3.17.41 \(\int \frac {3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3)} \, dx\) [1641]

Optimal. Leaf size=111 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )}{d^{3/4}} \]

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

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Rubi [F]
time = 11.61, 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^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(12*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*
b^2*d - 2*a*b*(1 + b/(2*a))*d*x^4 + a*(1 + (2*b)/a)*d*x^8 + (1 - d)*x^12)), x], x, x^(1/4)])/(-((a - x)*(b - x
)^2*x))^(1/4) - (4*(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*b^2*d - 2*a*b*(1 + b/(2*a))*d*x^4 + a*(1 + (2*b)/a)*d*x^8 + (1 - d)*x^12)), x], x, x^(1/4
)])/(-((a - x)*(b - x)^2*x))^(1/4)

Rubi steps

\begin {align*} \int \frac {3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(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 {3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (-a b^2 d+b (2 a+b) d x-(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} (-3 a b+(a+2 b) x)}{\sqrt [4]{x} \sqrt [4]{-a+x} \left (-a b^2 d+b (2 a+b) d x-(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 ) \text {Subst}\left (\int \frac {x^2 \sqrt {-b+x^4} \left (-3 a b+(a+2 b) x^4\right )}{\sqrt [4]{-a+x^4} \left (-a b^2 d+b (2 a+b) d x^4-(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 ) \text {Subst}\left (\int \left (\frac {3 a b x^2 \sqrt {-b+x^4}}{\sqrt [4]{-a+x^4} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )}+\frac {(-a-2 b) x^6 \sqrt {-b+x^4}}{\sqrt [4]{-a+x^4} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d 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 (-a-2 b) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt {-b+x^4}}{\sqrt [4]{-a+x^4} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d x^8+(1-d) x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (12 a b \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-b+x^4}}{\sqrt [4]{-a+x^4} \left (a b^2 d-2 a b \left (1+\frac {b}{2 a}\right ) d x^4+a \left (1+\frac {2 b}{a}\right ) d 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 [A]
time = 38.48, size = 170, normalized size = 1.53 \begin {gather*} \frac {2 \sqrt [4]{-1+\frac {a}{x}} (b-x) \left (\text {ArcTan}\left (\frac {-1+\sqrt {d} \sqrt {-1+\frac {a}{x}} \left (-1+\frac {b}{x}\right )}{\sqrt [4]{d} \sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}}}{1+\sqrt {d} \sqrt {-1+\frac {a}{x}} \left (-1+\frac {b}{x}\right )}\right )\right )}{d^{3/4} \sqrt {-2+\frac {2 b}{x}} \sqrt [4]{(b-x)^2 x (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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