\(\int \frac {1}{x^{7/2} (a+b x^2) (c+d x^2)} \, dx\) [793]

Optimal result

Integrand size = 24, antiderivative size = 367 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {2}{5 a c x^{5/2}}+\frac {2 (b c+a d)}{a^2 c^2 \sqrt {x}}-\frac {b^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {b^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {d^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{9/4} (b c-a d)}-\frac {d^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{9/4} (b c-a d)}-\frac {b^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {d^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{9/4} (b c-a d)} \] Output:

-2/5/a/c/x^(5/2)+2*(a*d+b*c)/a^2/c^2/x^(1/2)-1/2*b^(9/4)*arctan(1-2^(1/2)* 
b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(9/4)/(-a*d+b*c)+1/2*b^(9/4)*arctan(1+2 
^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(9/4)/(-a*d+b*c)+1/2*d^(9/4)*arc 
tan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(9/4)/(-a*d+b*c)-1/2*d^(9 
/4)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(9/4)/(-a*d+b*c)-1 
/2*b^(9/4)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^ 
(1/2)/a^(9/4)/(-a*d+b*c)+1/2*d^(9/4)*arctanh(2^(1/2)*c^(1/4)*d^(1/4)*x^(1/ 
2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(9/4)/(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {2 \left (-a c+5 b c x^2+5 a d x^2\right )}{5 a^2 c^2 x^{5/2}}+\frac {b^{9/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} a^{9/4} (-b c+a d)}+\frac {d^{9/4} \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt {2} c^{9/4} (b c-a d)}+\frac {b^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{9/4} (-b c+a d)}+\frac {d^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{9/4} (b c-a d)} \] Input:

Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x]
 

Output:

(2*(-(a*c) + 5*b*c*x^2 + 5*a*d*x^2))/(5*a^2*c^2*x^(5/2)) + (b^(9/4)*ArcTan 
[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(9/4 
)*(-(b*c) + a*d)) + (d^(9/4)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4) 
*d^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(9/4)*(b*c - a*d)) + (b^(9/4)*ArcTanh[(Sqrt 
[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(9/4)*(-(b 
*c) + a*d)) + (d^(9/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] 
+ Sqrt[d]*x)])/(Sqrt[2]*c^(9/4)*(b*c - a*d))
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.50, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {368, 980, 27, 1053, 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.

\(\displaystyle \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle 2 \int \frac {1}{x^3 \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}\)

\(\Big \downarrow \) 980

\(\displaystyle 2 \left (\frac {\int -\frac {5 \left (b d x^2+b c+a d\right )}{x \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{5 a c}-\frac {1}{5 a c x^{5/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (-\frac {\int \frac {b d x^2+b c+a d}{x \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {1}{5 a c x^{5/2}}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 \left (-\frac {-\frac {\int \frac {x \left (b^2 c^2+a b d c+a^2 d^2+b d (b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {a d+b c}{a c \sqrt {x}}}{a c}-\frac {1}{5 a c x^{5/2}}\right )\)

\(\Big \downarrow \) 1054

\(\displaystyle 2 \left (-\frac {-\frac {\int \left (\frac {c^2 x b^3}{(b c-a d) \left (b x^2+a\right )}+\frac {a^2 d^3 x}{(a d-b c) \left (d x^2+c\right )}\right )d\sqrt {x}}{a c}-\frac {a d+b c}{a c \sqrt {x}}}{a c}-\frac {1}{5 a c x^{5/2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {-\frac {\frac {a^2 d^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {a^2 d^{9/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {a^2 d^{9/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {a^2 d^{9/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {b^{9/4} c^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {b^{9/4} c^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {b^{9/4} c^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {b^{9/4} c^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)}}{a c}-\frac {a d+b c}{a c \sqrt {x}}}{a c}-\frac {1}{5 a c x^{5/2}}\right )\)

Input:

Int[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x]
 

Output:

