\(\int (c+\frac {d}{x^2})^{3/2} x^5 (a+b x) \, dx\) [15]

Optimal result

Integrand size = 20, antiderivative size = 142 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\frac {a d^2 \sqrt {c+\frac {d}{x^2}} x^2}{16 c}+\frac {7}{24} a d \sqrt {c+\frac {d}{x^2}} x^4-\frac {2 b d \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{35 c^2}+\frac {1}{6} a c \sqrt {c+\frac {d}{x^2}} x^6+\frac {b \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{7 c}-\frac {a d^3 \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{16 c^{3/2}} \] Output:

1/16*a*d^2*(c+d/x^2)^(1/2)*x^2/c+7/24*a*d*(c+d/x^2)^(1/2)*x^4-2/35*b*d*(c+ 
d/x^2)^(5/2)*x^5/c^2+1/6*a*c*(c+d/x^2)^(1/2)*x^6+1/7*b*(c+d/x^2)^(5/2)*x^7 
/c-1/16*a*d^3*arctanh((c+d/x^2)^(1/2)/c^(1/2))/c^(3/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.87 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (\sqrt {d+c x^2} \left (-48 b \left (2 d-5 c x^2\right ) \left (d+c x^2\right )^2+35 a c x \left (3 d^2+14 c d x^2+8 c^2 x^4\right )\right )+105 a \sqrt {c} d^3 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{1680 c^2 \sqrt {d+c x^2}} \] Input:

Integrate[(c + d/x^2)^(3/2)*x^5*(a + b*x),x]
 

Output:

(Sqrt[c + d/x^2]*x*(Sqrt[d + c*x^2]*(-48*b*(2*d - 5*c*x^2)*(d + c*x^2)^2 + 
 35*a*c*x*(3*d^2 + 14*c*d*x^2 + 8*c^2*x^4)) + 105*a*Sqrt[c]*d^3*Log[-(Sqrt 
[c]*x) + Sqrt[d + c*x^2]]))/(1680*c^2*Sqrt[d + c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.57 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1892, 1803, 539, 25, 539, 27, 534, 243, 51, 51, 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.

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

\(\Big \downarrow \) 1892

\(\displaystyle \int x^6 \left (\frac {a}{x}+b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}dx\)

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 539

\(\displaystyle \frac {\int -\left (c+\frac {d}{x^2}\right )^{3/2} \left (7 a c-\frac {2 b d}{x}\right ) x^7d\frac {1}{x}}{7 c}+\frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {\int \left (c+\frac {d}{x^2}\right )^{3/2} \left (7 a c-\frac {2 b d}{x}\right ) x^7d\frac {1}{x}}{7 c}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {\int c d \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {7 a}{x}+12 b\right ) x^6d\frac {1}{x}}{6 c}-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \int \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {7 a}{x}+12 b\right ) x^6d\frac {1}{x}-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \left (7 a \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5d\frac {1}{x}-\frac {12 b x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}\right )-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \left (\frac {7}{2} a \int \left (c+\frac {d}{x^2}\right )^{3/2} x^3d\frac {1}{x^2}-\frac {12 b x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}\right )-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \left (\frac {7}{2} a \left (\frac {3}{4} d \int \sqrt {c+\frac {d}{x^2}} x^2d\frac {1}{x^2}-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {12 b x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}\right )-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \left (\frac {7}{2} a \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}-x \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {12 b x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}\right )-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \left (\frac {7}{2} a \left (\frac {3}{4} d \left (\int \frac {1}{\frac {\sqrt {c+\frac {d}{x^2}}}{d}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}-x \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {12 b x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}\right )-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {-\frac {1}{6} d \left (\frac {7}{2} a \left (\frac {3}{4} d \left (x \left (-\sqrt {c+\frac {d}{x^2}}\right )-\frac {d \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {12 b x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{5 c}\right )-\frac {7}{6} a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}\)

Input:

Int[(c + d/x^2)^(3/2)*x^5*(a + b*x),x]
 

Output:

(b*(c + d/x^2)^(5/2)*x^7)/(7*c) - ((-7*a*(c + d/x^2)^(5/2)*x^6)/6 - (d*((- 
12*b*(c + d/x^2)^(5/2)*x^5)/(5*c) + (7*a*(-1/2*((c + d/x^2)^(3/2)*x^2) + ( 
3*d*(-(Sqrt[c + d/x^2]*x) - (d*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/Sqrt[c])) 
/4))/2))/6)/(7*c)
 

