\(\int \frac {x^2 (c+d x^2)^{3/2}}{(a+b x^2)^2} \, dx\) [742]

Optimal result

Integrand size = 24, antiderivative size = 149 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {d x \sqrt {c+d x^2}}{b^2}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(b c-4 a d) \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^3}+\frac {\sqrt {d} (3 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {478, 542, 537, 223, 212, 385, 211} \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-4 a d) \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^3}+\frac {\sqrt {d} (3 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {d x \sqrt {c+d x^2}}{b^2} \]

[In]

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

[Out]

(d*x*Sqrt[c + d*x^2])/b^2 - (x*(c + d*x^2)^(3/2))/(2*b*(a + b*x^2)) + ((b*c - 4*a*d)*Sqrt[b*c - a*d]*ArcTan[(S
qrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*b^3) + (Sqrt[d]*(3*b*c - 4*a*d)*ArcTanh[(Sqrt[d]*x)/S
qrt[c + d*x^2]])/(2*b^3)

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 542

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\sqrt {c+d x^2} \left (c+4 d x^2\right )}{a+b x^2} \, dx}{2 b} \\ & = \frac {d x \sqrt {c+d x^2}}{b^2}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {2 c (b c-2 a d)+2 d (3 b c-4 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{4 b^2} \\ & = \frac {d x \sqrt {c+d x^2}}{b^2}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(d (3 b c-4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 b^3}+\frac {((b c-4 a d) (b c-a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b^3} \\ & = \frac {d x \sqrt {c+d x^2}}{b^2}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(d (3 b c-4 a d)) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 b^3}+\frac {((b c-4 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^3} \\ & = \frac {d x \sqrt {c+d x^2}}{b^2}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(b c-4 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 \sqrt {a} b^3}+\frac {\sqrt {d} (3 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1365\) vs. \(2(149)=298\).

Time = 5.97 (sec) , antiderivative size = 1365, normalized size of antiderivative = 9.16 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {b x \left (-b c+2 a d+b d x^2\right ) \left (64 c^4+144 c^3 d x^2+104 c^2 d^2 x^4+25 c d^3 x^6+d^4 x^8-64 c^{7/2} \sqrt {c+d x^2}-112 c^{5/2} d x^2 \sqrt {c+d x^2}-56 c^{3/2} d^2 x^4 \sqrt {c+d x^2}-7 \sqrt {c} d^3 x^6 \sqrt {c+d x^2}\right )}{\left (a+b x^2\right ) \left (-64 c^{7/2}-112 c^{5/2} d x^2-56 c^{3/2} d^2 x^4-7 \sqrt {c} d^3 x^6+64 c^3 \sqrt {c+d x^2}+80 c^2 d x^2 \sqrt {c+d x^2}+24 c d^2 x^4 \sqrt {c+d x^2}+d^3 x^6 \sqrt {c+d x^2}\right )}+\frac {b^{3/2} c^{3/2} \sqrt {b c-a d} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {4 \sqrt {a} \sqrt {b} \sqrt {c} d \sqrt {b c-a d} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {4 \sqrt {a} \sqrt {b} \sqrt {c} d \sqrt {b c-a d} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {b c (b c-a d) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {4 \sqrt {a} d (-b c+a d) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {b^{3/2} c^{3/2} \sqrt {b c-a d} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {b c (b c-a d) \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {a} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {4 \sqrt {a} d (-b c+a d) \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+8 a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )+6 b c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{2 b^3} \]

[In]

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

[Out]

