3.740 \(\int \frac{x^4 (c+d x^2)^{3/2}}{(a+b x^2)^2} \, dx\)
Optimal. Leaf size=197 \[ \frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4 \sqrt{d}}+\frac{3 x \sqrt{c+d x^2} (3 b c-4 a d)}{8 b^3}-\frac{3 \sqrt{a} (b c-2 a d) \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^4}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2} \]
[Out]
(3*(3*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^3) + (3*d*x^3*Sqrt[c + d*x^2])/(4*b^2) - (x^3*(c + d*x^2)^(3/2))/(2
*b*(a + b*x^2)) - (3*Sqrt[a]*(b*c - 2*a*d)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2]
)])/(2*b^4) + (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^4*Sqrt[d])
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Rubi [A] time = 0.337645, antiderivative size = 197, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{467, 581, 582, 523, 217, 206, 377, 205} \[ \frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4 \sqrt{d}}+\frac{3 x \sqrt{c+d x^2} (3 b c-4 a d)}{8 b^3}-\frac{3 \sqrt{a} (b c-2 a d) \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^4}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2} \]
Antiderivative was successfully verified.
[In]
Int[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]
[Out]
(3*(3*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^3) + (3*d*x^3*Sqrt[c + d*x^2])/(4*b^2) - (x^3*(c + d*x^2)^(3/2))/(2
*b*(a + b*x^2)) - (3*Sqrt[a]*(b*c - 2*a*d)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2]
)])/(2*b^4) + (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^4*Sqrt[d])
Rule 467
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 581
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*g*(m + n*(p + q + 1) + 1)), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
Rule 582
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q
+ 1) + 1)), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
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{x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \sqrt{c+d x^2} \left (3 c+6 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \left (6 c (2 b c-3 a d)+6 d (3 b c-4 a d) x^2\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{8 b^2}\\ &=\frac{3 (3 b c-4 a d) x \sqrt{c+d x^2}}{8 b^3}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac{\int \frac{6 a c d (3 b c-4 a d)-6 d \left (b^2 c^2-8 a b c d+8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{16 b^3 d}\\ &=\frac{3 (3 b c-4 a d) x \sqrt{c+d x^2}}{8 b^3}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac{(3 a (b c-2 a d) (b c-a d)) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 b^4}+\frac{\left (3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{8 b^4}\\ &=\frac{3 (3 b c-4 a d) x \sqrt{c+d x^2}}{8 b^3}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac{(3 a (b c-2 a d) (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{2 b^4}+\frac{\left (3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{8 b^4}\\ &=\frac{3 (3 b c-4 a d) x \sqrt{c+d x^2}}{8 b^3}+\frac{3 d x^3 \sqrt{c+d x^2}}{4 b^2}-\frac{x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac{3 \sqrt{a} (b c-2 a d) \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^4}+\frac{3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4 \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.169172, size = 192, normalized size = 0.97 \[ \frac{\frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}-\frac{12 \sqrt{a} \left (2 a^2 d^2-3 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{b c-a d}}+\frac{b \sqrt{c+d x^2} \left (-12 a^2 d x+a b \left (9 c x-6 d x^3\right )+b^2 x^3 \left (5 c+2 d x^2\right )\right )}{a+b x^2}}{8 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]
[Out]
((b*Sqrt[c + d*x^2]*(-12*a^2*d*x + b^2*x^3*(5*c + 2*d*x^2) + a*b*(9*c*x - 6*d*x^3)))/(a + b*x^2) - (12*Sqrt[a]
*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[b*c - a*d] + (3
*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(8*b^4)
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Maple [B] time = 0.021, size = 4795, normalized size = 24.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x)
[Out]
-1/4/b^2*a/(a*d-b*c)/(x-1/b*(-a*b)^(1/2))*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a
*d-b*c)/b)^(5/2)+3/4/b^4*a^3*d^(5/2)/(a*d-b*c)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-
a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/4/b^2*a/(a*d-b*c)/(x+1/b*(-a*b)^
(1/2))*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)+3/4/b^4*a^3*d^(5/2
)/(a*d-b*c)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b
