3.1.68 \(\int \frac {\sqrt {e+f x^2}}{(a+b x^2) (c+d x^2)^{3/2}} \, dx\) [68]

Optimal. Leaf size=209 \[ -\frac {\sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d) \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

b*e^(3/2)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticPi(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),1-b*e/a/f,(1-
d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/a/c/(-a*d+b*c)/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-(1/(1+d
*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*d^(1/2)*(f*x
^2+e)^(1/2)/(-a*d+b*c)/c^(1/2)/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {555, 553, 422} \begin {gather*} \frac {b e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c \sqrt {f} \sqrt {e+f x^2} (b c-a d) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {d} \sqrt {e+f x^2} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

-((Sqrt[d]*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)*Sqrt[
c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) + (b*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*c*(b*c - a*d)*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sq
rt[e + f*x^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 553

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[c*(Sqrt[e +
f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), Ar
cTan[Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 555

Int[Sqrt[(e_) + (f_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[b/(b*c -
a*d), Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] - Dist[d/(b*c - a*d), Int[Sqrt[e + f*x^2]/(c +
d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]

Rubi steps

\begin {align*} \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx &=\frac {b \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b c-a d}-\frac {d \int \frac {\sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{b c-a d}\\ &=-\frac {\sqrt {d} \sqrt {e+f x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} (b c-a d) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d) \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.79, size = 347, normalized size = 1.66 \begin {gather*} \frac {\sqrt {\frac {d}{c}} \left (a d \sqrt {\frac {d}{c}} e x+a d \sqrt {\frac {d}{c}} f x^3+i a d e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i a (-d e+c f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i b c e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a c f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (\frac {b c}{a d};i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{a d (-b c+a d) \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e + f*x^2]/((a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

(Sqrt[d/c]*(a*d*Sqrt[d/c]*e*x + a*d*Sqrt[d/c]*f*x^3 + I*a*d*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elliptic
E[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[
I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b*c*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d),
I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*c*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d)
, I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*d*(-(b*c) + a*d)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]
time = 0.16, size = 390, normalized size = 1.87

method	result	size
default	\(\frac {\left (\sqrt {-\frac {d}{c}}\, a d f \,x^{3}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a c f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b c e +\sqrt {-\frac {d}{c}}\, a d e x \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{c a \sqrt {-\frac {d}{c}}\, \left (a d -b c \right ) \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\)	\(390\)
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+d e \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}-\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right ) f}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, \left (a d -b c \right )}+\frac {d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\left (a d -b c \right ) c \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\left (a d -b c \right ) c \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) f}{\left (a d -b c \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) e}{\left (a d -b c \right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(601\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((-d/c)^(1/2)*a*d*f*x^3-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c*
f+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*d*e-((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*d*e+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*El
lipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*c*f-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellip
ticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b*c*e+(-d/c)^(1/2)*a*d*e*x)*(d*x^2+c)^(1/2)*(f*x^2+e)^
(1/2)/c/a/(-d/c)^(1/2)/(a*d-b*c)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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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((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(f*x^2 + e)/((b*x^2 + a)*(d*x^2 + c)^(3/2)), x)

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Fricas [F(-1)] Timed out
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((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e + f x^{2}}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e)**(1/2)/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

Integral(sqrt(e + f*x**2)/((a + b*x**2)*(c + d*x**2)**(3/2)), 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((f*x^2+e)^(1/2)/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(f*x^2 + e)/((b*x^2 + a)*(d*x^2 + c)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {f\,x^2+e}}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x^2)^(1/2)/((a + b*x^2)*(c + d*x^2)^(3/2)),x)

[Out]

int((e + f*x^2)^(1/2)/((a + b*x^2)*(c + d*x^2)^(3/2)), x)

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