\(\int \frac {x^2}{(a+b x^2)^2 (c+d x^2)^{3/2}} \, dx\) [769]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {482, 541, 12, 385, 211} \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {(2 a d+b c) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}}-\frac {x}{2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}-\frac {3 d x}{2 \sqrt {c+d x^2} (b c-a d)^2} \]

[In]

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

[Out]

(-3*d*x)/(2*(b*c - a*d)^2*Sqrt[c + d*x^2]) - x/(2*(b*c - a*d)*(a + b*x^2)*Sqrt[c + d*x^2]) + ((b*c + 2*a*d)*Ar
cTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*(b*c - a*d)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 482

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 + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 541

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\int \frac {c-2 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{2 (b c-a d)} \\ & = -\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\int \frac {c (b c+2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 c (b c-a d)^2} \\ & = -\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 a d) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)^2} \\ & = -\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 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 c-a d)^2} \\ & = -\frac {3 d x}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(b c+2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{5/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1437\) vs. \(2(123)=246\).

Time = 9.77 (sec) , antiderivative size = 1437, normalized size of antiderivative = 11.68 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (-\frac {2 d x \left (-16 c^{5/2}-20 c^{3/2} d x^2-5 \sqrt {c} d^2 x^4+16 c^2 \sqrt {c+d x^2}+12 c d x^2 \sqrt {c+d x^2}+d^2 x^4 \sqrt {c+d x^2}\right )}{(b c-a d)^2 \left (c+d x^2-\sqrt {c} \sqrt {c+d x^2}\right ) \left (8 c^2+8 c d x^2+d^2 x^4-8 c^{3/2} \sqrt {c+d x^2}-4 \sqrt {c} d x^2 \sqrt {c+d x^2}\right )}+\frac {b x \left (32 c^{7/2}+64 c^{5/2} d x^2+38 c^{3/2} d^2 x^4+6 \sqrt {c} d^3 x^6-32 c^3 \sqrt {c+d x^2}-48 c^2 d x^2 \sqrt {c+d x^2}-18 c d^2 x^4 \sqrt {c+d x^2}-d^3 x^6 \sqrt {c+d x^2}\right )}{(b c-a d)^2 \left (a+b x^2\right ) \left (2 c+d x^2-2 \sqrt {c} \sqrt {c+d x^2}\right ) \left (8 c^2+8 c d x^2+d^2 x^4-8 c^{3/2} \sqrt {c+d x^2}-4 \sqrt {c} d x^2 \sqrt {c+d x^2}\right )}+\frac {b^{3/2} c^{3/2} \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} (b c-a d)^{5/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {c} 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 )}{(b c-a d)^{5/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {b c \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} (b c-a d)^2 \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 \sqrt {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 )}{(b c-a d)^2 \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {b^{3/2} c^{3/2} \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} (b c-a d)^{5/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {c} 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 )}{(b c-a d)^{5/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {b c \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} (b c-a d)^2 \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 \sqrt {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 )}{(b c-a d)^2 \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}\right ) \]

[In]

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

[Out]

