 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^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

(12*a*x^(4/3)*(-a + x)^(2/3)*Defer[Subst][Defer[Int][x^19/((-a + x^3)^(2/3)*(a^2 - 2*a*x^3 + x^6 - d*x^24)), x
], x, x^(1/3)])/(-((a - x)*x^2))^(2/3) + (9*x^(4/3)*(-a + x)^(2/3)*Defer[Subst][Defer[Int][x^22/((-a + x^3)^(2
/3)*(-a^2 + 2*a*x^3 - x^6 + d*x^24)), x], x, x^(1/3)])/(-((a - x)*x^2))^(2/3)

Rubi steps

\begin {align*} \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx &=\frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \frac {x^{17/3} (-4 a+3 x)}{(-a+x)^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx}{\left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{19} \left (-4 a+3 x^3\right )}{\left (-a+x^3\right )^{2/3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\left (3 x^{4/3} (-a+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {4 a x^{19}}{\left (-a+x^3\right )^{2/3} \left (a^2-2 a x^3+x^6-d x^{24}\right )}+\frac {3 x^{22}}{\left (-a+x^3\right )^{2/3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\left (9 x^{4/3} (-a+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{22}}{\left (-a+x^3\right )^{2/3} \left (-a^2+2 a x^3-x^6+d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}}+\frac {\left (12 a x^{4/3} (-a+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{19}}{\left (-a+x^3\right )^{2/3} \left (a^2-2 a x^3+x^6-d x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 1.01, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x-x^2+d x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

Integrate[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)), x]

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IntegrateAlgebraic [A]  time = 0.84, size = 207, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2+2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{a-x}\right )}{d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} x^4+\frac {\left (-a x^2+x^3\right )^{2/3}}{\sqrt [6]{d}}}{x^2 \sqrt [3]{-a x^2+x^3}}\right )}{2 d^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]

[Out]

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

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fricas [B]  time = 0.52, size = 431, normalized size = 2.08 \begin {gather*} -\sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} d^{4} \frac {1}{d^{5}}^{\frac {5}{6}} x^{2} \sqrt {\frac {d^{2} \frac {1}{d^{5}}^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{4}}} + 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{4} \frac {1}{d^{5}}^{\frac {5}{6}} + \sqrt {3} x^{2}}{3 \, x^{2}}\right ) - \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} d^{4} \frac {1}{d^{5}}^{\frac {5}{6}} x^{2} \sqrt {\frac {d^{2} \frac {1}{d^{5}}^{\frac {1}{3}} x^{4} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{4}}} + 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{4} \frac {1}{d^{5}}^{\frac {5}{6}} - \sqrt {3} x^{2}}{3 \, x^{2}}\right ) - \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (-\frac {d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {d^{2} \frac {1}{d^{5}}^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{4}}\right ) + \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left (\frac {d^{2} \frac {1}{d^{5}}^{\frac {1}{3}} x^{4} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d \frac {1}{d^{5}}^{\frac {1}{6}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="fricas")

[Out]

-sqrt(3)*(d^(-5))^(1/6)*arctan(1/3*(2*sqrt(3)*d^4*(d^(-5))^(5/6)*x^2*sqrt((d^2*(d^(-5))^(1/3)*x^4 + (-a*x^2 +
x^3)^(1/3)*d*(d^(-5))^(1/6)*x^2 + (-a*x^2 + x^3)^(2/3))/x^4) + 2*sqrt(3)*(-a*x^2 + x^3)^(1/3)*d^4*(d^(-5))^(5/
6) + sqrt(3)*x^2)/x^2) - sqrt(3)*(d^(-5))^(1/6)*arctan(1/3*(2*sqrt(3)*d^4*(d^(-5))^(5/6)*x^2*sqrt((d^2*(d^(-5)
)^(1/3)*x^4 - (-a*x^2 + x^3)^(1/3)*d*(d^(-5))^(1/6)*x^2 + (-a*x^2 + x^3)^(2/3))/x^4) + 2*sqrt(3)*(-a*x^2 + x^3
)^(1/3)*d^4*(d^(-5))^(5/6) - sqrt(3)*x^2)/x^2) - 1/2*(d^(-5))^(1/6)*log(-(d*(d^(-5))^(1/6)*x^2 + (-a*x^2 + x^3
)^(1/3))/x^2) + 1/2*(d^(-5))^(1/6)*log((d*(d^(-5))^(1/6)*x^2 - (-a*x^2 + x^3)^(1/3))/x^2) - 1/4*(d^(-5))^(1/6)
*log((d^2*(d^(-5))^(1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d*(d^(-5))^(1/6)*x^2 + (-a*x^2 + x^3)^(2/3))/x^4) + 1/4*(d
^(-5))^(1/6)*log((d^2*(d^(-5))^(1/3)*x^4 - (-a*x^2 + x^3)^(1/3)*d*(d^(-5))^(1/6)*x^2 + (-a*x^2 + x^3)^(2/3))/x
^4)

