3.52 \(\int \frac{\sqrt{a+b x^2}}{(c+d x^2)^4} \, dx\)
Optimal. Leaf size=208 \[ \frac{a \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{48 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (4 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}+\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3} \]
[Out]
(x*Sqrt[a + b*x^2])/(6*c*(c + d*x^2)^3) + ((4*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(24*c^2*(b*c - a*d)*(c + d*x^2)^
2) + ((2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(48*c^3*(b*c - a*d)^2*(c + d*x^2)) + (a*(8*b^2*c^2 -
12*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(5/2))
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Rubi [A] time = 0.213217, antiderivative size = 208, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used =
{412, 527, 12, 377, 208} \[ \frac{a \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{48 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (4 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}+\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + b*x^2]/(c + d*x^2)^4,x]
[Out]
(x*Sqrt[a + b*x^2])/(6*c*(c + d*x^2)^3) + ((4*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(24*c^2*(b*c - a*d)*(c + d*x^2)^
2) + ((2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(48*c^3*(b*c - a*d)^2*(c + d*x^2)) + (a*(8*b^2*c^2 -
12*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(5/2))
Rule 412
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*(p
+ 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p,
-1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
Rule 527
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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{\sqrt{a+b x^2}}{\left (c+d x^2\right )^4} \, dx &=\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3}-\frac{\int \frac{-5 a-4 b x^2}{\sqrt{a+b x^2} \left (c+d x^2\right )^3} \, dx}{6 c}\\ &=\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac{(4 b c-5 a d) x \sqrt{a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}-\frac{\int \frac{-a (16 b c-15 a d)-2 b (4 b c-5 a d) x^2}{\sqrt{a+b x^2} \left (c+d x^2\right )^2} \, dx}{24 c^2 (b c-a d)}\\ &=\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac{(4 b c-5 a d) x \sqrt{a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac{(2 b c-5 a d) (4 b c-3 a d) x \sqrt{a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}-\frac{\int -\frac{3 a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{48 c^3 (b c-a d)^2}\\ &=\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac{(4 b c-5 a d) x \sqrt{a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac{(2 b c-5 a d) (4 b c-3 a d) x \sqrt{a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2} \left (c+d x^2\right )} \, dx}{16 c^3 (b c-a d)^2}\\ &=\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac{(4 b c-5 a d) x \sqrt{a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac{(2 b c-5 a d) (4 b c-3 a d) x \sqrt{a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 c^3 (b c-a d)^2}\\ &=\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac{(4 b c-5 a d) x \sqrt{a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac{(2 b c-5 a d) (4 b c-3 a d) x \sqrt{a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac{a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.957091, size = 227, normalized size = 1.09 \[ \frac{x \sqrt{a+b x^2} \left ((b c-a d) \left (a^2 d^2 \left (33 c^2+40 c d x^2+15 d^2 x^4\right )-2 a b c d \left (30 c^2+35 c d x^2+13 d^2 x^4\right )+8 b^2 c^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )\right )+\frac{3 a \left (c+d x^2\right )^3 \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}} \tanh ^{-1}\left (\sqrt{\frac{x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{x^2}\right )}{48 c^3 \left (c+d x^2\right )^3 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b*x^2]/(c + d*x^2)^4,x]
[Out]
(x*Sqrt[a + b*x^2]*((b*c - a*d)*(8*b^2*c^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4) - 2*a*b*c*d*(30*c^2 + 35*c*d*x^2 + 13
*d^2*x^4) + a^2*d^2*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)) + (3*a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Sqrt[((b*c
- a*d)*x^2)/(c*(a + b*x^2))]*(c + d*x^2)^3*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/x^2))/(48*c^3*(b
*c - a*d)^3*(c + d*x^2)^3)
