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

Optimal. Leaf size=157 \[ -\frac {\log \left (a^2 d^{2/3} x^4+a^2 \sqrt [3]{d} x^2 \sqrt [3]{x^3-a x^2}+a^2 \left (x^3-a x^2\right )^{2/3}\right )}{2 d^{2/3}}+\frac {\log \left (a \sqrt [3]{x^3-a x^2}-a \sqrt [3]{d} x^2\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{2 \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} x^2}\right )}{d^{2/3}} \]

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Rubi [F]  time = 2.22, 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-x+d 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 - x + d*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 - x^3 + d*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 - x^3
+ d*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-x+d 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-x+d 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-x^3+d 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-x^3+d x^{12}\right )}+\frac {3 x^{10}}{\left (-a+x^3\right )^{2/3} \left (a-x^3+d 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-x^3+d 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-x^3+d 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 = 0.52, 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-x+d 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 - x + d*x^4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.57, size = 157, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \sqrt [3]{-a x^2+x^3}}\right )}{d^{2/3}}+\frac {\log \left (-a \sqrt [3]{d} x^2+a \sqrt [3]{-a x^2+x^3}\right )}{d^{2/3}}-\frac {\log \left (a^2 d^{2/3} x^4+a^2 \sqrt [3]{d} x^2 \sqrt [3]{-a x^2+x^3}+a^2 \left (-a x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 158, normalized size = 1.01 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} d x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d^{2} x^{2}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {1}{3}} d x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d^{2}} \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)/(d*x^4+a-x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.50, size = 1, normalized size = 0.01 \begin {gather*} 0 \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)/(d*x^4+a-x),x, algorithm="giac")

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

Timed out

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