3.2.65 \(\int \frac {1}{(d+e x^2) \sqrt {-a+c x^4}} \, dx\) [165]

Optimal. Leaf size=73 \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt {-a+c x^4}} \]

[Out]

a^(1/4)*EllipticPi(c^(1/4)*x/a^(1/4),-e*a^(1/2)/d/c^(1/2),I)*(1-c*x^4/a)^(1/2)/c^(1/4)/d/(c*x^4-a)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1233, 1232} \begin {gather*} \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt {c x^4-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)
*d*Sqrt[-a + c*x^4])

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1233

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a)]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+c x^4}} \, dx &=\frac {\sqrt {1-\frac {c x^4}{a}} \int \frac {1}{\left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}}} \, dx}{\sqrt {-a+c x^4}}\\ &=\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt {-a+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.09, size = 92, normalized size = 1.26 \begin {gather*} -\frac {i \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d \sqrt {-a+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(
Sqrt[-(Sqrt[c]/Sqrt[a])]*d*Sqrt[-a + c*x^4])

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Maple [A]
time = 0.13, size = 99, normalized size = 1.36

method	result	size
default	\(\frac {\sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, \frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\)	\(99\)
elliptic	\(\frac {\sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, \frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

integral(sqrt(c*x^4 - a)/(c*d*x^4 - a*d + (c*x^6 - a*x^2)*e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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