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

Optimal. Leaf size=151 \[ -\frac {\log \left (a^2 d^{2/3} \left (x^3-a x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2 \sqrt [3]{x^3-a x^2}+a^2 x^4\right )}{2 \sqrt [3]{d}}+\frac {\log \left (a x^2-a \sqrt [3]{d} \sqrt [3]{x^3-a x^2}\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{d} \sqrt [3]{x^3-a x^2}+x^2}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 2.33, 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 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-12*a*x^(4/3)*(-a + x)^(2/3)*Defer[Subst][Defer[Int][x^7/((-a + x^3)^(2/3)*(a*d - d*x^3 + x^12)), x], x, x^(1
/3)])/(-((a - x)*x^2))^(2/3) + (9*x^(4/3)*(-a + x)^(2/3)*Defer[Subst][Defer[Int][x^10/((-a + x^3)^(2/3)*(a*d -
d*x^3 + x^12)), x], x, x^(1/3)])/(-((a - x)*x^2))^(2/3)

Rubi steps

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

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.53, size = 151, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a x^2-a \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 x^4+a^2 \sqrt [3]{d} x^2 \sqrt [3]{-a x^2+x^3}+a^2 d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 338, normalized size = 2.24 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {x^{4} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x^{2} - 2 \, a d + 2 \, d x + \sqrt {3} {\left (d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}}}{x^{4} + a d - d x}\right ) + 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(3)*d*sqrt(-1/d^(2/3))*log(-(x^4 - 3*(-a*x^2 + x^3)^(1/3)*d^(1/3)*x^2 - 2*a*d + 2*d*x + sqrt(3)*(d^(
1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d^(2/3)*x^2 - 2*(-a*x^2 + x^3)^(2/3)*d)*sqrt(-1/d^(2/3)))/(x^4 + a*d - d*x)) +
2*d^(2/3)*log((d^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*d)/x^2) - d^(2/3)*log((d^(1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d
^(2/3)*x^2 + (-a*x^2 + x^3)^(2/3)*d)/x^4))/d, -1/2*(2*sqrt(3)*d^(2/3)*arctan(1/3*sqrt(3)*(d^(1/3)*x^2 + 2*(-a*
x^2 + x^3)^(1/3)*d^(2/3))/(d^(1/3)*x^2)) - 2*d^(2/3)*log((d^(2/3)*x^2 - (-a*x^2 + x^3)^(1/3)*d)/x^2) + d^(2/3)
*log((d^(1/3)*x^4 + (-a*x^2 + x^3)^(1/3)*d^(2/3)*x^2 + (-a*x^2 + x^3)^(2/3)*d)/x^4))/d]

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giac [A]  time = 0.27, size = 235, normalized size = 1.56 \begin {gather*} -\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a {\left | d \right |}^{\frac {2}{3}} - \sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a {\left | d \right |}^{\frac {2}{3}} - d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) - \frac {1}{2} \, \left (-\frac {1}{d}\right )^{\frac {1}{3}} \log \left (\frac {3}{4} \, a^{2} d^{\frac {4}{3}} + \frac {1}{4} \, {\left (2 \, d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {2}{3}} - 2 \, d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) + \frac {1}{2} \, \left (-\frac {1}{d}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a {\left | d \right |}^{\frac {2}{3}} - \sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left (d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a {\left | d \right |}^{\frac {2}{3}} - d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (-4 a +3 x \right )}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (x^{4}+a d -d x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(x^4+a*d-d*x),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^{3}}{{\left (x^{4} + a d - d x\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^3*(-4*a+3*x)/(x^2*(-a+x))^(2/3)/(x^4+a*d-d*x),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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