Defintions of rubi rules used

rule 25

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

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 51

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

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 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 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 534

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

rule 539

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

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 = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {\left (240 b \,c^{3} x^{6}+280 a \,c^{3} x^{5}+384 b \,c^{2} d \,x^{4}+490 a \,c^{2} d \,x^{3}+48 b c \,d^{2} x^{2}+105 a c \,d^{2} x -96 b \,d^{3}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{1680 c^{2}}-\frac {a \,d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{16 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}}\)	\(132\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (240 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,x^{2}+280 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} a x -96 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} b d -70 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d x -105 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, a \,d^{2} x -105 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a c \,d^{3}\right )}{1680 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c^{\frac {5}{2}}}\)	\(141\)

Input:

int((c+d/x^2)^(3/2)*x^5*(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/1680*(240*b*c^3*x^6+280*a*c^3*x^5+384*b*c^2*d*x^4+490*a*c^2*d*x^3+48*b*c 
*d^2*x^2+105*a*c*d^2*x-96*b*d^3)/c^2*x*((c*x^2+d)/x^2)^(1/2)-1/16*a/c^(3/2 
)*d^3*ln(c^(1/2)*x+(c*x^2+d)^(1/2))/(c*x^2+d)^(1/2)*x*((c*x^2+d)/x^2)^(1/2 
)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.82 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\left [\frac {105 \, a \sqrt {c} d^{3} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (240 \, b c^{3} x^{7} + 280 \, a c^{3} x^{6} + 384 \, b c^{2} d x^{5} + 490 \, a c^{2} d x^{4} + 48 \, b c d^{2} x^{3} + 105 \, a c d^{2} x^{2} - 96 \, b d^{3} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3360 \, c^{2}}, \frac {105 \, a \sqrt {-c} d^{3} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (240 \, b c^{3} x^{7} + 280 \, a c^{3} x^{6} + 384 \, b c^{2} d x^{5} + 490 \, a c^{2} d x^{4} + 48 \, b c d^{2} x^{3} + 105 \, a c d^{2} x^{2} - 96 \, b d^{3} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{1680 \, c^{2}}\right ] \] Input:

integrate((c+d/x^2)^(3/2)*x^5*(b*x+a),x, algorithm="fricas")
 

Output:

[1/3360*(105*a*sqrt(c)*d^3*log(-2*c*x^2 + 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x 
^2) - d) + 2*(240*b*c^3*x^7 + 280*a*c^3*x^6 + 384*b*c^2*d*x^5 + 490*a*c^2* 
d*x^4 + 48*b*c*d^2*x^3 + 105*a*c*d^2*x^2 - 96*b*d^3*x)*sqrt((c*x^2 + d)/x^ 
2))/c^2, 1/1680*(105*a*sqrt(-c)*d^3*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x 
^2)/(c*x^2 + d)) + (240*b*c^3*x^7 + 280*a*c^3*x^6 + 384*b*c^2*d*x^5 + 490* 
a*c^2*d*x^4 + 48*b*c*d^2*x^3 + 105*a*c*d^2*x^2 - 96*b*d^3*x)*sqrt((c*x^2 + 
 d)/x^2))/c^2]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (131) = 262\).

Time = 8.18 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.90 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\frac {a c^{2} x^{7}}{6 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {11 a c \sqrt {d} x^{5}}{24 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {17 a d^{\frac {3}{2}} x^{3}}{48 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {5}{2}} x}{16 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{16 c^{\frac {3}{2}}} + \frac {15 b c^{6} d^{\frac {9}{2}} x^{10} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {33 b c^{5} d^{\frac {11}{2}} x^{8} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {17 b c^{4} d^{\frac {13}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {3 b c^{3} d^{\frac {15}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {12 b c^{2} d^{\frac {17}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {8 b c d^{\frac {19}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {b d^{\frac {3}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {b d^{\frac {5}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c} - \frac {2 b d^{\frac {7}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c^{2}} \] Input:

integrate((c+d/x**2)**(3/2)*x**5*(b*x+a),x)
 

