\(\int \frac {x^{9/2}}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [812]

Optimal result

Integrand size = 24, antiderivative size = 474 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {(b c+a d) x^{3/2}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a x^{3/2}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {a^{3/4} (7 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{3/4} (b c-a d)^3}-\frac {a^{3/4} (7 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{3/4} (b c-a d)^3}-\frac {c^{3/4} (b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{3/4} (b c-a d)^3}+\frac {c^{3/4} (b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{3/4} (b c-a d)^3}+\frac {a^{3/4} (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} b^{3/4} (b c-a d)^3}-\frac {c^{3/4} (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} d^{3/4} (b c-a d)^3} \] Output:

1/2*(a*d+b*c)*x^(3/2)/b/(-a*d+b*c)^2/(d*x^2+c)+1/2*a*x^(3/2)/b/(-a*d+b*c)/ 
(b*x^2+a)/(d*x^2+c)+1/8*a^(3/4)*(a*d+7*b*c)*arctan(1-2^(1/2)*b^(1/4)*x^(1/ 
2)/a^(1/4))*2^(1/2)/b^(3/4)/(-a*d+b*c)^3-1/8*a^(3/4)*(a*d+7*b*c)*arctan(1+ 
2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/b^(3/4)/(-a*d+b*c)^3-1/8*c^(3/4)* 
(7*a*d+b*c)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/d^(3/4)/(-a* 
d+b*c)^3+1/8*c^(3/4)*(7*a*d+b*c)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4)) 
*2^(1/2)/d^(3/4)/(-a*d+b*c)^3+1/8*a^(3/4)*(a*d+7*b*c)*arctanh(2^(1/2)*a^(1 
/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/b^(3/4)/(-a*d+b*c)^3-1/8* 
c^(3/4)*(7*a*d+b*c)*arctanh(2^(1/2)*c^(1/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/ 
2)*x))*2^(1/2)/d^(3/4)/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.72 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {1}{8} \left (\frac {4 x^{3/2} \left (2 a c+b c x^2+a d x^2\right )}{(b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} a^{3/4} (7 b c+a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4} (b c-a d)^3}+\frac {\sqrt {2} c^{3/4} (b c+7 a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{3/4} (-b c+a d)^3}+\frac {\sqrt {2} a^{3/4} (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4} (b c-a d)^3}+\frac {\sqrt {2} c^{3/4} (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{3/4} (-b c+a d)^3}\right ) \] Input:

Integrate[x^(9/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]
 

Output:

((4*x^(3/2)*(2*a*c + b*c*x^2 + a*d*x^2))/((b*c - a*d)^2*(a + b*x^2)*(c + d 
*x^2)) + (Sqrt[2]*a^(3/4)*(7*b*c + a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt 
[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(b^(3/4)*(b*c - a*d)^3) + (Sqrt[2]*c^(3/4)* 
(b*c + 7*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x 
])])/(d^(3/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*a^(3/4)*(7*b*c + a*d)*ArcTanh[( 
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(b^(3/4)*(b*c - a 
*d)^3) + (Sqrt[2]*c^(3/4)*(b*c + 7*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*S 
qrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(d^(3/4)*(-(b*c) + a*d)^3))/8
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.36, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {368, 970, 1049, 27, 1054, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle 2 \int \frac {x^5}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 970

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\int \frac {x \left (3 a c-(4 b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1049

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {\int \frac {4 b c x \left (6 a c-(b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {x^{3/2} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \int \frac {x \left (6 a c-(b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{b c-a d}-\frac {x^{3/2} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1054

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \int \left (\frac {a (7 b c+a d) x}{(b c-a d) \left (b x^2+a\right )}-\frac {c (b c+7 a d) x}{(b c-a d) \left (d x^2+c\right )}\right )d\sqrt {x}}{b c-a d}-\frac {x^{3/2} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}-\frac {\frac {b \left (-\frac {a^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (a d+7 b c)}{2 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (a d+7 b c)}{2 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{3/4} (a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} b^{3/4} (b c-a d)}-\frac {a^{3/4} (a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} b^{3/4} (b c-a d)}+\frac {c^{3/4} (7 a d+b c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} (7 a d+b c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} (7 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} d^{3/4} (b c-a d)}+\frac {c^{3/4} (7 a d+b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} d^{3/4} (b c-a d)}\right )}{b c-a d}-\frac {x^{3/2} (a d+b c)}{\left (c+d x^2\right ) (b c-a d)}}{4 b (b c-a d)}\right )\)

Input:

Int[x^(9/2)/((a + b*x^2)^2*(c + d*x^2)^2),x]
 

Output:

