\(\int \frac {x^{9/2}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [821]

Optimal result

Integrand size = 24, antiderivative size = 560 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {(b c+2 a d) x^{3/2}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a x^{3/2}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(5 b c+19 a d) x^{3/2}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {a^{3/4} \sqrt [4]{b} (7 b c+5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {a^{3/4} \sqrt [4]{b} (7 b c+5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\left (5 b^2 c^2+70 a b c d+21 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^4}+\frac {\left (5 b^2 c^2+70 a b c d+21 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^4}+\frac {a^{3/4} \sqrt [4]{b} (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\left (5 b^2 c^2+70 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{32 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)^4} \] Output:

1/4*(2*a*d+b*c)*x^(3/2)/b/(-a*d+b*c)^2/(d*x^2+c)^2+1/2*a*x^(3/2)/b/(-a*d+b 
*c)/(b*x^2+a)/(d*x^2+c)^2+1/16*(19*a*d+5*b*c)*x^(3/2)/(-a*d+b*c)^3/(d*x^2+ 
c)+1/8*a^(3/4)*b^(1/4)*(5*a*d+7*b*c)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^(1 
/4))*2^(1/2)/(-a*d+b*c)^4-1/8*a^(3/4)*b^(1/4)*(5*a*d+7*b*c)*arctan(1+2^(1/ 
2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/(-a*d+b*c)^4-1/64*(21*a^2*d^2+70*a*b*c 
*d+5*b^2*c^2)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(1/4)/d^ 
(3/4)/(-a*d+b*c)^4+1/64*(21*a^2*d^2+70*a*b*c*d+5*b^2*c^2)*arctan(1+2^(1/2) 
*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(1/4)/d^(3/4)/(-a*d+b*c)^4+1/8*a^(3/4) 
*b^(1/4)*(5*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)/(-a*d+b*c)^4-1/64*(21*a^2*d^2+70*a*b*c*d+5*b^2*c^2)*arct 
anh(2^(1/2)*c^(1/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(1/4)/d 
^(3/4)/(-a*d+b*c)^4
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.69 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {\frac {4 (b c-a d) x^{3/2} \left (b^2 c x^2 \left (9 c+5 d x^2\right )+a^2 d \left (7 c+11 d x^2\right )+a b \left (17 c^2+28 c d x^2+19 d^2 x^4\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^2}+8 \sqrt {2} a^{3/4} \sqrt [4]{b} (7 b c+5 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} \left (5 b^2 c^2+70 a b c d+21 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{c} d^{3/4}}+8 \sqrt {2} a^{3/4} \sqrt [4]{b} (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )-\frac {\sqrt {2} \left (5 b^2 c^2+70 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{c} d^{3/4}}}{64 (b c-a d)^4} \] Input:

Integrate[x^(9/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]
 

Output:

((4*(b*c - a*d)*x^(3/2)*(b^2*c*x^2*(9*c + 5*d*x^2) + a^2*d*(7*c + 11*d*x^2 
) + a*b*(17*c^2 + 28*c*d*x^2 + 19*d^2*x^4)))/((a + b*x^2)*(c + d*x^2)^2) + 
 8*Sqrt[2]*a^(3/4)*b^(1/4)*(7*b*c + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(S 
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - (Sqrt[2]*(5*b^2*c^2 + 70*a*b*c*d + 21*a 
^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/( 
c^(1/4)*d^(3/4)) + 8*Sqrt[2]*a^(3/4)*b^(1/4)*(7*b*c + 5*a*d)*ArcTanh[(Sqrt 
[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] - (Sqrt[2]*(5*b^2*c^2 
+ 70*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt 
[c] + Sqrt[d]*x)])/(c^(1/4)*d^(3/4)))/(64*(b*c - a*d)^4)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.35, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 970, 1049, 27, 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.

