3.1.52 \(\int \frac {a+b x^2}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx\) [52]

Optimal. Leaf size=191 \[ \frac {b x \sqrt {2+d x^2}}{d \sqrt {3+f x^2}}-\frac {\sqrt {2} b \sqrt {2+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}+\frac {a \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} \sqrt {f} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}} \]

[Out]

b*x*(d*x^2+2)^(1/2)/d/(f*x^2+3)^(1/2)+1/2*a*(1/(3*f*x^2+9))^(1/2)*(3*f*x^2+9)^(1/2)*EllipticF(x*f^(1/2)*3^(1/2
)/(3*f*x^2+9)^(1/2),1/2*(4-6*d/f)^(1/2))*(d*x^2+2)^(1/2)*2^(1/2)/f^(1/2)/((d*x^2+2)/(f*x^2+3))^(1/2)/(f*x^2+3)
^(1/2)-b*(1/(3*f*x^2+9))^(1/2)*(3*f*x^2+9)^(1/2)*EllipticE(x*f^(1/2)*3^(1/2)/(3*f*x^2+9)^(1/2),1/2*(4-6*d/f)^(
1/2))*2^(1/2)*(d*x^2+2)^(1/2)/d/f^(1/2)/((d*x^2+2)/(f*x^2+3))^(1/2)/(f*x^2+3)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {545, 429, 506, 422} \begin {gather*} \frac {a \sqrt {d x^2+2} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{\sqrt {2} \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\frac {\sqrt {2} b \sqrt {d x^2+2} E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {b x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*Sqrt[2 + d*x^2])/(d*Sqrt[3 + f*x^2]) - (Sqrt[2]*b*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]],
1 - (3*d)/(2*f)])/(d*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]) + (a*Sqrt[2 + d*x^2]*EllipticF[Arc
Tan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(Sqrt[2]*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + 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 429

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

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 2.74, size = 81, normalized size = 0.42 \begin {gather*} -\frac {i \left (3 b E\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+(-3 b+a f) F\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )\right )}{\sqrt {3} \sqrt {d} f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(3*b*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + (-3*b + a*f)*EllipticF[I*ArcSinh[(Sqrt[d]*
x)/Sqrt[2]], (2*f)/(3*d)]))/(Sqrt[3]*Sqrt[d]*f)

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Maple [A]
time = 0.13, size = 105, normalized size = 0.55

method	result	size
default	\(\frac {\sqrt {2}\, \left (\EllipticF \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) a d -2 \EllipticF \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) b +2 \EllipticE \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) b \right )}{2 \sqrt {-f}\, d}\)	\(105\)
elliptic	\(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (d \,x^{2}+2\right )}\, \left (\frac {a \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{2 \sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}-\frac {b \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, \left (\EllipticF \left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}\, d}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*2^(1/2)*(EllipticF(1/3*x*3^(1/2)*(-f)^(1/2),1/2*2^(1/2)*3^(1/2)*(d/f)^(1/2))*a*d-2*EllipticF(1/3*x*3^(1/2)
*(-f)^(1/2),1/2*2^(1/2)*3^(1/2)*(d/f)^(1/2))*b+2*EllipticE(1/3*x*3^(1/2)*(-f)^(1/2),1/2*2^(1/2)*3^(1/2)*(d/f)^
(1/2))*b)/(-f)^(1/2)/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((b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {b\,x^2+a}{\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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