2*(-1/5*1/(a*c*x^(5/2)) - (-((b*c + a*d)/(a*c*Sqrt[x])) - (-1/2*(b^(9/4)*c 
^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - 
a*d)) + (b^(9/4)*c^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqr 
t[2]*a^(1/4)*(b*c - a*d)) + (a^2*d^(9/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[ 
x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) - (a^2*d^(9/4)*ArcTan[1 + (S 
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) + (b^(9/ 
4)*c^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt 
[2]*a^(1/4)*(b*c - a*d)) - (b^(9/4)*c^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1 
/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)) - (a^2*d^(9/4)*L 
og[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(4*Sqrt[2]*c^(1 
/4)*(b*c - a*d)) + (a^2*d^(9/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt 
[x] + Sqrt[d]*x])/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)))/(a*c))/(a*c))
 

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 980

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

rule 1053

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b 
*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( 
m + 1))   Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) 
- e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 
) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 
0] && LtQ[m, -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 = 0.55 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(-\frac {2}{5 a c \,x^{\frac {5}{2}}}-\frac {2 \left (-a d -b c \right )}{a^{2} c^{2} \sqrt {x}}-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(269\)
default	\(-\frac {2}{5 a c \,x^{\frac {5}{2}}}-\frac {2 \left (-a d -b c \right )}{a^{2} c^{2} \sqrt {x}}-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(269\)
risch	\(-\frac {2 \left (-5 a d \,x^{2}-5 x^{2} b c +a c \right )}{5 a^{2} c^{2} x^{\frac {5}{2}}}+\frac {\frac {a^{2} d^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {b^{2} c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{2} c^{2}}\)	\(275\)

Input:

int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

-2/5/a/c/x^(5/2)-2/a^2/c^2*(-a*d-b*c)/x^(1/2)-1/4*b^2/a^2/(a*d-b*c)/(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))+1/4*d^2/c^2/(a*d-b*c)/(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*arcta 
n(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.53 (sec) , antiderivative size = 1548, normalized size of antiderivative = 4.22 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")
 

Output:

1/10*(5*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^1 
2*b*c*d^3 + a^13*d^4))^(1/4)*a^2*c^2*x^3*log(b^7*sqrt(x) + (a^7*b^3*c^3 - 
3*a^8*b^2*c^2*d + 3*a^9*b*c*d^2 - a^10*d^3)*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^ 
3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) - 5*(-b^ 
9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + 
a^13*d^4))^(1/4)*a^2*c^2*x^3*log(b^7*sqrt(x) - (a^7*b^3*c^3 - 3*a^8*b^2*c^ 
2*d + 3*a^9*b*c*d^2 - a^10*d^3)*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6* 
a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) + 5*I*(-b^9/(a^9*b^4 
*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4)) 
^(1/4)*a^2*c^2*x^3*log(b^7*sqrt(x) - (I*a^7*b^3*c^3 - 3*I*a^8*b^2*c^2*d + 
3*I*a^9*b*c*d^2 - I*a^10*d^3)*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^ 
11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) - 5*I*(-b^9/(a^9*b^4*c 
^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^( 
1/4)*a^2*c^2*x^3*log(b^7*sqrt(x) - (-I*a^7*b^3*c^3 + 3*I*a^8*b^2*c^2*d - 3 
*I*a^9*b*c*d^2 + I*a^10*d^3)*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^1 
1*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) - 5*(-d^9/(b^4*c^13 - 4 
*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4 
)*a^2*c^2*x^3*log(d^7*sqrt(x) + (b^3*c^10 - 3*a*b^2*c^9*d + 3*a^2*b*c^8*d^ 
2 - a^3*c^7*d^3)*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4 
*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(3/4)) + 5*(-d^9/(b^4*c^13 - 4*a*b^3*c^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x**(7/2)/(b*x**2+a)/(d*x**2+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {b^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c^{3} - a c^{2} d\right )}} + \frac {2 \, {\left (5 \, {\left (b c + a d\right )} x^{2} - a c\right )}}{5 \, a^{2} c^{2} x^{\frac {5}{2}}} \] Input:

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")
 

Output:

