\(\int (a+\frac {b}{x^2}) (c+\frac {d}{x^2})^{3/2} x^2 \, dx\) [166]

Optimal result

Integrand size = 22, antiderivative size = 118 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=-\frac {d (3 b c+2 a d) \sqrt {c+\frac {d}{x^2}}}{6 c x}+\frac {1}{3} (3 b c+2 a d) \sqrt {c+\frac {d}{x^2}} x+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {1}{2} \sqrt {d} (3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \] Output:

-1/6*d*(2*a*d+3*b*c)*(c+d/x^2)^(1/2)/c/x+1/3*(2*a*d+3*b*c)*(c+d/x^2)^(1/2) 
*x+1/3*a*(c+d/x^2)^(5/2)*x^3/c-1/2*d^(1/2)*(2*a*d+3*b*c)*arctanh(d^(1/2)/( 
c+d/x^2)^(1/2)/x)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {d+c x^2} \left (-3 b d+6 b c x^2+8 a d x^2+2 a c x^4\right )-3 \sqrt {d} (3 b c+2 a d) x^2 \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )\right )}{6 x \sqrt {d+c x^2}} \] Input:

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

Output:

(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(-3*b*d + 6*b*c*x^2 + 8*a*d*x^2 + 2*a*c* 
x^4) - 3*Sqrt[d]*(3*b*c + 2*a*d)*x^2*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]]))/(6 
*x*Sqrt[d + c*x^2])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 773, 247, 211, 224, 219}

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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 773

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 773

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 
2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] &&  !IntegerQ[p]
 

rule 955

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), 
 x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1))   Int[(e 
*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* 
c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || 
(LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {d b \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{2 x}+\frac {\left (-\frac {\sqrt {d}\, \left (2 a d +3 c b \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right )}{2}+\sqrt {c \,x^{2}+d}\, b c +a \,c^{2} \left (\frac {x^{2} \sqrt {c \,x^{2}+d}}{3 c}-\frac {2 d \sqrt {c \,x^{2}+d}}{3 c^{2}}\right )+2 a d \sqrt {c \,x^{2}+d}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{\sqrt {c \,x^{2}+d}}\)	\(147\)
default	\(-\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x \left (6 d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) a \,x^{2}+9 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b c \,x^{2}-2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d \,x^{2}-3 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b c \,x^{2}+3 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b -6 \sqrt {c \,x^{2}+d}\, a \,d^{2} x^{2}-9 \sqrt {c \,x^{2}+d}\, b c d \,x^{2}\right )}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d}\)	\(170\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.56 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\left [\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (2 \, a c x^{4} + 2 \, {\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x}, \frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + {\left (2 \, a c x^{4} + 2 \, {\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x}\right ] \] Input:

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

Output:

[1/12*(3*(3*b*c + 2*a*d)*sqrt(d)*x*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + 
 d)/x^2) + 2*d)/x^2) + 2*(2*a*c*x^4 + 2*(3*b*c + 4*a*d)*x^2 - 3*b*d)*sqrt( 
(c*x^2 + d)/x^2))/x, 1/6*(3*(3*b*c + 2*a*d)*sqrt(-d)*x*arctan(sqrt(-d)*x*s 
qrt((c*x^2 + d)/x^2)/d) + (2*a*c*x^4 + 2*(3*b*c + 4*a*d)*x^2 - 3*b*d)*sqrt 
((c*x^2 + d)/x^2))/x]
 

Sympy [A] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.71 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {a \sqrt {c} d x}{\sqrt {1 + \frac {d}{c x^{2}}}} + \frac {a c \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {a d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3} - a d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {a d^{2}}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{\frac {3}{2}} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - \frac {b \sqrt {c} d \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} + \frac {b \sqrt {c} d}{x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.38 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {1}{6} \, {\left (2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + 6 \, \sqrt {c + \frac {d}{x^{2}}} d x + 3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} a + \frac {1}{4} \, {\left (4 \, \sqrt {c + \frac {d}{x^{2}}} c x - \frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c d x}{{\left (c + \frac {d}{x^{2}}\right )} x^{2} - d} + 3 \, c \sqrt {d} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \] Input:

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

Output:

1/6*(2*(c + d/x^2)^(3/2)*x^3 + 6*sqrt(c + d/x^2)*d*x + 3*d^(3/2)*log((sqrt 
(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*a + 1/4*(4*sqrt(c 
 + d/x^2)*c*x - 2*sqrt(c + d/x^2)*c*d*x/((c + d/x^2)*x^2 - d) + 3*c*sqrt(d 
)*log((sqrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*b
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {1}{6} \, c {\left (\frac {3 \, {\left (3 \, b c d \mathrm {sgn}\left (x\right ) + 2 \, a d^{2} \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{c \sqrt {-d}} - \frac {3 \, \sqrt {c x^{2} + d} b d \mathrm {sgn}\left (x\right )}{c x^{2}} + \frac {2 \, {\left ({\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{2} \mathrm {sgn}\left (x\right ) + 3 \, \sqrt {c x^{2} + d} b c^{3} \mathrm {sgn}\left (x\right ) + 3 \, \sqrt {c x^{2} + d} a c^{2} d \mathrm {sgn}\left (x\right )\right )}}{c^{3}}\right )} \] Input:

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

Output:

1/6*c*(3*(3*b*c*d*sgn(x) + 2*a*d^2*sgn(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d) 
)/(c*sqrt(-d)) - 3*sqrt(c*x^2 + d)*b*d*sgn(x)/(c*x^2) + 2*((c*x^2 + d)^(3/ 
2)*a*c^2*sgn(x) + 3*sqrt(c*x^2 + d)*b*c^3*sgn(x) + 3*sqrt(c*x^2 + d)*a*c^2 
*d*sgn(x))/c^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.58 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {2 \sqrt {c \,x^{2}+d}\, a c \,x^{4}+8 \sqrt {c \,x^{2}+d}\, a d \,x^{2}+6 \sqrt {c \,x^{2}+d}\, b c \,x^{2}-3 \sqrt {c \,x^{2}+d}\, b d +6 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) a d \,x^{2}+9 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) b c \,x^{2}-6 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) a d \,x^{2}-9 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) b c \,x^{2}}{6 x^{2}} \] Input:

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

Output:

(2*sqrt(c*x**2 + d)*a*c*x**4 + 8*sqrt(c*x**2 + d)*a*d*x**2 + 6*sqrt(c*x**2 
 + d)*b*c*x**2 - 3*sqrt(c*x**2 + d)*b*d + 6*sqrt(d)*log((sqrt(c*x**2 + d) 
+ sqrt(c)*x - sqrt(d))/sqrt(d))*a*d*x**2 + 9*sqrt(d)*log((sqrt(c*x**2 + d) 
 + sqrt(c)*x - sqrt(d))/sqrt(d))*b*c*x**2 - 6*sqrt(d)*log((sqrt(c*x**2 + d 
) + sqrt(c)*x + sqrt(d))/sqrt(d))*a*d*x**2 - 9*sqrt(d)*log((sqrt(c*x**2 + 
d) + sqrt(c)*x + sqrt(d))/sqrt(d))*b*c*x**2)/(6*x**2)