\(\int (c+\frac {d}{x^2})^{3/2} x^2 (a+b x) \, dx\) [18]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 (a+b x) \, dx=\frac {1}{8} d \sqrt {c+\frac {d}{x^2}} \left (3 b+\frac {8 a}{x}\right ) x^2+\frac {1}{12} \left (c+\frac {d}{x^2}\right )^{3/2} \left (3 b+\frac {4 a}{x}\right ) x^4+\frac {3 b d^2 \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 \sqrt {c}}-a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \] Output:

1/8*d*(c+d/x^2)^(1/2)*(3*b+8*a/x)*x^2+1/12*(c+d/x^2)^(3/2)*(3*b+4*a/x)*x^4 
+3/8*b*d^2*arctanh((c+d/x^2)^(1/2)/c^(1/2))/c^(1/2)-a*d^(3/2)*arctanh(d^(1 
/2)/(c+d/x^2)^(1/2)/x)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.20 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 (a+b x) \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (\sqrt {c} \sqrt {d+c x^2} \left (32 a d+15 b d x+8 a c x^2+6 b c x^3\right )+48 a \sqrt {c} d^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {d+c x^2}}{\sqrt {d}}\right )-9 b d^2 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{24 \sqrt {c} \sqrt {d+c x^2}} \] Input:

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

Output:

(Sqrt[c + d/x^2]*x*(Sqrt[c]*Sqrt[d + c*x^2]*(32*a*d + 15*b*d*x + 8*a*c*x^2 
 + 6*b*c*x^3) + 48*a*Sqrt[c]*d^(3/2)*ArcTanh[(Sqrt[c]*x - Sqrt[d + c*x^2]) 
/Sqrt[d]] - 9*b*d^2*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]]))/(24*Sqrt[c]*Sqrt 
[d + c*x^2])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1892, 1803, 537, 25, 537, 25, 538, 224, 219, 243, 73, 221}

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.

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

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 537

\(\displaystyle \frac {1}{4} d \int -\sqrt {c+\frac {d}{x^2}} \left (\frac {4 a}{x}+3 b\right ) x^3d\frac {1}{x}+\frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \int \sqrt {c+\frac {d}{x^2}} \left (\frac {4 a}{x}+3 b\right ) x^3d\frac {1}{x}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (-\frac {1}{2} d \int -\frac {\left (\frac {8 a}{x}+3 b\right ) x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \int \frac {\left (\frac {8 a}{x}+3 b\right ) x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \left (8 a \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+3 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \left (8 a \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+3 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \left (3 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {8 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}\right )-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \left (\frac {3}{2} b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}+\frac {8 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}\right )-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \left (\frac {3 b \int \frac {1}{\frac {\sqrt {c+\frac {d}{x^2}}}{d}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{d}+\frac {8 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}\right )-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{12} x^4 \left (\frac {4 a}{x}+3 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{4} d \left (\frac {1}{2} d \left (\frac {8 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {1}{2} x^2 \left (\frac {8 a}{x}+3 b\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

Input:

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

Output:

((c + d/x^2)^(3/2)*(3*b + (4*a)/x)*x^4)/12 - (d*(-1/2*(Sqrt[c + d/x^2]*(3* 
b + (8*a)/x)*x^2) + (d*((-3*b*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/Sqrt[c] + 
(8*a*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/Sqrt[d]))/2))/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 221

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

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 537

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

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1892

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^ 
(p_.), x_Symbol] :> Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; F 
reeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n 
2] ||  !IntegerQ[p])
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.25

method	result	size
default	\(-\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (-6 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} b x +24 \sqrt {c}\, d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) a -8 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} a -9 \sqrt {c}\, \sqrt {c \,x^{2}+d}\, b d x -24 \sqrt {c}\, \sqrt {c \,x^{2}+d}\, a d -9 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b \,d^{2}\right )}{24 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c}}\)	\(148\)

Input:

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

Output:

-1/24*((c*x^2+d)/x^2)^(3/2)*x^3*(-6*c^(1/2)*(c*x^2+d)^(3/2)*b*x+24*c^(1/2) 
*d^(3/2)*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*a-8*c^(1/2)*(c*x^2+d)^(3/2)*a 
-9*c^(1/2)*(c*x^2+d)^(1/2)*b*d*x-24*c^(1/2)*(c*x^2+d)^(1/2)*a*d-9*ln(c^(1/ 
2)*x+(c*x^2+d)^(1/2))*b*d^2)/(c*x^2+d)^(3/2)/c^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 535, normalized size of antiderivative = 4.53 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 (a+b x) \, dx=\left [\frac {9 \, b \sqrt {c} d^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 24 \, a c d^{\frac {3}{2}} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (6 \, b c^{2} x^{4} + 8 \, a c^{2} x^{3} + 15 \, b c d x^{2} + 32 \, a c d x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, c}, -\frac {9 \, b \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 12 \, a c d^{\frac {3}{2}} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - {\left (6 \, b c^{2} x^{4} + 8 \, a c^{2} x^{3} + 15 \, b c d x^{2} + 32 \, a c d x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{24 \, c}, \frac {48 \, a c \sqrt {-d} d \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + 9 \, b \sqrt {c} d^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (6 \, b c^{2} x^{4} + 8 \, a c^{2} x^{3} + 15 \, b c d x^{2} + 32 \, a c d x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, c}, -\frac {9 \, b \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 24 \, a c \sqrt {-d} d \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) - {\left (6 \, b c^{2} x^{4} + 8 \, a c^{2} x^{3} + 15 \, b c d x^{2} + 32 \, a c d x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{24 \, c}\right ] \] Input:

