3.17.82 \(\int \frac {-((2 a-3 b) b)+2 (a-2 b) x+x^2}{\sqrt [4]{(-a+x) (-b+x)^2} (-a^3-b^2 d+(3 a^2+2 b d) x-(3 a+d) x^2+x^3)} \, dx\) [1682]

Optimal. Leaf size=113 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{a-x}\right )}{d^{3/4}} \]

[Out]

-2*arctan(d^(1/4)*(-a*b^2+(2*a*b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/4)/(a-x))/d^(3/4)+2*arctanh(d^(1/4)*(-a*b^2+(2*a*
b+b^2)*x+(-a-2*b)*x^2+x^3)^(1/4)/(a-x))/d^(3/4)

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Rubi [F]
time = 4.95, 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 {-((2 a-3 b) b)+2 (a-2 b) x+x^2}{\sqrt [4]{(-a+x) (-b+x)^2} \left (-a^3-b^2 d+\left (3 a^2+2 b d\right ) x-(3 a+d) x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.87, size = 151, normalized size = 1.34 \begin {gather*} \frac {\sqrt {2} \sqrt [4]{a-x} \sqrt {b-x} \left (\text {ArcTan}\left (\frac {(a-x)^{3/2}+\sqrt {d} (-b+x)}{\sqrt {2} \sqrt [4]{d} (a-x)^{3/4} \sqrt {b-x}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} (a-x)^{3/4} \sqrt {b-x}}{(a-x)^{3/2}+\sqrt {d} (b-x)}\right )\right )}{d^{3/4} \sqrt [4]{(b-x)^2 (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {-\left (2 a -3 b \right ) b +2 \left (a -2 b \right ) x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-a^{3}-b^{2} d +\left (3 a^{2}+2 b d \right ) x -\left (3 a +d \right ) x^{2}+x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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((-(2*a-3*b)*b+2*(a-2*b)*x+x^2)/((-a+x)*(-b+x)^2)^(1/4)/(-a^3-b^2*d+(3*a^2+2*b*d)*x-(3*a+d)*x^2+x^3),
x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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((-(2*a-3*b)*b+2*(a-2*b)*x+x**2)/((-a+x)*(-b+x)**2)**(1/4)/(-a**3-b**2*d+(3*a**2+2*b*d)*x-(3*a+d)*x**
2+x**3),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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