((-2*d*x*(-16*c^(5/2) - 20*c^(3/2)*d*x^2 - 5*Sqrt[c]*d^2*x^4 + 16*c^2*Sqrt[c + d*x^2] + 12*c*d*x^2*Sqrt[c + d*
x^2] + d^2*x^4*Sqrt[c + d*x^2]))/((b*c - a*d)^2*(c + d*x^2 - Sqrt[c]*Sqrt[c + d*x^2])*(8*c^2 + 8*c*d*x^2 + d^2
*x^4 - 8*c^(3/2)*Sqrt[c + d*x^2] - 4*Sqrt[c]*d*x^2*Sqrt[c + d*x^2])) + (b*x*(32*c^(7/2) + 64*c^(5/2)*d*x^2 + 3
8*c^(3/2)*d^2*x^4 + 6*Sqrt[c]*d^3*x^6 - 32*c^3*Sqrt[c + d*x^2] - 48*c^2*d*x^2*Sqrt[c + d*x^2] - 18*c*d^2*x^4*S
qrt[c + d*x^2] - d^3*x^6*Sqrt[c + d*x^2]))/((b*c - a*d)^2*(a + b*x^2)*(2*c + d*x^2 - 2*Sqrt[c]*Sqrt[c + d*x^2]
)*(8*c^2 + 8*c*d*x^2 + d^2*x^4 - 8*c^(3/2)*Sqrt[c + d*x^2] - 4*Sqrt[c]*d*x^2*Sqrt[c + d*x^2])) + (b^(3/2)*c^(3
/2)*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]*(b*c - a*d)^(5/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*Sqrt[a]*Sqrt[b]*Sqrt[c]*
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
*c - a*d)^(5/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (b*c*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqr
t[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(Sqrt[a]*(b*c - a*d)^2*Sqrt[2*b*c -
a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*Sqrt[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*c - a*d)^2*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[
b*c - a*d]]) + (b^(3/2)*c^(3/2)*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[a]*(b*c - a*d)^(5/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]])
+ (2*Sqrt[a]*Sqrt[b]*Sqrt[c]*d*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqr
t[c] + Sqrt[c + d*x^2]))])/((b*c - a*d)^(5/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (b*c*Ar
cTan[(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]*(b*c - a*d)^2*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*Sqrt[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*c - a*d)^2*Sqrt[2*b*c -
a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]))/2

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {-\frac {b \sqrt {d \,x^{2}+c}\, x}{2 \left (b \,x^{2}+a \right )}+\frac {\left (2 a d +b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{2 \sqrt {\left (a d -b c \right ) a}}-\frac {d x}{\sqrt {d \,x^{2}+c}}}{\left (a d -b c \right )^{2}}\)	\(97\)
default	\(\text {Expression too large to display}\)	\(1922\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (103) = 206\).

Time = 0.48 (sec) , antiderivative size = 744, normalized size of antiderivative = 6.05 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + a b c^{2} + 2 \, a^{2} c d + {\left (b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \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 ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 3 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{4} + {\left (a b^{4} c^{4} - 2 \, a^{2} b^{3} c^{3} d + 2 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{2}\right )}}, \frac {{\left ({\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + a b c^{2} + 2 \, a^{2} c d + {\left (b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (3 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 3 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{4} + {\left (a b^{4} c^{4} - 2 \, a^{2} b^{3} c^{3} d + 2 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/8*(((b^2*c*d + 2*a*b*d^2)*x^4 + a*b*c^2 + 2*a^2*c*d + (b^2*c^2 + 3*a*b*c*d + 2*a^2*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)) + 4*(3*(a*b^2*c*d - a^2*b*d^
2)*x^3 + (a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2)*x)*sqrt(d*x^2 + c))/(a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*
d^2 - a^5*c*d^3 + (a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 3*a^3*b^2*c*d^3 - a^4*b*d^4)*x^4 + (a*b^4*c^4 - 2*a^2*b^3
*c^3*d + 2*a^4*b*c*d^3 - a^5*d^4)*x^2), 1/4*(((b^2*c*d + 2*a*b*d^2)*x^4 + a*b*c^2 + 2*a^2*c*d + (b^2*c^2 + 3*a
*b*c*d + 2*a^2*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)) - 2*(3*(a*b^2*c*d - a^2*b*d^2)*x^3 + (a*b^2*c^2 + a
^2*b*c*d - 2*a^3*d^2)*x)*sqrt(d*x^2 + c))/(a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3 + (a*b^
4*c^3*d - 3*a^2*b^3*c^2*d^2 + 3*a^3*b^2*c*d^3 - a^4*b*d^4)*x^4 + (a*b^4*c^4 - 2*a^2*b^3*c^3*d + 2*a^4*b*c*d^3
- a^5*d^4)*x^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (103) = 206\).

Time = 0.88 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.43 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d x}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {d x^{2} + c}} - \frac {{\left (b c \sqrt {d} + 2 \, a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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