\(\displaystyle \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle 2 \int \frac {x^5}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^3}d\sqrt {x}\)

\(\Big \downarrow \) 970

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int \frac {x \left (3 a c-(4 b c+5 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1049

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\int \frac {4 b c x \left (9 a c-5 (b c+2 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}-\frac {x^{3/2} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \int \frac {x \left (9 a c-5 (b c+2 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{2 (b c-a d)}-\frac {x^{3/2} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1049

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\int \frac {c x \left (3 a (17 b c+7 a d)-b (5 b c+19 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {x^{3/2} (19 a d+5 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {x^{3/2} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\int \frac {x \left (3 a (17 b c+7 a d)-b (5 b c+19 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 (b c-a d)}-\frac {x^{3/2} (19 a d+5 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {x^{3/2} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 1054

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\int \left (\frac {8 a b (7 b c+5 a d) x}{(b c-a d) \left (b x^2+a\right )}-\frac {\left (5 b^2 c^2+70 a b d c+21 a^2 d^2\right ) x}{(b c-a d) \left (d x^2+c\right )}\right )d\sqrt {x}}{4 (b c-a d)}-\frac {x^{3/2} (19 a d+5 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {x^{3/2} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {a x^{3/2}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {-\frac {2 \sqrt {2} a^{3/4} \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (5 a d+7 b c)}{b c-a d}+\frac {2 \sqrt {2} a^{3/4} \sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (5 a d+7 b c)}{b c-a d}+\frac {\sqrt {2} a^{3/4} \sqrt [4]{b} (5 a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{b c-a d}-\frac {\sqrt {2} a^{3/4} \sqrt [4]{b} (5 a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{b c-a d}+\frac {\left (21 a^2 d^2+70 a b c d+5 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)}-\frac {\left (21 a^2 d^2+70 a b c d+5 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)}-\frac {\left (21 a^2 d^2+70 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)}+\frac {\left (21 a^2 d^2+70 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} d^{3/4} (b c-a d)}}{4 (b c-a d)}-\frac {x^{3/2} (19 a d+5 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {x^{3/2} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\)

Input:

Int[x^(9/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]
 

Output:

2*((a*x^(3/2))/(4*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (-1/2*((b*c + 
 2*a*d)*x^(3/2))/((b*c - a*d)*(c + d*x^2)^2) + (b*(-1/4*((5*b*c + 19*a*d)* 
x^(3/2))/((b*c - a*d)*(c + d*x^2)) + ((-2*Sqrt[2]*a^(3/4)*b^(1/4)*(7*b*c + 
 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(b*c - a*d) + (2*Sq 
rt[2]*a^(3/4)*b^(1/4)*(7*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x]) 
/a^(1/4)])/(b*c - a*d) + ((5*b^2*c^2 + 70*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - 
 (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*d^(3/4)*(b*c - a*d 
)) - ((5*b^2*c^2 + 70*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sq 
rt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*d^(3/4)*(b*c - a*d)) + (Sqrt[2]*a^(3/4 
)*b^(1/4)*(7*b*c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + 
Sqrt[b]*x])/(b*c - a*d) - (Sqrt[2]*a^(3/4)*b^(1/4)*(7*b*c + 5*a*d)*Log[Sqr 
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(b*c - a*d) - ((5*b^2 
*c^2 + 70*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt 
[x] + Sqrt[d]*x])/(4*Sqrt[2]*c^(1/4)*d^(3/4)*(b*c - a*d)) + ((5*b^2*c^2 + 
70*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + S 
qrt[d]*x])/(4*Sqrt[2]*c^(1/4)*d^(3/4)*(b*c - a*d)))/(4*(b*c - a*d))))/(2*( 
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 = 3.05 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(\frac {\frac {2 \left (\left (-\frac {11}{32} a^{2} d^{3}+\frac {3}{16} a c \,d^{2} b +\frac {5}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}-\frac {c \left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}\right )}{\left (x^{2} d +c \right )^{2}}+\frac {\left (\frac {21}{32} a^{2} d^{2}+\frac {35}{16} a b c d +\frac {5}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{\left (a d -b c \right )^{4}}-\frac {2 a b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 a d}{4}+\frac {7 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\)	\(364\)
default	\(\frac {\frac {2 \left (\left (-\frac {11}{32} a^{2} d^{3}+\frac {3}{16} a c \,d^{2} b +\frac {5}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}-\frac {c \left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}\right )}{\left (x^{2} d +c \right )^{2}}+\frac {\left (\frac {21}{32} a^{2} d^{2}+\frac {35}{16} a b c d +\frac {5}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{\left (a d -b c \right )^{4}}-\frac {2 a b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 a d}{4}+\frac {7 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\)	\(364\)

