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

Optimal. Leaf size=87 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{\sqrt [4]{d}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{\sqrt [4]{d}} \]

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Rubi [F]  time = 9.95, 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^2 b+a (2 a+b) x-3 a x^2+x^3}{\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[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(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^2*Sqrt[-a + x^2])/(Sqrt[-b + 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][Sqrt[-a + x^2]/(Sqrt[-b + 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]*Defer[Su
bst][Defer[Int][(x^4*Sqrt[-a + x^2])/(Sqrt[-b + 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 {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\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 {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x} \sqrt {-a+x} \sqrt {-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}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \left (a b-2 a x+x^2\right )}{\sqrt {x} \sqrt {-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}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \left (a b-2 a x^2+x^4\right )}{\sqrt {-b+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^2 \sqrt {-a+x^2}}{\sqrt {-b+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 \sqrt {-a+x^2}}{\sqrt {-b+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^4 \sqrt {-a+x^2}}{\sqrt {-b+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^4 \sqrt {-a+x^2}}{\sqrt {-b+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^2 \sqrt {-a+x^2}}{\sqrt {-b+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 {\sqrt {-a+x^2}}{\sqrt {-b+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 = 8.08, size = 5341, normalized size = 61.39 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(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 = 1.22, size = 90, normalized size = 1.03 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt [4]{d} x (-b+x)}\right )}{\sqrt [4]{d}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{\sqrt [4]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 1.57, size = 340, normalized size = 3.91 \begin {gather*} \frac {\arctan \left (-\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{{\left (b x - x^{2}\right )} d^{\frac {1}{4}}}\right )}{d^{\frac {1}{4}}} - \frac {\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 (\frac {b d x - d x^{2}}{d^{\frac {1}{4}}} + \frac {a d - d x}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\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 (\frac {b d x - d x^{2}}{d^{\frac {1}{4}}} + \frac {a d - d x}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right )}{4 \, d^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/((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 = 246, normalized size = 2.83

method	result	size
default	\(b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-2 b d \,\textit {\_Z}^{3}+\left (b^{2} d -1\right ) \textit {\_Z}^{2}+2 a \textit {\_Z} -a^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +a^{2} b \right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha -a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\)	\(246\)
elliptic	\(b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-2 b d \,\textit {\_Z}^{3}+\left (b^{2} d -1\right ) \textit {\_Z}^{2}+2 a \textit {\_Z} -a^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +a^{2} b \right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha -a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\)	\(246\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/(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,meth
od=_RETURNVERBOSE)

[Out]

b*sum((-_alpha^3+3*_alpha^2*a-2*_alpha*a^2-_alpha*a*b+a^2*b)/(-2*_alpha^3*d+3*_alpha^2*b*d-_alpha*b^2*d+_alpha
-a)*(_alpha^3*d-_alpha^2*b*d-_alpha+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)*EllipticPi((-(-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=RootOf(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 {a^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\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((-a^2*b+a*(2*a+b)*x-3*a*x^2+x^3)/(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, algorithm="maxima")

[Out]

integrate((a^2*b - (2*a + b)*a*x + 3*a*x^2 - x^3)/((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 [B]  time = 7.32, size = 135, normalized size = 1.55 \begin {gather*} \frac {\ln \left (\frac {a-x+2\,d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-\sqrt {d}\,x^2+b\,\sqrt {d}\,x}{a-x+\sqrt {d}\,x^2-b\,\sqrt {d}\,x}\right )}{2\,d^{1/4}}+\frac {\ln \left (\frac {x-a-\sqrt {d}\,x^2+b\,\sqrt {d}\,x+d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,2{}\mathrm {i}}{a-x-\sqrt {d}\,x^2+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(-(a^2*b + 3*a*x^2 - x^3 - a*x*(2*a + b))/((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]

log((a - x + 2*d^(1/4)*(x*(a - x)*(b - x))^(1/2) - d^(1/2)*x^2 + b*d^(1/2)*x)/(a - x + d^(1/2)*x^2 - b*d^(1/2)
*x))/(2*d^(1/4)) + (log((x - a + d^(1/4)*(x*(a - x)*(b - x))^(1/2)*2i - d^(1/2)*x^2 + b*d^(1/2)*x)/(a - x - d^
(1/2)*x^2 + 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**2*b+a*(2*a+b)*x-3*a*x**2+x**3)/(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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