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

Optimal. Leaf size=91 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right )}{d \sqrt{f}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} d \sqrt{d e-c f}} \]

[Out]

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

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Rubi [A]  time = 0.0481703, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {523, 217, 206, 377, 205} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right )}{d \sqrt{f}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} d \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.109592, size = 88, normalized size = 0.97 \[ \frac{\frac{(a d-b c) \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right )}{\sqrt{f}}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.009, size = 646, normalized size = 7.1 \begin{align*}{\frac{b}{d}\ln \left ( x\sqrt{f}+\sqrt{f{x}^{2}+e} \right ){\frac{1}{\sqrt{f}}}}-{\frac{a}{2}\ln \left ({ \left ( -2\,{\frac{cf-de}{d}}+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}+{\frac{bc}{2\,d}\ln \left ({ \left ( -2\,{\frac{cf-de}{d}}+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}+{\frac{a}{2}\ln \left ({ \left ( -2\,{\frac{cf-de}{d}}-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}-{\frac{bc}{2\,d}\ln \left ({ \left ( -2\,{\frac{cf-de}{d}}-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*b*ln(x*f^(1/2)+(f*x^2+e)^(1/2))/f^(1/2)-1/2/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d+2*f*(-c*d
)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*f+2*f*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)
/d)-(c*f-d*e)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*a+1/2/(-c*d)^(1/2)/d/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d+2*f*(
-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*f+2*f*(-c*d)^(1/2)/d*(x-(-c*d)^(
1/2)/d)-(c*f-d*e)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*b*c+1/2/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d-2
*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*f-2*f*(-c*d)^(1/2)/d*(x+(-c*
d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(x+(-c*d)^(1/2)/d))*a-1/2/(-c*d)^(1/2)/d/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)
/d-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*f-2*f*(-c*d)^(1/2)/d*(x+
(-c*d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(x+(-c*d)^(1/2)/d))*b*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.24016, size = 1596, normalized size = 17.54 \begin{align*} \left [\frac{\sqrt{-c d e + c^{2} f}{\left (b c - a d\right )} f \log \left (\frac{{\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \,{\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2} - 4 \,{\left ({\left (d e - 2 \, c f\right )} x^{3} - c e x\right )} \sqrt{-c d e + c^{2} f} \sqrt{f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 2 \,{\left (b c d e - b c^{2} f\right )} \sqrt{f} \log \left (-2 \, f x^{2} - 2 \, \sqrt{f x^{2} + e} \sqrt{f} x - e\right )}{4 \,{\left (c d^{2} e f - c^{2} d f^{2}\right )}}, -\frac{\sqrt{c d e - c^{2} f}{\left (b c - a d\right )} f \arctan \left (\frac{\sqrt{c d e - c^{2} f}{\left ({\left (d e - 2 \, c f\right )} x^{2} - c e\right )} \sqrt{f x^{2} + e}}{2 \,{\left ({\left (c d e f - c^{2} f^{2}\right )} x^{3} +{\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right ) -{\left (b c d e - b c^{2} f\right )} \sqrt{f} \log \left (-2 \, f x^{2} - 2 \, \sqrt{f x^{2} + e} \sqrt{f} x - e\right )}{2 \,{\left (c d^{2} e f - c^{2} d f^{2}\right )}}, \frac{\sqrt{-c d e + c^{2} f}{\left (b c - a d\right )} f \log \left (\frac{{\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \,{\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2} - 4 \,{\left ({\left (d e - 2 \, c f\right )} x^{3} - c e x\right )} \sqrt{-c d e + c^{2} f} \sqrt{f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 4 \,{\left (b c d e - b c^{2} f\right )} \sqrt{-f} \arctan \left (\frac{\sqrt{-f} x}{\sqrt{f x^{2} + e}}\right )}{4 \,{\left (c d^{2} e f - c^{2} d f^{2}\right )}}, -\frac{\sqrt{c d e - c^{2} f}{\left (b c - a d\right )} f \arctan \left (\frac{\sqrt{c d e - c^{2} f}{\left ({\left (d e - 2 \, c f\right )} x^{2} - c e\right )} \sqrt{f x^{2} + e}}{2 \,{\left ({\left (c d e f - c^{2} f^{2}\right )} x^{3} +{\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right ) + 2 \,{\left (b c d e - b c^{2} f\right )} \sqrt{-f} \arctan \left (\frac{\sqrt{-f} x}{\sqrt{f x^{2} + e}}\right )}{2 \,{\left (c d^{2} e f - c^{2} d f^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.66463, size = 159, normalized size = 1.75 \begin{align*} \frac{{\left (b c \sqrt{f} - a d \sqrt{f}\right )} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{\sqrt{-c^{2} f^{2} + c d f e} d} - \frac{b \log \left ({\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2}\right )}{2 \, d \sqrt{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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