3.1.3 \(\int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx\)

Optimal. Leaf size=239 \[ \frac {\sqrt {\sqrt {4 a c+b^2}-b} \tanh ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}-b} \left (\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}-\frac {\sqrt {\sqrt {4 a c+b^2}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}+b} \left (-\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \]

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Rubi [A]  time = 0.18, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2072} \begin {gather*} \frac {\sqrt {\sqrt {4 a c+b^2}-b} \tanh ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}-b} \left (\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}-\frac {\sqrt {\sqrt {4 a c+b^2}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}+b} \left (-\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b + Sqrt[b^2 + 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 + 4*a*c]]*x*(b - Sqrt[b^2 + 4*a*c] - 2*c*x^2))/(2*Sqrt
[2]*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*d) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*A
rcTanh[(Sqrt[-b + Sqrt[b^2 + 4*a*c]]*x*(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2))/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a +
b*x^2 - c*x^4])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*d)

Rule 2072

Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^4), x_Symbol] :> With[{q = Sqrt[b^2 - 4*a*c]},
-Simp[(a*Sqrt[b + q]*ArcTan[(Sqrt[b + q]*x*(b - q + 2*c*x^2))/(2*Sqrt[2]*Rt[-(a*c), 2]*Sqrt[a + b*x^2 + c*x^4
])])/(2*Sqrt[2]*Rt[-(a*c), 2]*d), x] + Simp[(a*Sqrt[-b + q]*ArcTanh[(Sqrt[-b + q]*x*(b + q + 2*c*x^2))/(2*Sqrt
[2]*Rt[-(a*c), 2]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[2]*Rt[-(a*c), 2]*d), x]] /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c*d + a*e, 0] && NegQ[a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx &=-\frac {\sqrt {b+\sqrt {b^2+4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x \left (b-\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}+\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \tanh ^{-1}\left (\frac {\sqrt {-b+\sqrt {b^2+4 a c}} x \left (b+\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}\\ \end {align*}

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Mathematica [C]  time = 0.68, size = 432, normalized size = 1.81 \begin {gather*} \frac {\sqrt {\frac {4 c x^2}{\sqrt {4 a c+b^2}-b}+2} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (2 i \sqrt {a} \sqrt {c} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+\left (b-2 i \sqrt {a} \sqrt {c}\right ) \Pi \left (-\frac {i \left (b+\sqrt {b^2+4 a c}\right )}{2 \sqrt {a} \sqrt {c}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-\left (b+2 i \sqrt {a} \sqrt {c}\right ) \Pi \left (\frac {i \left (b+\sqrt {b^2+4 a c}\right )}{2 \sqrt {a} \sqrt {c}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )\right )}{4 \sqrt {a} \sqrt {c} d \sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \sqrt {a+b x^2-c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + (4*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*((2*I)*Sqrt[a]*Sqrt[
c]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 +
4*a*c])] + (b - (2*I)*Sqrt[a]*Sqrt[c])*EllipticPi[((-1/2*I)*(b + Sqrt[b^2 + 4*a*c]))/(Sqrt[a]*Sqrt[c]), I*ArcS
inh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] - (b + (2*
I)*Sqrt[a]*Sqrt[c])*EllipticPi[((I/2)*(b + Sqrt[b^2 + 4*a*c]))/(Sqrt[a]*Sqrt[c]), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(
b + Sqrt[b^2 + 4*a*c]))]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])]))/(4*Sqrt[a]*Sqrt[c]*Sqrt[-(c/(b
+ Sqrt[b^2 + 4*a*c]))]*d*Sqrt[a + b*x^2 - c*x^4])

