3.11.5 \(\int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} (a d+a e x^2+c d x^4)} \, dx\) [1005]

Optimal. Leaf size=53 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b d+a e} x}{\sqrt {d} \sqrt {a-b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {b d+a e}} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2137, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {x \sqrt {a e+b d}}{\sqrt {d} \sqrt {a-b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {a e+b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2137

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 10.53, size = 416, normalized size = 7.85 \begin {gather*} \frac {i \sqrt {2+\frac {4 c x^2}{-b+\sqrt {b^2-4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \left (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 )-\Pi \left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) d}{-a e+\sqrt {a} \sqrt {-4 c d^2+a e^2}};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 )-\Pi \left (\frac {\left (-b+\sqrt {b^2-4 a c}\right ) d}{a e+\sqrt {a} \sqrt {-4 c d^2+a e^2}};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 )}{2 \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} d \sqrt {a-b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/2)*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])]*(EllipticF[I*A
rcSinh[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])] - Ellipti
cPi[((b - Sqrt[b^2 - 4*a*c])*d)/(-(a*e) + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), 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])] - EllipticPi[((-b + Sqrt[b^2 - 4*a*c])*d)
/(a*e + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), 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])]))/(Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a - b*x^2 + c*x^4])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.08, size = 517, normalized size = 9.75

method	result	size
elliptic	\(-\frac {\arctan \left (\frac {d \sqrt {c \,x^{4}-b \,x^{2}+a}}{x \sqrt {\left (a e +b d \right ) d}}\right )}{\sqrt {\left (a e +b d \right ) d}}\)	\(46\)
default	\(-\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 d \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}-b \,x^{2}+a}}-\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} e -2 d \right ) \left (-\frac {\arctanh \left (\frac {2 c \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-b \,\underline {\hspace {1.25 ex}}\alpha ^{2}-b \,x^{2}+2 a}{2 \sqrt {-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (a e +b d \right )}{d}}\, \sqrt {c \,x^{4}-b \,x^{2}+a}}\right )}{\sqrt {-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (a e +b d \right )}{d}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c d +a e \right ) \sqrt {2-\frac {b \,x^{2}}{a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {2-\frac {b \,x^{2}}{a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{a}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {-\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {-4 a c +b^{2}}\, c d +\underline {\hspace {1.25 ex}}\alpha ^{2} b c d -\sqrt {-4 a c +b^{2}}\, a e +a b e}{2 a c d}, \frac {\sqrt {-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{d a \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}-b \,x^{2}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} c d +a e \right )}\right )}{4 d}\)	\(517\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(-b+(-4*a*c+b^2)
^(1/2))/a*x^2)^(1/2)/(c*x^4-b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*
b*(-b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/4*a/d*sum((-_alpha^2*e-2*d)/_alpha/(2*_alpha^2*c*d+a*e)*(-1/(-_alpha^2
*(a*e+b*d)/d)^(1/2)*arctanh(1/2*(2*_alpha^2*c*x^2-_alpha^2*b-b*x^2+2*a)/(-_alpha^2*(a*e+b*d)/d)^(1/2)/(c*x^4-b
*x^2+a)^(1/2))+1/d/a*2^(1/2)*_alpha*(_alpha^2*c*d+a*e)/((b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(2-b*x^2/a-x^2/a*(-4*a
*c+b^2)^(1/2))^(1/2)*(2-b*x^2/a+x^2/a*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4-b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)
*((b+(-4*a*c+b^2)^(1/2))/a)^(1/2),-1/2*(-_alpha^2*(-4*a*c+b^2)^(1/2)*c*d+_alpha^2*b*c*d-(-4*a*c+b^2)^(1/2)*a*e
+a*b*e)/a/c/d,(-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))/a)^(1/2))),_alpha=RootOf(
_Z^4*c*d+_Z^2*a*e+a*d))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 8.15, size = 319, normalized size = 6.02 \begin {gather*} \left [-\frac {\sqrt {-b d^{2} - a d e} \log \left (-\frac {c^{2} d^{2} x^{8} - 8 \, b c d^{2} x^{6} + 2 \, {\left (4 \, b^{2} + a c\right )} d^{2} x^{4} + a^{2} x^{4} e^{2} - 8 \, a b d^{2} x^{2} + a^{2} d^{2} + 4 \, {\left (c d x^{5} - 2 \, b d x^{3} - a x^{3} e + a d x\right )} \sqrt {c x^{4} - b x^{2} + a} \sqrt {-b d^{2} - a d e} - 2 \, {\left (3 \, a c d x^{6} - 4 \, a b d x^{4} + 3 \, a^{2} d x^{2}\right )} e}{c^{2} d^{2} x^{8} + 2 \, a c d^{2} x^{4} + a^{2} x^{4} e^{2} + a^{2} d^{2} + 2 \, {\left (a c d x^{6} + a^{2} d x^{2}\right )} e}\right )}{4 \, {\left (b d^{2} + a d e\right )}}, -\frac {\arctan \left (-\frac {2 \, \sqrt {c x^{4} - b x^{2} + a} \sqrt {b d^{2} + a d e} x}{c d x^{4} - 2 \, b d x^{2} - a x^{2} e + a d}\right )}{2 \, \sqrt {b d^{2} + a d e}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*sqrt(-b*d^2 - a*d*e)*log(-(c^2*d^2*x^8 - 8*b*c*d^2*x^6 + 2*(4*b^2 + a*c)*d^2*x^4 + a^2*x^4*e^2 - 8*a*b*d
^2*x^2 + a^2*d^2 + 4*(c*d*x^5 - 2*b*d*x^3 - a*x^3*e + a*d*x)*sqrt(c*x^4 - b*x^2 + a)*sqrt(-b*d^2 - a*d*e) - 2*
(3*a*c*d*x^6 - 4*a*b*d*x^4 + 3*a^2*d*x^2)*e)/(c^2*d^2*x^8 + 2*a*c*d^2*x^4 + a^2*x^4*e^2 + a^2*d^2 + 2*(a*c*d*x
^6 + a^2*d*x^2)*e))/(b*d^2 + a*d*e), -1/2*arctan(-2*sqrt(c*x^4 - b*x^2 + a)*sqrt(b*d^2 + a*d*e)*x/(c*d*x^4 - 2
*b*d*x^2 - a*x^2*e + a*d))/sqrt(b*d^2 + a*d*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a-c\,x^4}{\left (c\,d\,x^4+a\,e\,x^2+a\,d\right )\,\sqrt {c\,x^4-b\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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