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

Output:

[1/48*(9*b*sqrt(c)*d^2*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) 
- d) + 24*a*c*d^(3/2)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2* 
d)/x^2) + 2*(6*b*c^2*x^4 + 8*a*c^2*x^3 + 15*b*c*d*x^2 + 32*a*c*d*x)*sqrt(( 
c*x^2 + d)/x^2))/c, -1/24*(9*b*sqrt(-c)*d^2*arctan(sqrt(-c)*x^2*sqrt((c*x^ 
2 + d)/x^2)/(c*x^2 + d)) - 12*a*c*d^(3/2)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt(( 
c*x^2 + d)/x^2) + 2*d)/x^2) - (6*b*c^2*x^4 + 8*a*c^2*x^3 + 15*b*c*d*x^2 + 
32*a*c*d*x)*sqrt((c*x^2 + d)/x^2))/c, 1/48*(48*a*c*sqrt(-d)*d*arctan(sqrt( 
-d)*x*sqrt((c*x^2 + d)/x^2)/d) + 9*b*sqrt(c)*d^2*log(-2*c*x^2 - 2*sqrt(c)* 
x^2*sqrt((c*x^2 + d)/x^2) - d) + 2*(6*b*c^2*x^4 + 8*a*c^2*x^3 + 15*b*c*d*x 
^2 + 32*a*c*d*x)*sqrt((c*x^2 + d)/x^2))/c, -1/24*(9*b*sqrt(-c)*d^2*arctan( 
sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - 24*a*c*sqrt(-d)*d*arctan 
(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/d) - (6*b*c^2*x^4 + 8*a*c^2*x^3 + 15*b*c 
*d*x^2 + 32*a*c*d*x)*sqrt((c*x^2 + d)/x^2))/c]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (104) = 208\).

Time = 4.13 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.00 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 (a+b x) \, dx=\frac {a \sqrt {c} d x}{\sqrt {1 + \frac {d}{c x^{2}}}} + \frac {a c \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {a d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3} - a d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {a d^{2}}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{2} x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b c \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {b d^{\frac {3}{2}} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {b d^{\frac {3}{2}} x}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 \sqrt {c}} \] Input:

integrate((c+d/x**2)**(3/2)*x**2*(b*x+a),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**2*x**5/(4*sqrt(d)*sqrt(c 
*x**2/d + 1)) + 3*b*c*sqrt(d)*x**3/(8*sqrt(c*x**2/d + 1)) + b*d**(3/2)*x*s 
qrt(c*x**2/d + 1)/2 + b*d**(3/2)*x/(8*sqrt(c*x**2/d + 1)) + 3*b*d**2*asinh 
(sqrt(c)*x/sqrt(d))/(8*sqrt(c))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.47 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 (a+b x) \, dx=\frac {1}{6} \, {\left (2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + 6 \, \sqrt {c + \frac {d}{x^{2}}} d x + 3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} a - \frac {1}{16} \, {\left (\frac {3 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c + c^{2}}\right )} b \] Input:

integrate((c+d/x^2)^(3/2)*x^2*(b*x+a),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 - 1/16*(3*d^2*l 
og((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/sqrt(c) - 2*(5 
*(c + d/x^2)^(3/2)*d^2 - 3*sqrt(c + d/x^2)*c*d^2)/((c + d/x^2)^2 - 2*(c + 
d/x^2)*c + c^2))*b
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.29 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2 (a+b x) \, dx=\frac {8 \sqrt {c \,x^{2}+d}\, a \,c^{2} x^{2}+32 \sqrt {c \,x^{2}+d}\, a c d +6 \sqrt {c \,x^{2}+d}\, b \,c^{2} x^{3}+15 \sqrt {c \,x^{2}+d}\, b c d x +9 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b \,d^{2}+24 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) a c d -24 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) a c d}{24 c} \] Input:

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

Output:

(8*sqrt(c*x**2 + d)*a*c**2*x**2 + 32*sqrt(c*x**2 + d)*a*c*d + 6*sqrt(c*x** 
2 + d)*b*c**2*x**3 + 15*sqrt(c*x**2 + d)*b*c*d*x + 9*sqrt(c)*log((sqrt(c*x 
**2 + d) + sqrt(c)*x)/sqrt(d))*b*d**2 + 24*sqrt(d)*log((sqrt(c*x**2 + d) + 
 sqrt(c)*x - sqrt(d))/sqrt(d))*a*c*d - 24*sqrt(d)*log((sqrt(c*x**2 + d) + 
sqrt(c)*x + sqrt(d))/sqrt(d))*a*c*d)/(24*c)