3.550 \(\int \frac{1}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}} \]

[Out]

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

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Rubi [A]  time = 0.0674658, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2156, 205, 377} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2156

Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[c, Int[u/(c^2 - a*e
^2 + c*d*x^n), x], x] - Dist[a*e, Int[u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d
, e, n}, x] && EqQ[b*c - a*d, 0]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.100649, size = 83, normalized size = 0.81 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right )-\tan ^{-1}\left (\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right )}{\sqrt{b} \sqrt{a c^2-d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.019, size = 1995, normalized size = 19.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d*c^2*b/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2)
)^(1/2))*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)+1/2*d*c^2*b^(1/2)/((-a*b)^(1/2)*c^2+
(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((b*(x+(-a*b)^(1/2)/b)-(-a*b)^(1/
2))/b^(1/2)+((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2))-1/2*d*c^4*b/((-a*b)^(1/2)*c^2+(-
c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*((x-(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^
2)^(1/2)-1/2*d*c^2*b^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2
))^(1/2))*ln(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+b*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))/b^(1/2)+((x-(-c^2*b*(a*c^
2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2
))+1/2*c^2*b/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^
2*b*(a*c^2-d^2))^(1/2)*d^3/(1/c^2*d^2)^(1/2)*ln((2/c^2*d^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-
d^2))^(1/2)/c^2/b)+2*(1/c^2*d^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/
c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))+1/2*d*c^4*b/(
(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2
))^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1
/2)/c^2/b)+1/c^2*d^2)^(1/2)-1/2*d*c^2*b^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2
+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+b*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))/b^(1/
2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^
2/b)+1/c^2*d^2)^(1/2))-1/2*c^2*b/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c
^2-d^2))^(1/2))/(-c^2*b*(a*c^2-d^2))^(1/2)*d^3/(1/c^2*d^2)^(1/2)*ln((2/c^2*d^2-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^
2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(1/c^2*d^2)^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b
*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+1/c^2*d^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c
^2/b))+1/2*d*c^2*b/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^
2-d^2))^(1/2))*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2)+1/2*d*c^2*b^(1/2)/((-a*b)^(1/2
)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/(-(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))*ln((b*(x-(-a*b)^(1/2)/b)+(-a*
b)^(1/2))/b^(1/2)+((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2))+1/(b*(a*c^2-d^2))^(1/2)*ar
ctan(x*c*b/(b*(a*c^2-d^2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b c x^{2} + a c + \sqrt{b x^{2} + a} d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*c*x^2 + a*c + sqrt(b*x^2 + a)*d), x)

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Fricas [B]  time = 1.35453, size = 1017, normalized size = 9.87 \begin{align*} \left [-\frac{\sqrt{-a b c^{2} + b d^{2}} \log \left (\frac{a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} +{\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2} - 4 \, \sqrt{-a b c^{2} + b d^{2}}{\left ({\left (a b c^{2} d - 2 \, b d^{3}\right )} x^{3} +{\left (a^{2} c^{2} d - a d^{3}\right )} x\right )} \sqrt{b x^{2} + a}}{b^{2} c^{4} x^{4} + a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4} + 2 \,{\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right ) + 2 \, \sqrt{-a b c^{2} + b d^{2}} \log \left (\frac{b c^{2} x^{2} - a c^{2} - 2 \, \sqrt{-a b c^{2} + b d^{2}} c x + d^{2}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right )}{4 \,{\left (a b c^{2} - b d^{2}\right )}}, -\frac{2 \, \sqrt{a b c^{2} - b d^{2}} \arctan \left (-\frac{\sqrt{a b c^{2} - b d^{2}} c x}{a c^{2} - d^{2}}\right ) - \sqrt{a b c^{2} - b d^{2}} \arctan \left (\frac{{\left (a^{2} c^{2} - a d^{2} +{\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}\right )} \sqrt{a b c^{2} - b d^{2}} \sqrt{b x^{2} + a}}{2 \,{\left ({\left (a b^{2} c^{2} d - b^{2} d^{3}\right )} x^{3} +{\left (a^{2} b c^{2} d - a b d^{3}\right )} x\right )}}\right )}{2 \,{\left (a b c^{2} - b d^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(sqrt(-a*b*c^2 + b*d^2)*log((a^4*c^4 - 2*a^3*c^2*d^2 + a^2*d^4 + (a^2*b^2*c^4 - 8*a*b^2*c^2*d^2 + 8*b^2*
d^4)*x^4 + 2*(a^3*b*c^4 - 5*a^2*b*c^2*d^2 + 4*a*b*d^4)*x^2 - 4*sqrt(-a*b*c^2 + b*d^2)*((a*b*c^2*d - 2*b*d^3)*x
^3 + (a^2*c^2*d - a*d^3)*x)*sqrt(b*x^2 + a))/(b^2*c^4*x^4 + a^2*c^4 - 2*a*c^2*d^2 + d^4 + 2*(a*b*c^4 - b*c^2*d
^2)*x^2)) + 2*sqrt(-a*b*c^2 + b*d^2)*log((b*c^2*x^2 - a*c^2 - 2*sqrt(-a*b*c^2 + b*d^2)*c*x + d^2)/(b*c^2*x^2 +
a*c^2 - d^2)))/(a*b*c^2 - b*d^2), -1/2*(2*sqrt(a*b*c^2 - b*d^2)*arctan(-sqrt(a*b*c^2 - b*d^2)*c*x/(a*c^2 - d^
2)) - sqrt(a*b*c^2 - b*d^2)*arctan(1/2*(a^2*c^2 - a*d^2 + (a*b*c^2 - 2*b*d^2)*x^2)*sqrt(a*b*c^2 - b*d^2)*sqrt(
b*x^2 + a)/((a*b^2*c^2*d - b^2*d^3)*x^3 + (a^2*b*c^2*d - a*b*d^3)*x)))/(a*b*c^2 - b*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a c + b c x^{2} + d \sqrt{a + b x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a*c + b*c*x**2 + d*sqrt(a + b*x**2)), x)

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Giac [A]  time = 1.10828, size = 144, normalized size = 1.4 \begin{align*} \frac{\arctan \left (\frac{b c x}{\sqrt{a b c^{2} - b d^{2}}}\right )}{\sqrt{a b c^{2} - b d^{2}}} + \frac{\arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt{a c^{2} - d^{2}} d}\right )}{\sqrt{a c^{2} - d^{2}} \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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