*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-9/8/b^3*a*d^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/
2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c+3/4/b^3*a^2/(-a*b)^
(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d-3/4/b^2*a/(-a*b)^
(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c-3/8/b^3*a*d*((x-1
/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+3/4/b^2*a/(-a*b)^(1/2)*((x+1
/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c-3/8/b^3*a*d*((x+1/b*(-a*b)^(
1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-9/8/b^3*a*d^(1/2)*ln((-d*(-a*b)^(1/2)/b
+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(1/2))*c+3/4/b^5*a^3*d^3*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(
x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(
a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))+3/8/b^2*a*d/(a*d-b*c)*c*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*
(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-3/4/b^3*a*d*(-a*b)^(1/2)/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a
*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c-3/4/b^3*a^2/(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d-3/4/b^4*a^2*d^2*(-a*b)^(1/2)/(a*d-b*c)*((x-1/b*(-a*b)^
(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/4/b^2*a*d/(a*d-b*c)*((x-1/b*(-a*b)^(1/
2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+3/8/b^2*a*d^(1/2)/(a*d-b*c)*c^2*ln((d*(-a
*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-
(a*d-b*c)/b)^(1/2))+3/8/b^2*c*x*(d*x^2+c)^(1/2)+3/8/b^2*c^2/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/4/b^2*a/(-
a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+3/4/b^4*a^2*d^
(3/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b
*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/4/b^2*a/(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*
(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+3/4/b^4*a^2*d^(3/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x
+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/4/b^2*x*(d*x^2+c)^(3/2)+3
/2/b^3*a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a
*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*
(-a*b)^(1/2)))*d*c+3/4/b^3*a*d*(-a*b)^(1/2)/(a*d-b*c)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*
b)^(1/2))-(a*d-b*c)/b)^(1/2)*c-3/4/b^5*a^3*d^3*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+
2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x
-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))+3/8/b^2*a*d/(a*d-b*c)*c*((x-1/b*(-a*b)^(1/2))^2*d
+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-3/2/b^4*a^2*d^2*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*
c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(
1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*c+3/4/b^3*a*d*(-a*
b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*
c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b
)^(1/2)))*c^2+3/2/b^4*a^2*d^2*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/
b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2)
)-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c-3/4/b^3*a*d*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(
a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)
^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c^2-3/2/b^3*a^2/(-a*b)^(1/2)/(-(a*d-b*
c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(
1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d*c+1/4/b^3*a*d*(-
a*b)^(1/2)/(a*d-b*c)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-3/8/
b^3*a^2*d^2/(a*d-b*c)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-9
/8/b^3*a^2*d^(3/2)/(a*d-b*c)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c+3/4/b^4*a^3/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((
-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-
a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d^2+3/4/b^2*a/(-a*b)^(1/2)/(-(a*d-
b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)
^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c^2-3/4/b^4*a^3/
(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^
(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2
)))*d^2-3/4/b^2*a/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))
+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/
(x+1/b*(-a*b)^(1/2)))*c^2-9/8/b^3*a^2*d^(3/2)/(a*d-b*c)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+
((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c+3/4/b^4*a^2*d^2*(-a*b)
