\(\int \frac {x^4 (c+d x^2)^{3/2}}{(a+b x^2)^2} \, dx\) [740]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {3 (3 b c-4 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x^3 \sqrt {c+d x^2}}{4 b^2}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} (b c-2 a d) \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^4}+\frac {3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4 \sqrt {d}} \]

[Out]

-1/2*x^3*(d*x^2+c)^(3/2)/b/(b*x^2+a)+3/8*(8*a^2*d^2-8*a*b*c*d+b^2*c^2)*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/b^4/
d^(1/2)-3/2*(-2*a*d+b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))*a^(1/2)*(-a*d+b*c)^(1/2)/b^4+3/8*(
-4*a*d+3*b*c)*x*(d*x^2+c)^(1/2)/b^3+3/4*d*x^3*(d*x^2+c)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {478, 595, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4 \sqrt {d}}-\frac {3 \sqrt {a} (b c-2 a d) \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^4}+\frac {3 x \sqrt {c+d x^2} (3 b c-4 a d)}{8 b^3}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {3 d x^3 \sqrt {c+d x^2}}{4 b^2} \]

[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 211

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 385

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 478

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 537

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 595

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 596

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]

Rubi steps \begin{align*} \text {integral}& = -\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)) \text {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 ) \text {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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3517\) vs. \(2(197)=394\).

Time = 14.75 (sec) , antiderivative size = 3517, normalized size of antiderivative = 17.85 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Result too large to show} \]

[In]

Integrate[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]

[Out]