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giac [B]  time = 0.96, size = 399, normalized size = 1.93 \begin {gather*} \frac {1}{2} \, \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{{\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) + \frac {1}{2} \, \sqrt {3} \frac {1}{d^{5}}^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{{\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) + \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) - \frac {1}{4} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left (\sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - \sqrt {3} a d^{\frac {1}{6}} - \sqrt {3} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) + \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{2} \, \frac {1}{d^{5}}^{\frac {1}{6}} \log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {1}{6}} - {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="giac")

[Out]

1/2*sqrt(3)*(d^(-5))^(1/6)*arctan((sqrt(3)*(-a/x + 1)^(4/3) - sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))/((
-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))) + 1/2*sqrt(3)*(d^(-5))^(1/6)*arctan(-(sqrt(3)*(-a/x + 1)^(4/3
) + sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))/((-a/x + 1)^(4/3) - a*d^(1/6) - (-a/x + 1)^(1/3))) + 1/4*(d^
(-5))^(1/6)*log((sqrt(3)*(-a/x + 1)^(4/3) + sqrt(3)*a*d^(1/6) - sqrt(3)*(-a/x + 1)^(1/3))^2 + ((-a/x + 1)^(4/3
) - a*d^(1/6) - (-a/x + 1)^(1/3))^2) - 1/4*(d^(-5))^(1/6)*log((sqrt(3)*(-a/x + 1)^(4/3) - sqrt(3)*a*d^(1/6) -
sqrt(3)*(-a/x + 1)^(1/3))^2 + ((-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))^2) + 1/2*(d^(-5))^(1/6)*log(ab
s((-a/x + 1)^(4/3) + a*d^(1/6) - (-a/x + 1)^(1/3))) - 1/2*(d^(-5))^(1/6)*log(abs((-a/x + 1)^(4/3) - a*d^(1/6)
- (-a/x + 1)^(1/3)))

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{7} \left (-4 a +3 x \right )}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (d \,x^{8}-a^{2}+2 a x -x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x)

[Out]

int(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (4 \, a - 3 \, x\right )} x^{7}}{{\left (d x^{8} - a^{2} + 2 \, a x - x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(d*x^8-a^2+2*a*x-x^2),x, algorithm="maxima")

[Out]

-integrate((4*a - 3*x)*x^7/((d*x^8 - a^2 + 2*a*x - x^2)*(-(a - x)*x^2)^(2/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x^7\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,a\,x+d\,x^8-x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^7*(4*a - 3*x))/((-x^2*(a - x))^(2/3)*(2*a*x + d*x^8 - a^2 - x^2)),x)

[Out]

int(-(x^7*(4*a - 3*x))/((-x^2*(a - x))^(2/3)*(2*a*x + d*x^8 - a^2 - x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} \left (- 4 a + 3 x\right )}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a^{2} + 2 a x + d x^{8} - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(-4*a+3*x)/(x**2*(-a+x))**(2/3)/(d*x**8-a**2+2*a*x-x**2),x)

[Out]

Integral(x**7*(-4*a + 3*x)/((x**2*(-a + x))**(2/3)*(-a**2 + 2*a*x + d*x**8 - x**2)), x)

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