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

Optimal. Leaf size=76 \[ \frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt [4]{d}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt [4]{d}} \]

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Rubi [F]  time = 21.99, 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 x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {a b x-x^3}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx &=\int \frac {x \left (a b-x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {x} \left (a b-x^2\right )}{\sqrt {-a+x} \sqrt {-b+x} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+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 \left (a b-x^4\right )}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a^2 b^2 d-2 a b (a+b) d x^2+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^4-2 (a+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 {x^6}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^2+\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4+2 a \left (1+\frac {b}{a}\right ) d x^6-d x^8\right )}+\frac {a b x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^2-\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4-2 a \left (1+\frac {b}{a}\right ) 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 {-a+x^2} \sqrt {-b+x^2} \left (-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^2+\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4+2 a \left (1+\frac {b}{a}\right ) 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 {-a+x^2} \sqrt {-b+x^2} \left (a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^2-\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4-2 a \left (1+\frac {b}{a}\right ) 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 = 14.42, size = 32987, normalized size = 434.04 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.40, size = 76, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.95, size = 434, normalized size = 5.71 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{{\left (a b - {\left (a + b\right )} x + x^{2}\right )} d^{\frac {1}{4}}}\right )}{d^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {1}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {1}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} - \frac {\log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d^{\frac {1}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {1}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right )}{4 \, d^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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maple [C]  time = 0.09, size = 304, normalized size = 4.00

method	result	size
default	\(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b}\)	\(304\)
elliptic	\(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b}\)	\(304\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*sum(_alpha*(_alpha^2-a*b)/(-2*_alpha^3*d+3*_alpha^2*a*d+3*_alpha^2*b*d-_alpha*a^2*d-4*_alpha*a*b*d-_alpha
*b^2*d+a^2*b*d+a*b^2*d+_alpha)*(-_alpha^3*d+2*_alpha^2*a*d+_alpha^2*b*d-_alpha*a^2*d-2*_alpha*a*b*d+a^2*b*d+_a
lpha+b)*(-(-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)*EllipticPi((-(-b+x)/b)
^(1/2),(-_alpha^3*d+2*_alpha^2*a*d+_alpha^2*b*d-_alpha*a^2*d-2*_alpha*a*b*d+a^2*b*d+_alpha+b)/b,(b/(-a+b))^(1/
2)),_alpha=RootOf(d*_Z^4+(-2*a*d-2*b*d)*_Z^3+(a^2*d+4*a*b*d+b^2*d-1)*_Z^2+(-2*a^2*b*d-2*a*b^2*d)*_Z+a^2*b^2*d)
)

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

[Out]

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

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mupad [B]  time = 7.78, size = 175, normalized size = 2.30 \begin {gather*} \frac {\ln \left (\frac {x+2\,d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}{x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x}\right )}{2\,d^{1/4}}+\frac {\ln \left (\frac {x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x-d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,2{}\mathrm {i}}{x+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}\right )\,1{}\mathrm {i}}{2\,d^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x + 2*d^(1/4)*(x*(a - x)*(b - x))^(1/2) + d^(1/2)*x^2 + a*b*d^(1/2) - a*d^(1/2)*x - b*d^(1/2)*x)/(x - d^(
1/2)*x^2 - a*b*d^(1/2) + a*d^(1/2)*x + b*d^(1/2)*x))/(2*d^(1/4)) + (log((x - d^(1/4)*(x*(a - x)*(b - x))^(1/2)
*2i - d^(1/2)*x^2 - a*b*d^(1/2) + a*d^(1/2)*x + b*d^(1/2)*x)/(x + d^(1/2)*x^2 + a*b*d^(1/2) - a*d^(1/2)*x - b*
d^(1/2)*x))*1i)/(2*d^(1/4))

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

[Out]

Timed out

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