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IntegrateAlgebraic [C]  time = 0.72, size = 167, normalized size = 0.70 \begin {gather*} \frac {i \sqrt {-b-2 i \sqrt {a} \sqrt {c}} \tan ^{-1}\left (\frac {x \sqrt {-b-2 i \sqrt {a} \sqrt {c}}}{\sqrt {a+b x^2-c x^4}}\right )}{4 \sqrt {a} \sqrt {c} d}-\frac {i \sqrt {-b+2 i \sqrt {a} \sqrt {c}} \tan ^{-1}\left (\frac {x \sqrt {-b+2 i \sqrt {a} \sqrt {c}}}{\sqrt {a+b x^2-c x^4}}\right )}{4 \sqrt {a} \sqrt {c} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*Sqrt[-b - (2*I)*Sqrt[a]*Sqrt[c]]*ArcTan[(Sqrt[-b - (2*I)*Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^2 - c*x^4]])/
(Sqrt[a]*Sqrt[c]*d) - ((I/4)*Sqrt[-b + (2*I)*Sqrt[a]*Sqrt[c]]*ArcTan[(Sqrt[-b + (2*I)*Sqrt[a]*Sqrt[c]]*x)/Sqrt
[a + b*x^2 - c*x^4]])/(Sqrt[a]*Sqrt[c]*d)

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fricas [B]  time = 9.52, size = 669, normalized size = 2.80 \begin {gather*} -\frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) - \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)
) + sqrt(-c*x^4 + b*x^2 + a)*x^2 + (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) - a*d*x)*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4))
- b)/(a*c*d^2)))/(c*x^4 + a)) + 1/8*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(-c*x^4 + b*
x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) + sqrt(-c*x^4 + b*x^2 + a)*x^2 - (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) - a*d*x)*sq
rt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2)))/(c*x^4 + a)) - 1/8*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/
(a*c*d^2))*log((sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) - sqrt(-c*x^4 + b*x^2 + a)*x^2 + (a*c*d^3*x^
3*sqrt(-1/(a*c*d^4)) + a*d*x)*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2)))/(c*x^4 + a)) + 1/8*sqrt(-(2
*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2))*log((sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) - sqrt(-c*x
^4 + b*x^2 + a)*x^2 - (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) + a*d*x)*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d
^2)))/(c*x^4 + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.11, size = 568, normalized size = 2.38 \begin {gather*} \frac {\sqrt {2}\, \arctan \left (\frac {-\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{4 \sqrt {-b +\sqrt {4 a c +b^{2}}}\, d}-\frac {\sqrt {2}\, \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{4 \sqrt {-b +\sqrt {4 a c +b^{2}}}\, d}-\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\sqrt {4 a c +b^{2}}\right )}{32 a c d}+\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{32 a c d}+\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\sqrt {4 a c +b^{2}}\right )}{32 a c d}-\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{32 a c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)/a/c*b*ln((-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^(1/
2)-(-c*x^4+b*x^2+a)/x^2-(4*a*c+b^2)^(1/2))-1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)/a/c*(4*a*c+b^2)^(1/2)*ln
((-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^(1/2)-(-c*x^4+b*x^2+a)/x^2-(4*a*c+b^2)^(1/2))+1/4/d*2^
(1/2)/(-b+(4*a*c+b^2)^(1/2))^(1/2)*arctan(1/2*(2*(b+(4*a*c+b^2)^(1/2))^(1/2)-2*(-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/
x)/(-b+(4*a*c+b^2)^(1/2))^(1/2))-1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)/a/c*b*ln((-c*x^4+b*x^2+a)/x^2+(-c*
x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^(1/2)+(4*a*c+b^2)^(1/2))+1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2
))^(1/2)/a/c*(4*a*c+b^2)^(1/2)*ln((-c*x^4+b*x^2+a)/x^2+(-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^
(1/2)+(4*a*c+b^2)^(1/2))-1/4/d*2^(1/2)/(-b+(4*a*c+b^2)^(1/2))^(1/2)*arctan(1/2*(2*(-c*x^4+b*x^2+a)^(1/2)*2^(1/
2)/x+2*(b+(4*a*c+b^2)^(1/2))^(1/2))/(-b+(4*a*c+b^2)^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {a + b x^{2} - c x^{4}}}{a + c x^{4}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*x**2 - c*x**4)/(a + c*x**4), x)/d

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