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Maple [B] time = 0.026, size = 7922, normalized size = 38.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x^2+a)^(1/2)/(d*x^2+c)^4,x)
[Out]
result too large to display
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a}}{{\left (d x^{2} + c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^4,x, algorithm="maxima")
[Out]
integrate(sqrt(b*x^2 + a)/(d*x^2 + c)^4, x)
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Fricas [B] time = 6.19139, size = 2485, normalized size = 11.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^4,x, algorithm="fricas")
[Out]
[1/192*(3*(8*a*b^2*c^5 - 12*a^2*b*c^4*d + 5*a^3*c^3*d^2 + (8*a*b^2*c^2*d^3 - 12*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 +
3*(8*a*b^2*c^3*d^2 - 12*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + 3*(8*a*b^2*c^4*d - 12*a^2*b*c^3*d^2 + 5*a^3*c^2*d^
3)*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)) + 4*((8*b
^3*c^4*d^2 - 34*a*b^2*c^3*d^3 + 41*a^2*b*c^2*d^4 - 15*a^3*c*d^5)*x^5 + 2*(12*b^3*c^5*d - 47*a*b^2*c^4*d^2 + 55
*a^2*b*c^3*d^3 - 20*a^3*c^2*d^4)*x^3 + 3*(8*b^3*c^6 - 28*a*b^2*c^5*d + 31*a^2*b*c^4*d^2 - 11*a^3*c^3*d^3)*x)*s
qrt(b*x^2 + a))/(b^3*c^10 - 3*a*b^2*c^9*d + 3*a^2*b*c^8*d^2 - a^3*c^7*d^3 + (b^3*c^7*d^3 - 3*a*b^2*c^6*d^4 + 3
*a^2*b*c^5*d^5 - a^3*c^4*d^6)*x^6 + 3*(b^3*c^8*d^2 - 3*a*b^2*c^7*d^3 + 3*a^2*b*c^6*d^4 - a^3*c^5*d^5)*x^4 + 3*
(b^3*c^9*d - 3*a*b^2*c^8*d^2 + 3*a^2*b*c^7*d^3 - a^3*c^6*d^4)*x^2), -1/96*(3*(8*a*b^2*c^5 - 12*a^2*b*c^4*d + 5
*a^3*c^3*d^2 + (8*a*b^2*c^2*d^3 - 12*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 3*(8*a*b^2*c^3*d^2 - 12*a^2*b*c^2*d^3 + 5*
a^3*c*d^4)*x^4 + 3*(8*a*b^2*c^4*d - 12*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqr
t(-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))
- 2*((8*b^3*c^4*d^2 - 34*a*b^2*c^3*d^3 + 41*a^2*b*c^2*d^4 - 15*a^3*c*d^5)*x^5 + 2*(12*b^3*c^5*d - 47*a*b^2*c^
4*d^2 + 55*a^2*b*c^3*d^3 - 20*a^3*c^2*d^4)*x^3 + 3*(8*b^3*c^6 - 28*a*b^2*c^5*d + 31*a^2*b*c^4*d^2 - 11*a^3*c^3
*d^3)*x)*sqrt(b*x^2 + a))/(b^3*c^10 - 3*a*b^2*c^9*d + 3*a^2*b*c^8*d^2 - a^3*c^7*d^3 + (b^3*c^7*d^3 - 3*a*b^2*c
^6*d^4 + 3*a^2*b*c^5*d^5 - a^3*c^4*d^6)*x^6 + 3*(b^3*c^8*d^2 - 3*a*b^2*c^7*d^3 + 3*a^2*b*c^6*d^4 - a^3*c^5*d^5
)*x^4 + 3*(b^3*c^9*d - 3*a*b^2*c^8*d^2 + 3*a^2*b*c^7*d^3 - a^3*c^6*d^4)*x^2)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x**2+a)**(1/2)/(d*x**2+c)**4,x)
[Out]
Timed out
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Giac [B] time = 21.9838, size = 1293, normalized size = 6.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^4,x, algorithm="giac")
[Out]
-1/16*(8*a*b^(5/2)*c^2 - 12*a^2*b^(3/2)*c*d + 5*a^3*sqrt(b)*d^2)*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^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(-b^2*c^2 + a*b*c*d)) - 1
/24*(24*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*c^2*d^3 - 36*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*c
*d^4 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*d^5 + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*c^3
*d^2 - 480*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*c^2*d^3 + 330*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/
2)*c*d^4 - 75*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4*sqrt(b)*d^5 - 256*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(11/2)*c
^5 + 1216*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(9/2)*c^4*d - 2016*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(7/2)*c
^3*d^2 + 1736*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(5/2)*c^2*d^3 - 800*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^
(3/2)*c*d^4 + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*sqrt(b)*d^5 - 384*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^
(9/2)*c^4*d + 1392*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(7/2)*c^3*d^2 - 1608*(sqrt(b)*x - sqrt(b*x^2 + a))^4*
a^4*b^(5/2)*c^2*d^3 + 780*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(3/2)*c*d^4 - 150*(sqrt(b)*x - sqrt(b*x^2 + a)
)^4*a^6*sqrt(b)*d^5 - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(7/2)*c^3*d^2 + 336*(sqrt(b)*x - sqrt(b*x^2 + a
))^2*a^5*b^(5/2)*c^2*d^3 - 300*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6*b^(3/2)*c*d^4 + 75*(sqrt(b)*x - sqrt(b*x^2
+ a))^2*a^7*sqrt(b)*d^5 - 8*a^6*b^(5/2)*c^2*d^3 + 26*a^7*b^(3/2)*c*d^4 - 15*a^8*sqrt(b)*d^5)/((b^2*c^5*d - 2*a
*b*c^4*d^2 + a^2*c^3*d^3)*((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)^3)