2*((a*x^(3/2))/(4*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) - (-(((b*c + a*d) 
*x^(3/2))/((b*c - a*d)*(c + d*x^2))) + (b*(-1/2*(a^(3/4)*(7*b*c + a*d)*Arc 
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(3/4)*(b*c - a*d)) 
+ (a^(3/4)*(7*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2 
*Sqrt[2]*b^(3/4)*(b*c - a*d)) + (c^(3/4)*(b*c + 7*a*d)*ArcTan[1 - (Sqrt[2] 
*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*d^(3/4)*(b*c - a*d)) - (c^(3/4)*(b* 
c + 7*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*d^(3/ 
4)*(b*c - a*d)) + (a^(3/4)*(7*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^( 
1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*b^(3/4)*(b*c - a*d)) - (a^(3/4)*(7*b 
*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*S 
qrt[2]*b^(3/4)*(b*c - a*d)) - (c^(3/4)*(b*c + 7*a*d)*Log[Sqrt[c] - Sqrt[2] 
*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(4*Sqrt[2]*d^(3/4)*(b*c - a*d)) + ( 
c^(3/4)*(b*c + 7*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt 
[d]*x])/(4*Sqrt[2]*d^(3/4)*(b*c - a*d))))/(b*c - a*d))/(4*b*(b*c - a*d)))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 368

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], 
 x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m 
] && IntegerQ[p]
 

rule 970

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) 
^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) 
/(b*n*(b*c - a*d)*(p + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d 
*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, 
n, p, q, x]
 

rule 1049

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m 
 + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) 
, x] + Simp[1/(a*n*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^n)^(p + 1)*( 
c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e 
- a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
 

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(-\frac {2 c \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{x^{2} d +c}+\frac {\left (\frac {7 a d}{4}+\frac {b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 a \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7 b c}{4}+\frac {a d}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}\)	\(302\)
default	\(-\frac {2 c \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{x^{2} d +c}+\frac {\left (\frac {7 a d}{4}+\frac {b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 a \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7 b c}{4}+\frac {a d}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}\)	\(302\)

Input:

int(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
 

Output:

-2*c/(a*d-b*c)^3*((-1/4*a*d+1/4*b*c)*x^(3/2)/(d*x^2+c)+1/8*(7/4*a*d+1/4*b* 
c)/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/( 
x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x 
^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))+2*a/(a*d-b*c)^3*((1/4* 
a*d-1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(7/4*b*c+1/4*a*d)/b/(a/b)^(1/4)*2^(1/2) 
*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^ 
(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/ 
2)/(a/b)^(1/4)*x^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 23.94 (sec) , antiderivative size = 5952, normalized size of antiderivative = 12.56 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**(9/2)/(b*x**2+a)**2/(d*x**2+c)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.19 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")
 

Output:

-1/16*(7*a*b*c + a^2*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^( 
1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sq 
rt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b 
)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2 
)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3 
/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a)) 
/(a^(1/4)*b^(3/4)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 
1/16*(b*c^2 + 7*a*c*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1 
/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqr 
t(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d) 
*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2) 
*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/ 
4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/ 
(c^(1/4)*d^(3/4)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1 
/2*((b*c + a*d)*x^(7/2) + 2*a*c*x^(3/2))/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3* 
c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2 
*d - a^2*b*c*d^2 + a^3*d^3)*x^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (362) = 724\).

Time = 0.29 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.94 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")
 

Output:

-1/4*(7*(a*b^3)^(3/4)*b*c + (a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2) 
*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^6*c^3 - 3*sqrt(2)*a*b^5* 
c^2*d + 3*sqrt(2)*a^2*b^4*c*d^2 - sqrt(2)*a^3*b^3*d^3) - 1/4*(7*(a*b^3)^(3 
/4)*b*c + (a*b^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2* 
sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^6*c^3 - 3*sqrt(2)*a*b^5*c^2*d + 3*sqrt(2) 
*a^2*b^4*c*d^2 - sqrt(2)*a^3*b^3*d^3) + 1/4*((c*d^3)^(3/4)*b*c + 7*(c*d^3) 
^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/ 
4))/(sqrt(2)*b^3*c^3*d^3 - 3*sqrt(2)*a*b^2*c^2*d^4 + 3*sqrt(2)*a^2*b*c*d^5 
 - sqrt(2)*a^3*d^6) + 1/4*((c*d^3)^(3/4)*b*c + 7*(c*d^3)^(3/4)*a*d)*arctan 
(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3* 
c^3*d^3 - 3*sqrt(2)*a*b^2*c^2*d^4 + 3*sqrt(2)*a^2*b*c*d^5 - sqrt(2)*a^3*d^ 
6) + 1/8*(7*(a*b^3)^(3/4)*b*c + (a*b^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/ 
b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^6*c^3 - 3*sqrt(2)*a*b^5*c^2*d + 3*sqr 
t(2)*a^2*b^4*c*d^2 - sqrt(2)*a^3*b^3*d^3) - 1/8*(7*(a*b^3)^(3/4)*b*c + (a* 
b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2) 
*b^6*c^3 - 3*sqrt(2)*a*b^5*c^2*d + 3*sqrt(2)*a^2*b^4*c*d^2 - sqrt(2)*a^3*b 
^3*d^3) - 1/8*((c*d^3)^(3/4)*b*c + 7*(c*d^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x 
)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^3*d^3 - 3*sqrt(2)*a*b^2*c^2* 
d^4 + 3*sqrt(2)*a^2*b*c*d^5 - sqrt(2)*a^3*d^6) + 1/8*((c*d^3)^(3/4)*b*c + 
7*(c*d^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/...
 