^(1/2)/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/4/b^2*
a*d/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+3/8/b^2*a
*d^(1/2)/(a*d-b*c)*c^2*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a
*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/4/b^3*a*d*(-a*b)^(1/2)/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))^
2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-3/8/b^3*a^2*d^2/(a*d-b*c)*((x+1/b*(-a*b)^(1/2))
^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate((d*x^2 + c)^(3/2)*x^4/(b*x^2 + a)^2, x)
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Fricas [A] time = 3.48654, size = 2678, normalized size = 13.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x, algorithm="fricas")
[Out]
[1/16*(3*(a*b^2*c^2 - 8*a^2*b*c*d + 8*a^3*d^2 + (b^3*c^2 - 8*a*b^2*c*d + 8*a^2*b*d^2)*x^2)*sqrt(d)*log(-2*d*x^
2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 6*(a*b*c*d - 2*a^2*d^2 + (b^2*c*d - 2*a*b*d^2)*x^2)*sqrt(-a*b*c + a^2*d
)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3
- a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(2*b^3*d^2*x^5 + (5*b^3*c*d -
6*a*b^2*d^2)*x^3 + 3*(3*a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(d*x^2 + c))/(b^5*d*x^2 + a*b^4*d), -1/8*(3*(a*b^2*c^2
- 8*a^2*b*c*d + 8*a^3*d^2 + (b^3*c^2 - 8*a*b^2*c*d + 8*a^2*b*d^2)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2
+ c)) + 3*(a*b*c*d - 2*a^2*d^2 + (b^2*c*d - 2*a*b*d^2)*x^2)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8
*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*s
qrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - (2*b^3*d^2*x^5 + (5*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(3*a*b^2*c*d
- 4*a^2*b*d^2)*x)*sqrt(d*x^2 + c))/(b^5*d*x^2 + a*b^4*d), -1/16*(12*(a*b*c*d - 2*a^2*d^2 + (b^2*c*d - 2*a*b*d
^2)*x^2)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*
d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 3*(a*b^2*c^2 - 8*a^2*b*c*d + 8*a^3*d^2 + (b^3*c^2 - 8*a*b^2*c*d +
8*a^2*b*d^2)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 2*(2*b^3*d^2*x^5 + (5*b^3*c*d - 6
*a*b^2*d^2)*x^3 + 3*(3*a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(d*x^2 + c))/(b^5*d*x^2 + a*b^4*d), -1/8*(6*(a*b*c*d -
2*a^2*d^2 + (b^2*c*d - 2*a*b*d^2)*x^2)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 -
a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 3*(a*b^2*c^2 - 8*a^2*b*c*d + 8*a^3*
d^2 + (b^3*c^2 - 8*a*b^2*c*d + 8*a^2*b*d^2)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (2*b^3*d^2*x^5
+ (5*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(3*a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(d*x^2 + c))/(b^5*d*x^2 + a*b^4*d)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (c + d x^{2}\right )^{\frac{3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4*(d*x**2+c)**(3/2)/(b*x**2+a)**2,x)
[Out]
Integral(x**4*(c + d*x**2)**(3/2)/(a + b*x**2)**2, x)
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Giac [B] time = 1.19291, size = 532, normalized size = 2.7 \begin{align*} \frac{1}{8} \, \sqrt{d x^{2} + c} x{\left (\frac{2 \, d x^{2}}{b^{2}} + \frac{5 \, b^{7} c d^{2} - 8 \, a b^{6} d^{3}}{b^{9} d^{2}}\right )} + \frac{3 \,{\left (a b^{2} c^{2} \sqrt{d} - 3 \, a^{2} b c d^{\frac{3}{2}} + 2 \, a^{3} d^{\frac{5}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt{a b c d - a^{2} d^{2}} b^{4}} - \frac{3 \,{\left (b^{2} c^{2} \sqrt{d} - 8 \, a b c d^{\frac{3}{2}} + 8 \, a^{2} d^{\frac{5}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{16 \, b^{4} d} - \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b^{2} c^{2} \sqrt{d} - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac{3}{2}} + 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{3} d^{\frac{5}{2}} - a b^{2} c^{3} \sqrt{d} + a^{2} b c^{2} d^{\frac{3}{2}}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x, algorithm="giac")
[Out]
1/8*sqrt(d*x^2 + c)*x*(2*d*x^2/b^2 + (5*b^7*c*d^2 - 8*a*b^6*d^3)/(b^9*d^2)) + 3/2*(a*b^2*c^2*sqrt(d) - 3*a^2*b
*c*d^(3/2) + 2*a^3*d^(5/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^
2))/(sqrt(a*b*c*d - a^2*d^2)*b^4) - 3/16*(b^2*c^2*sqrt(d) - 8*a*b*c*d^(3/2) + 8*a^2*d^(5/2))*log((sqrt(d)*x -
sqrt(d*x^2 + c))^2)/(b^4*d) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b^2*c^2*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 +
c))^2*a^2*b*c*d^(3/2) + 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^3*d^(5/2) - a*b^2*c^3*sqrt(d) + a^2*b*c^2*d^(3/2)
)/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c)
)^2*a*d + b*c^2)*b^4)