((b*x*(-(b*c) + 2*a*d + b*d*x^2)*(64*c^4 + 144*c^3*d*x^2 + 104*c^2*d^2*x^4 + 25*c*d^3*x^6 + d^4*x^8 - 64*c^(7/
2)*Sqrt[c + d*x^2] - 112*c^(5/2)*d*x^2*Sqrt[c + d*x^2] - 56*c^(3/2)*d^2*x^4*Sqrt[c + d*x^2] - 7*Sqrt[c]*d^3*x^
6*Sqrt[c + d*x^2]))/((a + b*x^2)*(-64*c^(7/2) - 112*c^(5/2)*d*x^2 - 56*c^(3/2)*d^2*x^4 - 7*Sqrt[c]*d^3*x^6 + 6
4*c^3*Sqrt[c + d*x^2] + 80*c^2*d*x^2*Sqrt[c + d*x^2] + 24*c*d^2*x^4*Sqrt[c + d*x^2] + d^3*x^6*Sqrt[c + d*x^2])
) + (b^(3/2)*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]))])/(Sqrt[a]*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (4*Sqrt[a]*S
qrt[b]*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]*(Sq
rt[c] - Sqrt[c + d*x^2]))])/Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] + (4*Sqrt[a]*Sqrt[b]*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]))])/Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] + (b*c*(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]))])/(Sqrt[a]*Sqrt[2*b*c - a*d
- 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (4*Sqrt[a]*d*(-(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]))])/Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c
- a*d]] + (b^(3/2)*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)/(S
qrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(Sqrt[a]*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (b*c*
(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]))])/(Sqrt[a]*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (4*Sqrt[a]*d*(-(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]))])/Sqrt[2*b*c -
a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] + 8*a*d^(3/2)*ArcTanh[(Sqrt[d]*x)/(Sqrt[c] - Sqrt[c + d*x^2])] + 6*b
*c*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(2*b^3)

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(-\frac {-\sqrt {d}\, \left (\sqrt {d \,x^{2}+c}\, b x \sqrt {d}-4 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) a d +3 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) b c \right )+\left (a d -b c \right ) \left (-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 b^{3}}\)	\(152\)
risch	\(\frac {d x \sqrt {d \,x^{2}+c}}{2 b^{2}}-\frac {\frac {\sqrt {d}\, \left (4 a d -3 b c \right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b}+\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} 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 {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} 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 {\left (-5 a^{2} d^{2}+6 a b c d -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 )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (5 a^{2} d^{2}-6 a b c d +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 )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{2 b^{2}}\)	\(957\)
default	\(\text {Expression too large to display}\)	\(3381\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 996, normalized size of antiderivative = 6.68 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {2 \, {\left (3 \, a b c - 4 \, a^{2} d + {\left (3 \, b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} d x^{3} - {\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, -\frac {4 \, {\left (3 \, a b c - 4 \, a^{2} d + {\left (3 \, b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} d x^{3} - {\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {{\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - {\left (3 \, a b c - 4 \, a^{2} d + {\left (3 \, b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (b^{2} d x^{3} - {\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, -\frac {2 \, {\left (3 \, a b c - 4 \, a^{2} d + {\left (3 \, b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - 2 \, {\left (b^{2} d x^{3} - {\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (123) = 246\).

Time = 0.31 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.26 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} d x}{2 \, b^{2}} - \frac {{\left (3 \, b c \sqrt {d} - 4 \, a d^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3}} - \frac {{\left (b^{2} c^{2} \sqrt {d} - 5 \, a b c d^{\frac {3}{2}} + 4 \, a^{2} d^{\frac {5}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{3}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {5}{2}} - b^{2} c^{3} \sqrt {d} + a b c^{2} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{3}} \]

[In]

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

[Out]

1/2*sqrt(d*x^2 + c)*d*x/b^2 - 1/4*(3*b*c*sqrt(d) - 4*a*d^(3/2))*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/b^3 - 1/2
*(b^2*c^2*sqrt(d) - 5*a*b*c*d^(3/2) + 4*a^2*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^3) + ((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^2*sqrt(d)
- 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c*d^(3/2) + 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(5/2) - b^2*c^3*sq
rt(d) + a*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^3)

Mupad [F(-1)]

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

[In]

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

[Out]

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