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

Optimal. Leaf size=153 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{b-x}\right )}{d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a b^3 x+x^3 \left (3 a b+3 b^2\right )+x^2 \left (-3 a b^2-b^3\right )+x^4 (-a-3 b)+x^5}}{b-x}\right )}{d^{3/4}} \]

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Rubi [F]  time = 7.49, 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 b x+x^2}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (b-(1+a d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*x*(1 - x/a)^(1/4)*(1 - x/b)^(3/4)*AppellF1[3/4, 1/4, 3/4, 7/4, x/a, x/b])/(3*d*((a - x)*(b - x)^3*x)^(1/4))
+ ((1 + a*d - 2*b*d + Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2])*x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*Defer[Int][1/(
x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*(-1 - a*d - Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2] + 2*d*x)), x])/(d*((a - x)
*(b - x)^3*x)^(1/4)) + ((1 + a*d - 2*b*d - Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2])*x^(1/4)*(-a + x)^(1/4)*(-b + x)^
(3/4)*Defer[Int][1/(x^(1/4)*(-a + x)^(1/4)*(-b + x)^(3/4)*(-1 - a*d + Sqrt[1 + 2*a*d - 4*b*d + a^2*d^2] + 2*d*
x)), x])/(d*((a - x)*(b - x)^3*x)^(1/4))

Rubi steps

\begin {align*} \int \frac {a b-2 b x+x^2}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (b-(1+a d) x+d x^2\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {a b-2 b x+x^2}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (b-(1+a d) x+d x^2\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 \left (\frac {1}{d \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}}-\frac {b-a b d-(1+a d-2 b d) x}{d \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (b+(-1-a d) x+d x^2\right )}\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 {1}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}} \, dx}{d \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-a b d-(1+a d-2 b d) x}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (b+(-1-a d) x+d x^2\right )} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=-\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \left (\frac {-1-a d+2 b d-\sqrt {1+2 a d-4 b d+a^2 d^2}}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (-1-a d-\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )}+\frac {-1-a d+2 b d+\sqrt {1+2 a d-4 b d+a^2 d^2}}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (-1-a d+\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )}\right ) \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}+\frac {\left (\sqrt [4]{x} (-b+x)^{3/4} \sqrt [4]{1-\frac {x}{a}}\right ) \int \frac {1}{\sqrt [4]{x} (-b+x)^{3/4} \sqrt [4]{1-\frac {x}{a}}} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=-\frac {\left (\left (-1-a d+2 b d-\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (-1-a d-\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}-\frac {\left (\left (-1-a d+2 b d+\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (-1-a d+\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{3/4}} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}\\ &=\frac {4 x \sqrt [4]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{3/4} F_1\left (\frac {3}{4};\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x}{a},\frac {x}{b}\right )}{3 d \sqrt [4]{(a-x) (b-x)^3 x}}-\frac {\left (\left (-1-a d+2 b d-\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (-1-a d-\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}-\frac {\left (\left (-1-a d+2 b d+\sqrt {1+2 a d-4 b d+a^2 d^2}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} (-b+x)^{3/4} \left (-1-a d+\sqrt {1+2 a d-4 b d+a^2 d^2}+2 d x\right )} \, dx}{d \sqrt [4]{x (-a+x) (-b+x)^3}}\\ \end {align*}

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Mathematica [F]  time = 6.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b-2 b x+x^2}{\sqrt [4]{x (-a+x) (-b+x)^3} \left (b-(1+a d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.40, size = 153, 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}}{b-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}}{b-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {a b -2 b x +x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )^{3}\right )^{\frac {1}{4}} \left (b -\left (a d +1\right ) x +d \,x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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