3.58 \(\int \frac{(a+b x^2)^{3/2}}{(c+d x^2)^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^2}-\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^2}-\frac{x \sqrt{a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0903201, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {413, 523, 217, 206, 377, 208} \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^2}-\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^2}-\frac{x \sqrt{a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/(c + d*x^2)^2,x]

[Out]

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

Rule 413

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

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

Mathematica [A]  time = 0.121052, size = 142, normalized size = 1.08 \[ \frac{\frac{\left (a^2 d^2+a b c d-2 b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{3/2} \sqrt{a d-b c}}+2 b^{3/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{d x \sqrt{a+b x^2} (b c-a d)}{c \left (c+d x^2\right )}}{2 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^2,x]

[Out]

(-((d*(b*c - a*d)*x*Sqrt[a + b*x^2])/(c*(c + d*x^2))) + ((-2*b^2*c^2 + a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[-(b*c)
+ a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(3/2)*Sqrt[-(b*c) + a*d]) + 2*b^(3/2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^
2]])/(2*d^2)

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Maple [B]  time = 0.016, size = 4621, normalized size = 35.3 \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)^(3/2)/(d*x^2+c)^2,x)

[Out]

-3/4/c/d*b*(-c*d)^(1/2)/(a*d-b*c)*((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)*a-1/12/(-c*d)^(1/2)/c*((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)^(3/2)-1/
4/d^2*b^(3/2)*ln((-b*(-c*d)^(1/2)/d+b*(x+(-c*d)^(1/2)/d))/b^(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))+1/12/(-c*d)^(1/2)/c*((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)^(3/2)-1/4/d^2*b^(3/2)*ln((b*(-c*d)^(1/2)/d+b*(x-(-c*d)^(1/2)/d))/b^(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))+3/4/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))*a^2-3/4/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))*a^2+3/8/d*b^
2/(a*d-b*c)*((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+9/8/d*b^(3/2)/(
a*d-b*c)*ln((b*(-c*d)^(1/2)/d+b*(x-(-c*d)^(1/2)/d))/b^(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))*a+3/4/d^2*b^2*(-c*d)^(1/2)/(a*d-b*c)*((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)-3/4*c/d^2*b^(5/2)/(a*d-b*c)*ln((b*(-c*d)^(1/2)/d+b*(x-(-c*d)^(1/2)/d))
/b^(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))-1/4/c*b/(a*d-b*c)*(
(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)^(3/2)*x-3/8/c*b^(1/2)/(a*d-b*c)*a^2*
ln((b*(-c*d)^(1/2)/d+b*(x-(-c*d)^(1/2)/d))/b^(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))-3/4/d^2*b^2*(-c*d)^(1/2)/(a*d-b*c)*((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)-3/4*c/d^2*b^(5/2)/(a*d-b*c)*ln((-b*(-c*d)^(1/2)/d+b*(x+(-c*d)^(1/2)/d))/b^(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))-1/4/c*b/(a*d-b*c)*((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)^(3/2)*x-3/8/c*b^(1/2)/(a*d-b*c)*a^2*ln((-b*(-c
*d)^(1/2)/d+b*(x+(-c*d)^(1/2)/d))/b^(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))+1/8/c*b/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)*x+3/8
/c/d*b^(1/2)*ln((-b*(-c*d)^(1/2)/d+b*(x+(-c*d)^(1/2)/d))/b^(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))*a+1/4/(-c*d)^(1/2)/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))*a^2+1/8/c*b/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)*x+3/8/c/d*b^(1/2)*ln((b*(-c*d)^(1/2)/d+b*(x-(-c*d)^(1/2)/d))/b^(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))*a-1/4/(-c*d)^(1/2)/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))*a^2+3/8/d*b^2/(a*d-b*c)*((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+9/8/d*b^(3/2)/(a*d-b*c)*ln((-b*(-c*d)^(1/2)/d+b*
(x+(-c*d)^(1/2)/d))/b^(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))*
a-1/4/(-c*d)^(1/2)/c*((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)*a+1/4/(-
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)*b+1/4/(-c*d)^(1/
2)/c*((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)*a-1/4/(-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)*b+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)^(5/2)+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)^(5/2)+1/4/(-c*d)^(1/2)*c
/d^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))*b^2-1/4/c/d*b*(
-c*d)^(1/2)/(a*d-b*c)*((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)^(3/2)-3/8/c*b
/(a*d-b*c)*a*((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+1/4/c/d*b*(-c*
d)^(1/2)/(a*d-b*c)*((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)^(3/2)-3/8/c*b/(a
*d-b*c)*a*((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+1/2/(-c*d)^(1/2)/
d/((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))*a*b-1/4/(-c*d)^(1/
2)*c/d^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))*b^2-1/2/(-c
*d)^(1/2)/d/((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))*a*b+3/4*
c/d^3*b^3*(-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))-3/2/d^2*b^2*(-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))*a+3/2/d^2*b^2*(-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))*a-3/4*c/d^3*b^3*(-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))+3/4/c/d*b*(-c*d)^(1/2)/(a*d-b*c)*((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)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)/(d*x^2 + c)^2, x)

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Fricas [A]  time = 2.7808, size = 1941, normalized size = 14.82 \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)^(3/2)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{\left (c + d x^{2}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**2,x)

[Out]

Integral((a + b*x**2)**(3/2)/(c + d*x**2)**2, x)

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Giac [B]  time = 1.16482, size = 428, normalized size = 3.27 \begin{align*} -\frac{b^{\frac{3}{2}} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{2 \, d^{2}} + \frac{{\left (2 \, b^{\frac{5}{2}} c^{2} - a b^{\frac{3}{2}} c d - a^{2} \sqrt{b} d^{2}\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 )}{2 \, \sqrt{-b^{2} c^{2} + a b c d} c d^{2}} - \frac{2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} b^{\frac{5}{2}} c^{2} - 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a b^{\frac{3}{2}} c d +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} \sqrt{b} d^{2} + a^{2} b^{\frac{3}{2}} c d - a^{3} \sqrt{b} d^{2}}{{\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 )} c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

-1/2*b^(3/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/d^2 + 1/2*(2*b^(5/2)*c^2 - a*b^(3/2)*c*d - a^2*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))/(sqrt(-b^2*c^2 + a*b*c*
d)*c*d^2) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c*d +
(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*d^2 + a^2*b^(3/2)*c*d - a^3*sqrt(b)*d^2)/(((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)*c*d^2)