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

Optimal. Leaf size=145 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{x}\right )}{d^{3/4}} \]

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Rubi [F]  time = 37.23, 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 {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(12*a*b*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x^2*(-b + x^4)^(5/4))/((-a + x^4)^(1/4)
*(-(a*b^3*d) + 3*a*b^2*(1 + b/(3*a))*d*x^4 - 3*a*b*(1 + b/a)*d*x^8 + (1 + (a + 3*b)*d)*x^12 - d*x^16)), x], x,
x^(1/4)])/((a - x)*(b - x)^3*x)^(1/4) + (8*b*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*Defer[Subst][Defer[Int][(x
^6*(-b + x^4)^(5/4))/((-a + x^4)^(1/4)*(a*b^3*d - 3*a*b^2*(1 + b/(3*a))*d*x^4 + 3*a*b*(1 + b/a)*d*x^8 - (1 + (
a + 3*b)*d)*x^12 + d*x^16)), x], x, x^(1/4)])/((a - x)*(b - x)^3*x)^(1/4) + (4*x^(1/4)*(-a + x)^(1/4)*(-b + x)
^(3/4)*Defer[Subst][Defer[Int][(x^10*(-b + x^4)^(5/4))/((-a + x^4)^(1/4)*(a*b^3*d - 3*a*b^2*(1 + b/(3*a))*d*x^
4 + 3*a*b*(1 + b/a)*d*x^8 - (1 + (a + 3*b)*d)*x^12 + d*x^16)), x], x, x^(1/4)])/((a - x)*(b - x)^3*x)^(1/4)

Rubi steps

\begin {align*} \int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {\sqrt [4]{-b+x} \left (3 a b^2+\left (-3 a b-2 b^2\right ) x+b x^2+x^3\right )}{\sqrt [4]{x} \sqrt [4]{-a+x} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {(-b+x)^{5/4} \left (-3 a b+2 b x+x^2\right )}{\sqrt [4]{x} \sqrt [4]{-a+x} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-b+x^4\right )^{5/4} \left (-3 a b+2 b x^4+x^8\right )}{\sqrt [4]{-a+x^4} \left (a b^3 d-b^2 (3 a+b) d x^4+3 b (a+b) d x^8-(1+a d+3 b d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {3 a b x^2 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (-a b^3 d+3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4-3 a b \left (1+\frac {b}{a}\right ) d x^8+(1+(a+3 b) d) x^{12}-d x^{16}\right )}+\frac {2 b x^6 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )}+\frac {x^{10} \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{10} \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}+\frac {\left (8 b \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (a b^3 d-3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4+3 a b \left (1+\frac {b}{a}\right ) d x^8-(1+(a+3 b) d) x^{12}+d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}+\frac {\left (12 a b \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-b+x^4\right )^{5/4}}{\sqrt [4]{-a+x^4} \left (-a b^3 d+3 a b^2 \left (1+\frac {b}{3 a}\right ) d x^4-3 a b \left (1+\frac {b}{a}\right ) d x^8+(1+(a+3 b) d) x^{12}-d x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^3}}\\ \end {align*}

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Mathematica [F]  time = 2.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.49, size = 145, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}{x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+\left (-3 a b^2-b^3\right ) x^2+\left (3 a b+3 b^2\right ) x^3+(-a-3 b) x^4+x^5}}{x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((-3*a*b^3+2*b^2*(3*a+b)*x-3*b*(a+b)*x^2+x^4)/(x*(-a+x)*(-b+x)^3)^(1/4)/(a*b^3*d-b^2*(3*a+b)*d*x+3*b*
(a+b)*d*x^2-(a*d+3*b*d+1)*x^3+d*x^4),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 {3 \, a b^{3} - 2 \, {\left (3 \, a + b\right )} b^{2} x + 3 \, {\left (a + b\right )} b x^{2} - x^{4}}{{\left (a b^{3} d - {\left (3 \, a + b\right )} b^{2} d x + 3 \, {\left (a + b\right )} b d x^{2} + d x^{4} - {\left (a d + 3 \, b d + 1\right )} x^{3}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}^{3} x\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-3 a \,b^{3}+2 b^{2} \left (3 a +b \right ) x -3 b \left (a +b \right ) x^{2}+x^{4}}{\left (x \left (-a +x \right ) \left (-b +x \right )^{3}\right )^{\frac {1}{4}} \left (a \,b^{3} d -b^{2} \left (3 a +b \right ) d x +3 b \left (a +b \right ) d \,x^{2}-\left (a d +3 b d +1\right ) x^{3}+d \,x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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