Mupad [B] (verification not implemented)

Time = 4.72 (sec) , antiderivative size = 31071, normalized size of antiderivative = 65.55 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^(9/2)/((a + b*x^2)^2*(c + d*x^2)^2),x)
 

Output:

2*atan(((-(a^7*d^4 + 2401*a^3*b^4*c^4 + 1372*a^4*b^3*c^3*d + 294*a^5*b^2*c 
^2*d^2 + 28*a^6*b*c*d^3)/(4096*b^15*c^12 + 4096*a^12*b^3*d^12 - 49152*a^11 
*b^4*c*d^11 + 270336*a^2*b^13*c^10*d^2 - 901120*a^3*b^12*c^9*d^3 + 2027520 
*a^4*b^11*c^8*d^4 - 3244032*a^5*b^10*c^7*d^5 + 3784704*a^6*b^9*c^6*d^6 - 3 
244032*a^7*b^8*c^5*d^7 + 2027520*a^8*b^7*c^4*d^8 - 901120*a^9*b^6*c^3*d^9 
+ 270336*a^10*b^5*c^2*d^10 - 49152*a*b^14*c^11*d))^(1/4)*((((32*a^3*b^18*c 
^18*d^3 - 10656*a^4*b^17*c^17*d^4 + 115104*a^5*b^16*c^16*d^5 - 561696*a^6* 
b^15*c^15*d^6 + 1597344*a^7*b^14*c^14*d^7 - 2848032*a^8*b^13*c^13*d^8 + 30 
73312*a^9*b^12*c^12*d^9 - 1365408*a^10*b^11*c^11*d^10 - 1365408*a^11*b^10* 
c^10*d^11 + 3073312*a^12*b^9*c^9*d^12 - 2848032*a^13*b^8*c^8*d^13 + 159734 
4*a^14*b^7*c^7*d^14 - 561696*a^15*b^6*c^6*d^15 + 115104*a^16*b^5*c^5*d^16 
- 10656*a^17*b^4*c^4*d^17 + 32*a^18*b^3*c^3*d^18)*1i)/(a^14*d^14 + b^14*c^ 
14 + 91*a^2*b^12*c^12*d^2 - 364*a^3*b^11*c^11*d^3 + 1001*a^4*b^10*c^10*d^4 
 - 2002*a^5*b^9*c^9*d^5 + 3003*a^6*b^8*c^8*d^6 - 3432*a^7*b^7*c^7*d^7 + 30 
03*a^8*b^6*c^6*d^8 - 2002*a^9*b^5*c^5*d^9 + 1001*a^10*b^4*c^4*d^10 - 364*a 
^11*b^3*c^3*d^11 + 91*a^12*b^2*c^2*d^12 - 14*a*b^13*c^13*d - 14*a^13*b*c*d 
^13) - (x^(1/2)*(-(a^7*d^4 + 2401*a^3*b^4*c^4 + 1372*a^4*b^3*c^3*d + 294*a 
^5*b^2*c^2*d^2 + 28*a^6*b*c*d^3)/(4096*b^15*c^12 + 4096*a^12*b^3*d^12 - 49 
152*a^11*b^4*c*d^11 + 270336*a^2*b^13*c^10*d^2 - 901120*a^3*b^12*c^9*d^3 + 
 2027520*a^4*b^11*c^8*d^4 - 3244032*a^5*b^10*c^7*d^5 + 3784704*a^6*b^9*...
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 2419, normalized size of antiderivative = 5.10 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^2,x)
 

Output:

( - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x 
)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c*d**2 - 2*b**(1/4)*a**(3/4)* 
sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a** 
(1/4)*sqrt(2)))*a**2*d**3*x**2 - 14*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/ 
4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b* 
c**2*d - 16*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2* 
sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c*d**2*x**2 - 2*b**(1/4) 
*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b* 
*(1/4)*a**(1/4)*sqrt(2)))*a*b*d**3*x**4 - 14*b**(1/4)*a**(3/4)*sqrt(2)*ata 
n((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 
2)))*b**2*c**2*d*x**2 - 14*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/ 
4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c*d**2*x 
**4 + 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt 
(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c*d**2 + 2*b**(1/4)*a**(3/4 
)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a 
**(1/4)*sqrt(2)))*a**2*d**3*x**2 + 14*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**( 
1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a* 
b*c**2*d + 16*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 
2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c*d**2*x**2 + 2*b**(1/ 
4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b)...