Input:

int(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

2/(a*d-b*c)^4*(((-11/32*a^2*d^3+3/16*a*c*d^2*b+5/32*b^2*c^2*d)*x^(7/2)-1/3 
2*c*(7*a^2*d^2+2*a*b*c*d-9*b^2*c^2)*x^(3/2))/(d*x^2+c)^2+1/8*(21/32*a^2*d^ 
2+35/16*a*b*c*d+5/32*b^2*c^2)/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*a 
rctan(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*b/(a*d-b*c)^4*((1/4*a*d-1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(5/4*a*d+7 
/4*b*c)/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 = 113.39 (sec) , antiderivative size = 8803, normalized size of antiderivative = 15.72 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(9/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.36 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

-1/16*(7*a*b^2*c + 5*a^2*b*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))*sqrt(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 + sq 
rt(a))/(a^(1/4)*b^(3/4)))/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4 
*a^3*b*c*d^3 + a^4*d^4) + 1/128*(5*b^2*c^2 + 70*a*b*c*d + 21*a^2*d^2)*(2*s 
qrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sq 
rt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(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^4*c^4 
 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 1/16*((5 
*b^2*c*d + 19*a*b*d^2)*x^(11/2) + (9*b^2*c^2 + 28*a*b*c*d + 11*a^2*d^2)*x^ 
(7/2) + (17*a*b*c^2 + 7*a^2*c*d)*x^(3/2))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3 
*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2* 
c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^ 
3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1221 vs. \(2 (444) = 888\).

Time = 0.37 (sec) , antiderivative size = 1221, normalized size of antiderivative = 2.18 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

1/2*a*b*x^(3/2)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^ 
2 + a)) - 1/4*(7*(a*b^3)^(3/4)*b*c + 5*(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^4 - 4*sqr 
t(2)*a*b^5*c^3*d + 6*sqrt(2)*a^2*b^4*c^2*d^2 - 4*sqrt(2)*a^3*b^3*c*d^3 + s 
qrt(2)*a^4*b^2*d^4) - 1/4*(7*(a*b^3)^(3/4)*b*c + 5*(a*b^3)^(3/4)*a*d)*arct 
an(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^ 
6*c^4 - 4*sqrt(2)*a*b^5*c^3*d + 6*sqrt(2)*a^2*b^4*c^2*d^2 - 4*sqrt(2)*a^3* 
b^3*c*d^3 + sqrt(2)*a^4*b^2*d^4) + 1/32*(5*(c*d^3)^(3/4)*b^2*c^2 + 70*(c*d 
^3)^(3/4)*a*b*c*d + 21*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)* 
(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^5*d^3 - 4*sqrt(2)*a*b 
^3*c^4*d^4 + 6*sqrt(2)*a^2*b^2*c^3*d^5 - 4*sqrt(2)*a^3*b*c^2*d^6 + sqrt(2) 
*a^4*c*d^7) + 1/32*(5*(c*d^3)^(3/4)*b^2*c^2 + 70*(c*d^3)^(3/4)*a*b*c*d + 2 
1*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt 
(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^5*d^3 - 4*sqrt(2)*a*b^3*c^4*d^4 + 6*sqrt( 
2)*a^2*b^2*c^3*d^5 - 4*sqrt(2)*a^3*b*c^2*d^6 + sqrt(2)*a^4*c*d^7) + 1/8*(7 
*(a*b^3)^(3/4)*b*c + 5*(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^4 - 4*sqrt(2)*a*b^5*c^3*d + 6*sqrt(2)*a^2* 
b^4*c^2*d^2 - 4*sqrt(2)*a^3*b^3*c*d^3 + sqrt(2)*a^4*b^2*d^4) - 1/8*(7*(a*b 
^3)^(3/4)*b*c + 5*(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^4 - 4*sqrt(2)*a*b^5*c^3*d + 6*sqrt(2)*a^2*b...
 

