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

Optimal. Leaf size=119 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{a-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{a-x}\right )}{d^{3/4}} \]

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Rubi [F]  time = 9.56, 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 {a b^2-2 (2 a-b) b x+(3 a-2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (a^3+\left (-3 a^2+b^2 d\right ) x+(3 a-2 b d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 4.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b^2-2 (2 a-b) b x+(3 a-2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (a^3+\left (-3 a^2+b^2 d\right ) x+(3 a-2 b d) x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.42, size = 119, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{a-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{a-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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