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

Optimal. Leaf size=88 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{d^{3/4}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{d^{3/4}} \]

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Rubi [F]  time = 16.93, 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 {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 15.22, size = 31196, normalized size = 354.50 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.99, size = 88, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.28, size = 384, normalized size = 4.36 \begin {gather*} -\frac {1}{d^{3}}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} d^{2} \frac {1}{d^{3}}^{\frac {3}{4}}}{b x - x^{2}}\right ) - \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (b d^{3} x - d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (a d - d x\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (b d^{3} x - d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (a d - d x\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2+2*a*x+(b^2*d-1)*x^2-2*b*d*x^3+d*x^4),x, algor
ithm="fricas")

[Out]

-(d^(-3))^(1/4)*arctan(-sqrt(a*b*x - (a + b)*x^2 + x^3)*d^2*(d^(-3))^(3/4)/(b*x - x^2)) - 1/4*(d^(-3))^(1/4)*l
og((2*b*d*x^3 - d*x^4 - (b^2*d + 1)*x^2 - a^2 + 2*a*x + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((b*d^3*x - d^3*x^2)
*(d^(-3))^(3/4) + (a*d - d*x)*(d^(-3))^(1/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(2*b*d
*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*a*x)) + 1/4*(d^(-3))^(1/4)*log((2*b*d*x^3 - d*x^4 - (b^2*d + 1)*x^2 -
a^2 + 2*a*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((b*d^3*x - d^3*x^2)*(d^(-3))^(3/4) + (a*d - d*x)*(d^(-3))^(1
/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(2*b*d*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*
a*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.11, size = 358, normalized size = 4.07

method	result	size
default	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-2 b d \,\textit {\_Z}^{3}+\left (b^{2} d -1\right ) \textit {\_Z}^{2}+2 a \textit {\_Z} -a^{2}\right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3} a d +\underline {\hspace {1.25 ex}}\alpha ^{3} b d +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d -\underline {\hspace {1.25 ex}}\alpha a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha a +a^{2}\right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha -a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\)	\(358\)
elliptic	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-2 b d \,\textit {\_Z}^{3}+\left (b^{2} d -1\right ) \textit {\_Z}^{2}+2 a \textit {\_Z} -a^{2}\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a d -\underline {\hspace {1.25 ex}}\alpha ^{3} b d -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d +\underline {\hspace {1.25 ex}}\alpha a \,b^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha a -a^{2}\right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-3 b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\)	\(360\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*b*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)*EllipticF((-(-b+x)/b)
^(1/2),(b/(-a+b))^(1/2))-1/d*b*sum((-2*_alpha^3*a*d+_alpha^3*b*d+3*_alpha^2*a*b*d-_alpha^2*b^2*d-_alpha*a*b^2*
d+_alpha^2-2*_alpha*a+a^2)/(-2*_alpha^3*d+3*_alpha^2*b*d-_alpha*b^2*d+_alpha-a)*(_alpha^3*d-_alpha^2*b*d-_alph
a+2*a-b)/(a^2-2*a*b+b^2)*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(x*(a*b-a*x-b*x+x^2))^(1/2)*Ellip
ticPi((-(-b+x)/b)^(1/2),-(_alpha^3*d-_alpha^2*b*d-_alpha+2*a-b)*b/(a^2-2*a*b+b^2),(b/(-a+b))^(1/2)),_alpha=Roo
tOf(d*_Z^4-2*b*d*_Z^3+(b^2*d-1)*_Z^2+2*a*_Z-a^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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