Mupad [B] (verification not implemented)

Time = 7.23 (sec) , antiderivative size = 43430, normalized size of antiderivative = 77.55 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^(9/2)/((a + b*x^2)^2*(c + d*x^2)^3),x)
 

Output:

2*atan(((-(625*a^7*b*d^4 + 2401*a^3*b^5*c^4 + 6860*a^4*b^4*c^3*d + 3500*a^ 
6*b^2*c*d^3 + 7350*a^5*b^3*c^2*d^2)/(4096*a^16*d^16 + 4096*b^16*c^16 + 491 
520*a^2*b^14*c^14*d^2 - 2293760*a^3*b^13*c^13*d^3 + 7454720*a^4*b^12*c^12* 
d^4 - 17891328*a^5*b^11*c^11*d^5 + 32800768*a^6*b^10*c^10*d^6 - 46858240*a 
^7*b^9*c^9*d^7 + 52715520*a^8*b^8*c^8*d^8 - 46858240*a^9*b^7*c^7*d^9 + 328 
00768*a^10*b^6*c^6*d^10 - 17891328*a^11*b^5*c^5*d^11 + 7454720*a^12*b^4*c^ 
4*d^12 - 2293760*a^13*b^3*c^3*d^13 + 491520*a^14*b^2*c^2*d^14 - 65536*a*b^ 
15*c^15*d - 65536*a^15*b*c*d^15))^(1/4)*(((((9261*a^24*b^4*c*d^24)/16 - (1 
25*a^3*b^25*c^22*d^3)/16 + (172241*a^4*b^24*c^21*d^4)/16 - (1133685*a^5*b^ 
23*c^20*d^5)/8 + (6654165*a^6*b^22*c^19*d^6)/8 - (44533245*a^7*b^21*c^18*d 
^7)/16 + (87114105*a^8*b^20*c^17*d^8)/16 - (9027123*a^9*b^19*c^16*d^9)/2 - 
 (13483605*a^10*b^18*c^15*d^10)/2 + (231998355*a^11*b^17*c^14*d^11)/8 - (4 
00586095*a^12*b^16*c^13*d^12)/8 + (213587465*a^13*b^15*c^12*d^13)/4 - (143 
925561*a^14*b^14*c^11*d^14)/4 + (98252895*a^15*b^13*c^10*d^15)/8 + (162780 
45*a^16*b^12*c^9*d^16)/8 - (8765595*a^17*b^11*c^8*d^17)/2 + (3548355*a^18* 
b^10*c^7*d^18)/2 + (1023351*a^19*b^9*c^6*d^19)/16 - (5183955*a^20*b^8*c^5* 
d^20)/16 + (929435*a^21*b^7*c^4*d^21)/8 - (82715*a^22*b^6*c^3*d^22)/8 - (4 
6305*a^23*b^5*c^2*d^23)/16)*1i)/(a^21*d^21 - b^21*c^21 - 210*a^2*b^19*c^19 
*d^2 + 1330*a^3*b^18*c^18*d^3 - 5985*a^4*b^17*c^17*d^4 + 20349*a^5*b^16*c^ 
16*d^5 - 54264*a^6*b^15*c^15*d^6 + 116280*a^7*b^14*c^14*d^7 - 203490*a^...
 

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 4335, normalized size of antiderivative = 7.74 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^(9/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)
 

Output:

(80*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**3*d**2 + 160*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**2*d**3*x**2 + 80*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**4*x**4 + 112*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**4*d + 
304*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**3*d**2*x**2 + 272*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**3*x**4 + 80*b**(1/4)*a**(3/4)*sqrt(2)*a 
tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr 
t(2)))*a*b*c*d**4*x**6 + 112*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**4*d 
*x**2 + 224*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**3*d**2*x**4 + 112*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**2*d**3*x**6 - 80*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**3*d**2 - 160*b**(1/4)*a**(3/4)*sqrt(2)*atan((b...