(-32*a*b*c^(9/2)*x + 6*a^2*Sqrt[c]*d^4*x^7 + 32*a*b*c^4*x*Sqrt[c + d*x^2] - a^2*d^4*x^7*Sqrt[c + d*x^2] - a*c^
3*x*Sqrt[c + d*x^2]*(32*a*d - 48*b*d*x^2) - a*c^(7/2)*x*(-32*a*d + 64*b*d*x^2) - a*c^2*x*Sqrt[c + d*x^2]*(48*a
*d^2*x^2 - 18*b*d^2*x^4) - a*c^(5/2)*x*(-64*a*d^2*x^2 + 38*b*d^2*x^4) - a*c*x*Sqrt[c + d*x^2]*(18*a*d^3*x^4 -
b*d^3*x^6) - a*c^(3/2)*x*(-38*a*d^3*x^4 + 6*b*d^3*x^6))/(64*b^3*c^3*(a + b*x^2) + 96*b^3*c^2*d*x^2*(a + b*x^2)
+ 36*b^3*c*d^2*x^4*(a + b*x^2) + 2*b^3*d^3*x^6*(a + b*x^2) - 64*b^3*c^(5/2)*(a + b*x^2)*Sqrt[c + d*x^2] - 64*
b^3*c^(3/2)*d*x^2*(a + b*x^2)*Sqrt[c + d*x^2] - 12*b^3*Sqrt[c]*d^2*x^4*(a + b*x^2)*Sqrt[c + d*x^2]) + (3*Sqrt[
a]*c^(3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c]
+ Sqrt[c + d*x^2]))])/(2*b^(5/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) - (3*a^(3/2)*Sqrt[c]*
d*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c
+ d*x^2]))])/(b^(7/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + ((-3*Sqrt[a]*c^2)/(2*b^2*Sqrt[
2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*a^(3/2)*c*d)/(2*b^3*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c
]*Sqrt[b*c - a*d]]))*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqr
t[c + d*x^2]))] + ((3*a^(3/2)*c*d)/(b^3*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) - (3*a^(5/2)*d^
2)/(b^4*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]))*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*S
qrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))] - (3*Sqrt[a]*c^(3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2
*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(2*b^(5/2)*Sqrt[2*
b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*a^(3/2)*Sqrt[c]*d*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b*c - a*
d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(b^(7/2)*Sqrt[2*b*c - a*d +
2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + ((-3*Sqrt[a]*c^2)/(2*b^2*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c
- a*d]]) + (3*a^(3/2)*c*d)/(2*b^3*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]))*ArcTan[(Sqrt[2*b*c -
a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))] + ((3*a^(3/2)*c*d)/(b^3*S
qrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) - (3*a^(5/2)*d^2)/(b^4*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt
[c]*Sqrt[b*c - a*d]]))*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + S
qrt[c + d*x^2]))] + (2*c^2*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(b^2*Sqrt[d]) - (8*a*c*Sqrt[d]*A
rcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^3 + (6*a^2*d^(3/2)*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c
+ d*x^2])])/b^4 + ((-8*c^(5/2)*x)/b^2 + (8*c^2*x*Sqrt[c + d*x^2])/b^2 - (16*c^3*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c]
+ Sqrt[c + d*x^2])])/(b^2*Sqrt[d]) + (16*c^(5/2)*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^
2])])/(b^2*Sqrt[d]))/((-Sqrt[c] - Sqrt[d]*x + Sqrt[c + d*x^2])*(-Sqrt[c] + Sqrt[d]*x + Sqrt[c + d*x^2])) + ((7
2*c^(7/2)*x)/b^2 + (24*c^(5/2)*d*x^3)/b^2 - (72*c^3*x*Sqrt[c + d*x^2])/b^2 + (12*c^2*d*x^3*Sqrt[c + d*x^2])/b^
2 + (144*c^4*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(b^2*Sqrt[d]) + (72*c^3*Sqrt[d]*x^2*ArcTanh[(S
qrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^2 - (144*c^(7/2)*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sq
rt[c + d*x^2])])/(b^2*Sqrt[d]))/((-Sqrt[c] - Sqrt[d]*x + Sqrt[c + d*x^2])^2*(-Sqrt[c] + Sqrt[d]*x + Sqrt[c + d
*x^2])^2) + ((8*a*c^(3/2)*d*x)/b^3 - (8*a*c*d*x*Sqrt[c + d*x^2])/b^3 + (16*a*c^2*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/(
-Sqrt[c] + Sqrt[c + d*x^2])])/b^3 - (16*a*c^(3/2)*Sqrt[d]*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt
[c + d*x^2])])/b^3)/((-Sqrt[c] - Sqrt[d]*x + Sqrt[c + d*x^2])*(-Sqrt[c] + Sqrt[d]*x + Sqrt[c + d*x^2])) + ((-2
4*a*c^(5/2)*d*x)/b^3 - (8*a*c^(3/2)*d^2*x^3)/b^3 + (24*a*c^2*d*x*Sqrt[c + d*x^2])/b^3 - (4*a*c*d^2*x^3*Sqrt[c
+ d*x^2])/b^3 - (48*a*c^3*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^3 - (24*a*c^2*d^(3/2)*x
^2*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^3 + (48*a*c^(5/2)*Sqrt[d]*Sqrt[c + d*x^2]*ArcTanh[(Sqr
t[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^3)/((-Sqrt[c] - Sqrt[d]*x + Sqrt[c + d*x^2])^2*(-Sqrt[c] + Sqrt[d]*x
+ Sqrt[c + d*x^2])^2) + ((-320*c^(9/2)*x)/b^2 - (560*c^(7/2)*d*x^3)/(3*b^2) + (16*c^(5/2)*d^2*x^5)/b^2 + (320*
c^4*x*Sqrt[c + d*x^2])/b^2 + (80*c^3*d*x^3*Sqrt[c + d*x^2])/(3*b^2) + (32*c^2*d^2*x^5*Sqrt[c + d*x^2])/(3*b^2)
- (640*c^5*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(b^2*Sqrt[d]) - (480*c^4*Sqrt[d]*x^2*ArcTanh[(S
qrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^2 + (640*c^(9/2)*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sq
rt[c + d*x^2])])/(b^2*Sqrt[d]) + (160*c^(7/2)*Sqrt[d]*x^2*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt
[c + d*x^2])])/b^2)/((-Sqrt[c] - Sqrt[d]*x + Sqrt[c + d*x^2])^3*(-Sqrt[c] + Sqrt[d]*x + Sqrt[c + d*x^2])^3) +
((560*c^(11/2)*x)/b^2 + (1400*c^(9/2)*d*x^3)/(3*b^2) + (56*c^(7/2)*d^2*x^5)/(3*b^2) + (16*c^(5/2)*d^3*x^7)/(3*
b^2) - (560*c^5*x*Sqrt[c + d*x^2])/b^2 - (560*c^4*d*x^3*Sqrt[c + d*x^2])/(3*b^2) + (14*c^3*d^2*x^5*Sqrt[c + d*
x^2])/(3*b^2) + (4*c^2*d^3*x^7*Sqrt[c + d*x^2])/b^2 + (1120*c^6*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2
])])/(b^2*Sqrt[d]) + (1120*c^5*Sqrt[d]*x^2*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^2 + (140*c^4*d
^(3/2)*x^4*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^2 - (1120*c^(11/2)*Sqrt[c + d*x^2]*ArcTanh[(Sq
rt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(b^2*Sqrt[d]) - (560*c^(9/2)*Sqrt[d]*x^2*Sqrt[c + d*x^2]*ArcTanh[(Sqrt
[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/b^2)/((-Sqrt[c] - Sqrt[d]*x + Sqrt[c + d*x^2])^4*(-Sqrt[c] + Sqrt[d]*x +
Sqrt[c + d*x^2])^4)

Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\frac {\frac {b \sqrt {d \,x^{2}+c}\, \left (-2 b d \,x^{2}+8 a d -5 b c \right ) x}{2}-\frac {3 \left (8 a^{2} d^{2}-8 a b c d +b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{2 \sqrt {d}}-2 \left (a d -b c \right ) a \left (-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {3 \left (2 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{4 b^{4}}\)	\(164\)
risch	\(-\frac {x \left (-2 b d \,x^{2}+8 a d -5 b c \right ) \sqrt {d \,x^{2}+c}}{8 b^{3}}+\frac {\frac {3 \left (8 a^{2} d^{2}-8 a b c d +b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {2 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}-\frac {2 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}-\frac {2 a \left (7 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {2 a \left (7 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 b^{3}}\)	\(991\)
default	\(\text {Expression too large to display}\)	\(3440\)

[In]

int(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4/b^4*(1/2*b*(d*x^2+c)^(1/2)*(-2*b*d*x^2+8*a*d-5*b*c)*x-3/2*(8*a^2*d^2-8*a*b*c*d+b^2*c^2)/d^(1/2)*arctanh((
d*x^2+c)^(1/2)/x/d^(1/2))-2*(a*d-b*c)*a*(-b*(d*x^2+c)^(1/2)*x/(b*x^2+a)-3*(2*a*d-b*c)/((a*d-b*c)*a)^(1/2)*arct
anh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 1249, normalized size of antiderivative = 6.34 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[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)]

Sympy [F]

\[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (165) = 330\).

Time = 0.32 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.97 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {1}{8} \, \sqrt {d x^{2} + c} x {\left (\frac {2 \, d x^{2}}{b^{2}} + \frac {5 \, b^{7} c d^{2} - 8 \, a b^{6} d^{3}}{b^{9} d^{2}}\right )} + \frac {3 \, {\left (a b^{2} c^{2} \sqrt {d} - 3 \, a^{2} b c d^{\frac {3}{2}} + 2 \, a^{3} d^{\frac {5}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{4}} - \frac {3 \, {\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{16 \, b^{4} \sqrt {d}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {5}{2}} - a b^{2} c^{3} \sqrt {d} + a^{2} b c^{2} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{4}} \]

[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 - 8*a*b*c*d + 8*a^2*d^2)*log((sqrt(d)*x - sqrt(d*x^2 + c))^2
)/(b^4*sqrt(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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

int((x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x)

[Out]

int((x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2, x)