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

Optimal. Leaf size=101 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}-\frac{d x \sqrt{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(d*x*Sqrt[a + b*x^2])/(2*c*(b*c - a*d)*(c + d*x^2)) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqr
t[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.0489142, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {382, 377, 208} \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}-\frac{d x \sqrt{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*x*Sqrt[a + b*x^2])/(2*c*(b*c - a*d)*(c + d*x^2)) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqr
t[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(3/2))

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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 208

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )^2} \, dx &=-\frac{d x \sqrt{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{(2 b c-a d) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac{d x \sqrt{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{(2 b c-a d) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 c (b c-a d)}\\ &=-\frac{d x \sqrt{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.286529, size = 126, normalized size = 1.25 \[ \frac{x \left (\frac{\left (c+d x^2\right ) (2 b c-a d) \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{c \sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}}-d \left (a+b x^2\right )\right )}{2 c \sqrt{a+b x^2} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.015, size = 809, normalized size = 8. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.7679, size = 957, normalized size = 9.48 \begin{align*} \left [-\frac{4 \,{\left (b c^{2} d - a c d^{2}\right )} \sqrt{b x^{2} + a} x -{\left (2 \, b c^{2} - a c d +{\left (2 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \,{\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{8 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} +{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}, -\frac{2 \,{\left (b c^{2} d - a c d^{2}\right )} \sqrt{b x^{2} + a} x +{\left (2 \, b c^{2} - a c d +{\left (2 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{-b c^{2} + a c d} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt{b x^{2} + a}}{2 \,{\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} +{\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} +{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 1.18589, size = 327, normalized size = 3.24 \begin{align*} \frac{1}{2} \, b^{\frac{3}{2}}{\left (\frac{{\left (2 \, b c - a d\right )} \arctan \left (-\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{2} c^{2} - a b c d\right )} \sqrt{-b^{2} c^{2} + a b c d}} - \frac{2 \,{\left (2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} b c -{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} d + 4 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} b c - 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}{\left (b^{2} c^{2} - a b c d\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^(3/2)*((2*b*c - a*d)*arctan(-1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c
*d))/((b^2*c^2 - a*b*c*d)*sqrt(-b^2*c^2 + a*b*c*d)) - 2*(2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - (sqrt(b)*x -
sqrt(b*x^2 + a))^2*a*d + a^2*d)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c -
2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)*(b^2*c^2 - a*b*c*d)))