1/4*b^3*(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)*l 
og(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4 
)))/(a^2*b*c - a^3*d) - 1/4*d^3*(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)*sqr 
t(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*c^3 - a*c^2*d) + 2/5*(5*(b*c + a*d)*x^2 - a 
*c)/(a^2*c^2*x^(5/2))
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b c - \sqrt {2} a^{4} d} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b c - \sqrt {2} a^{4} d} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c^{4} - \sqrt {2} a c^{3} d} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c^{4} - \sqrt {2} a c^{3} d} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b c - \sqrt {2} a^{4} d\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b c - \sqrt {2} a^{4} d\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{4} - \sqrt {2} a c^{3} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{4} - \sqrt {2} a c^{3} d\right )}} + \frac {2 \, {\left (5 \, b c x^{2} + 5 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{2} x^{\frac {5}{2}}} \] Input:

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.18 (sec) , antiderivative size = 4643, normalized size of antiderivative = 12.65 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x)
 

Output:

- 2*atan((32*a^11*b^10*c^13*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 
64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 2*a^11 
*b^6*d^9*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 
 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4) + 32*a^21*c^3*d^10*x^(1/2)* 
(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2* 
d^2 - 64*a^12*b*c*d^3))^(5/4) + 2*a^8*b^9*c^3*d^6*x^(1/2)*(-b^9/(16*a^13*d 
^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b* 
c*d^3))^(1/4) + 192*a^13*b^8*c^11*d^2*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9* 
b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4 
) - 128*a^14*b^7*c^10*d^3*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64 
*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 32*a^15* 
b^6*c^9*d^4*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3* 
d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 32*a^17*b^4*c^7*d^6*x^ 
(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^ 
2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) - 128*a^18*b^3*c^6*d^7*x^(1/2)*(-b^9/( 
16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 6 
4*a^12*b*c*d^3))^(5/4) + 192*a^19*b^2*c^5*d^8*x^(1/2)*(-b^9/(16*a^13*d^4 + 
 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^ 
3))^(5/4) - 128*a^12*b^9*c^12*d*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^ 
4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) -...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {10 \sqrt {x}\, b^{\frac {9}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{3} x^{2}-10 \sqrt {x}\, b^{\frac {9}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{3} x^{2}-10 \sqrt {x}\, d^{\frac {9}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {d}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{3} x^{2}+10 \sqrt {x}\, d^{\frac {9}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {d}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{3} x^{2}-5 \sqrt {x}\, b^{\frac {9}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c^{3} x^{2}+5 \sqrt {x}\, b^{\frac {9}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c^{3} x^{2}+5 \sqrt {x}\, d^{\frac {9}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}+\sqrt {d}\, x \right ) a^{3} x^{2}-5 \sqrt {x}\, d^{\frac {9}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}+\sqrt {d}\, x \right ) a^{3} x^{2}-8 a^{3} c^{2} d +40 a^{3} c \,d^{2} x^{2}+8 a^{2} b \,c^{3}-40 a \,b^{2} c^{3} x^{2}}{20 \sqrt {x}\, a^{3} c^{3} x^{2} \left (a d -b c \right )} \] Input:

int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x)
 

Output:

(10*sqrt(x)*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*x**2 - 10*sqrt(x)* 
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*x**2 - 10*sqrt(x)*d**(1/4)*c** 
(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/ 
4)*c**(1/4)*sqrt(2)))*a**3*d**2*x**2 + 10*sqrt(x)*d**(1/4)*c**(3/4)*sqrt(2 
)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)* 
sqrt(2)))*a**3*d**2*x**2 - 5*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt 
(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2*c**3*x**2 + 5*sq 
rt(x)*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sq 
rt(a) + sqrt(b)*x)*b**2*c**3*x**2 + 5*sqrt(x)*d**(1/4)*c**(3/4)*sqrt(2)*lo 
g( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*a**3*d**2*x* 
*2 - 5*sqrt(x)*d**(1/4)*c**(3/4)*sqrt(2)*log(sqrt(x)*d**(1/4)*c**(1/4)*sqr 
t(2) + sqrt(c) + sqrt(d)*x)*a**3*d**2*x**2 - 8*a**3*c**2*d + 40*a**3*c*d** 
2*x**2 + 8*a**2*b*c**3 - 40*a*b**2*c**3*x**2)/(20*sqrt(x)*a**3*c**3*x**2*( 
a*d - b*c))