\(\int \frac {1}{x^{3/2} (a+b x^2) (c+d x^2)} \, dx\) [791]

Optimal result

Integrand size = 24, antiderivative size = 345 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {2}{a c \sqrt {x}}+\frac {b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {d^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}+\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {d^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{5/4} (b c-a d)} \] Output:

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

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\frac {4 b}{a \sqrt {x}}-\frac {4 d}{c \sqrt {x}}-\frac {\sqrt {2} b^{5/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4}}+\frac {\sqrt {2} d^{5/4} \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{5/4}}-\frac {\sqrt {2} b^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4}}+\frac {\sqrt {2} d^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{5/4}}}{-2 b c+2 a d} \] Input:

Integrate[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.46, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {368, 980, 25, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 980

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]
 

Output:

2*(-(1/(a*c*Sqrt[x])) - (-1/2*(b^(5/4)*c*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[ 
x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)) + (b^(5/4)*c*ArcTan[1 + (Sqrt[ 
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)) + (a*d^(5/4) 
*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - 
a*d)) - (a*d^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[ 
2]*c^(1/4)*(b*c - a*d)) + (b^(5/4)*c*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^(5/4)*c*Log[Sq 
rt[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*d^(5/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + S 
qrt[d]*x])/(4*Sqrt[2]*c^(1/4)*(b*c - a*d)) + (a*d^(5/4)*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))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 = 245, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) a \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2}{a c \sqrt {x}}-\frac {d \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) c \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(245\)
default	\(\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) a \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2}{a c \sqrt {x}}-\frac {d \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) c \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(245\)
risch	\(-\frac {2}{a c \sqrt {x}}-\frac {\frac {a d \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {b c \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a c}\)	\(250\)

Input:

int(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 1481, normalized size of antiderivative = 4.29 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x**(3/2)/(b*x**2+a)/(d*x**2+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a b c - a^{2} d\right )}} + \frac {d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c^{2} - a c d\right )}} - \frac {2}{a c \sqrt {x}} \] Input:

integrate(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^{3/2} \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^{2} b^{2} c - \sqrt {2} a^{3} b 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^{2} b^{2} c - \sqrt {2} a^{3} b 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^{3} d - \sqrt {2} a c^{2} d^{2}} + \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^{3} d - \sqrt {2} a c^{2} d^{2}} + \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^{2} b^{2} c - \sqrt {2} a^{3} b 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^{2} b^{2} c - \sqrt {2} a^{3} b 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^{3} d - \sqrt {2} a c^{2} d^{2}\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^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} - \frac {2}{a c \sqrt {x}} \] Input:

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

Mupad [B] (verification not implemented)

Time = 2.20 (sec) , antiderivative size = 6038, normalized size of antiderivative = 17.50 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-2 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2}+2 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c^{2}+2 \sqrt {x}\, d^{\frac {5}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {d}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}-2 \sqrt {x}\, d^{\frac {5}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {d}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{2}+\sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c^{2}-\sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c^{2}-\sqrt {x}\, d^{\frac {5}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}+\sqrt {d}\, x \right ) a^{2}+\sqrt {x}\, d^{\frac {5}{4}} c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}+\sqrt {d}\, x \right ) a^{2}-8 a^{2} c d +8 a b \,c^{2}}{4 \sqrt {x}\, a^{2} c^{2} \left (a d -b c \right )} \] Input:

int(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x)
 

Output:

( - 2*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*c**2 + 2*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*c**2 + 2*sqrt(x)*d**(1/4)*c**(3/4)*sqrt(2)*at 
an((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt 
(2)))*a**2*d - 2*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**2*d + sqrt(x 
)*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqr 
t(a) + sqrt(b)*x)*b*c**2 - 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*c**2 - sqrt(x)*d**(1/4)* 
c**(3/4)*sqrt(2)*log( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt 
(d)*x)*a**2*d + sqrt(x)*d**(1/4)*c**(3/4)*sqrt(2)*log(sqrt(x)*d**(1/4)*c** 
(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*a**2*d - 8*a**2*c*d + 8*a*b*c**2)/(4* 
sqrt(x)*a**2*c**2*(a*d - b*c))