3.25 \(\int \frac{(a+b x^2) \sqrt{e+f x^2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=283 \[ -\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.181188, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ \frac{x \sqrt{c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} (3 a d f-2 b c f+b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 531

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]

Rule 418

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

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 411

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

Rubi steps

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

Mathematica [C]  time = 0.394991, size = 212, normalized size = 0.75 \[ \frac{-i b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (-3 a d f+2 b c f-b d e) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+b f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right )}{3 d f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.017, size = 394, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( 3\,df{x}^{4}+3\,cf{x}^{2}+3\,de{x}^{2}+3\,ce \right ) df}\sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c} \left ( \sqrt{-{\frac{d}{c}}}{x}^{5}bd{f}^{2}+\sqrt{-{\frac{d}{c}}}{x}^{3}bc{f}^{2}+\sqrt{-{\frac{d}{c}}}{x}^{3}bdef+\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef-\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+3\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) adef-2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcef+\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bd{e}^{2}+\sqrt{-{\frac{d}{c}}}xbcef \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)*((-d/c)^(1/2)*x^5*b*d*f^2+(-d/c)^(1/2)*x^3*b*c*f^2+(-d/c)^(1/2)*x^3*b*d*e*
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))*b*c*e*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))*b*d*e^2+3*((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*f-2*((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))*b*c*e*f+((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))*b*d*e^2+(-d/c)^(1/2)*x*b*c*e*f)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/d/(-d/c)^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + e}}{\sqrt{d x^{2} + c}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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