Output:

a*c**2*x**7/(6*sqrt(d)*sqrt(c*x**2/d + 1)) + 11*a*c*sqrt(d)*x**5/(24*sqrt( 
c*x**2/d + 1)) + 17*a*d**(3/2)*x**3/(48*sqrt(c*x**2/d + 1)) + a*d**(5/2)*x 
/(16*c*sqrt(c*x**2/d + 1)) - a*d**3*asinh(sqrt(c)*x/sqrt(d))/(16*c**(3/2)) 
 + 15*b*c**6*d**(9/2)*x**10*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c 
**4*d**5*x**2 + 105*c**3*d**6) + 33*b*c**5*d**(11/2)*x**8*sqrt(c*x**2/d + 
1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 17*b*c**4*d 
**(13/2)*x**6*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 
+ 105*c**3*d**6) + 3*b*c**3*d**(15/2)*x**4*sqrt(c*x**2/d + 1)/(105*c**5*d* 
*4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 12*b*c**2*d**(17/2)*x**2*s 
qrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6 
) + 8*b*c*d**(19/2)*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5 
*x**2 + 105*c**3*d**6) + b*d**(3/2)*x**4*sqrt(c*x**2/d + 1)/5 + b*d**(5/2) 
*x**2*sqrt(c*x**2/d + 1)/(15*c) - 2*b*d**(7/2)*sqrt(c*x**2/d + 1)/(15*c**2 
)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.23 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\frac {1}{96} \, {\left (\frac {3 \, d^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} + 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c d^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d^{3}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} c - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} c^{2} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} c^{3} - c^{4}}\right )} a + \frac {{\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5}\right )} b}{35 \, c^{2}} \] Input:

integrate((c+d/x^2)^(3/2)*x^5*(b*x+a),x, algorithm="maxima")
 

Output:

1/96*(3*d^3*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/c 
^(3/2) + 2*(3*(c + d/x^2)^(5/2)*d^3 + 8*(c + d/x^2)^(3/2)*c*d^3 - 3*sqrt(c 
 + d/x^2)*c^2*d^3)/((c + d/x^2)^3*c - 3*(c + d/x^2)^2*c^2 + 3*(c + d/x^2)* 
c^3 - c^4))*a + 1/35*(5*(c + d/x^2)^(7/2)*x^7 - 7*(c + d/x^2)^(5/2)*d*x^5) 
*b/c^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.03 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\frac {a d^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, c^{\frac {3}{2}}} - \frac {1}{1680} \, \sqrt {c x^{2} + d} {\left (\frac {96 \, b d^{3} \mathrm {sgn}\left (x\right )}{c^{2}} - {\left (\frac {105 \, a d^{2} \mathrm {sgn}\left (x\right )}{c} + 2 \, {\left (\frac {24 \, b d^{2} \mathrm {sgn}\left (x\right )}{c} + {\left (245 \, a d \mathrm {sgn}\left (x\right ) + 4 \, {\left (48 \, b d \mathrm {sgn}\left (x\right ) + 5 \, {\left (6 \, b c x \mathrm {sgn}\left (x\right ) + 7 \, a c \mathrm {sgn}\left (x\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} - \frac {{\left (35 \, a c d^{3} \log \left ({\left | d \right |}\right ) - 64 \, b \sqrt {c} d^{\frac {7}{2}}\right )} \mathrm {sgn}\left (x\right )}{1120 \, c^{\frac {5}{2}}} \] Input:

integrate((c+d/x^2)^(3/2)*x^5*(b*x+a),x, algorithm="giac")
 

Output:

1/16*a*d^3*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))*sgn(x)/c^(3/2) - 1/1680* 
sqrt(c*x^2 + d)*(96*b*d^3*sgn(x)/c^2 - (105*a*d^2*sgn(x)/c + 2*(24*b*d^2*s 
gn(x)/c + (245*a*d*sgn(x) + 4*(48*b*d*sgn(x) + 5*(6*b*c*x*sgn(x) + 7*a*c*s 
gn(x))*x)*x)*x)*x)*x) - 1/1120*(35*a*c*d^3*log(abs(d)) - 64*b*sqrt(c)*d^(7 
/2))*sgn(x)/c^(5/2)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 7.51 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.87 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\sqrt {c+\frac {d}{x^2}}\,\left (\frac {b\,c\,x^7}{7}+\frac {8\,b\,d\,x^5}{35}+\frac {b\,d^2\,x^3}{35\,c}-\frac {2\,b\,d^3\,x}{35\,c^2}\right )+\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{6}+\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{5/2}}{16\,c}-\frac {a\,c\,x^6\,\sqrt {c+\frac {d}{x^2}}}{16}+\frac {a\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c+\frac {d}{x^2}}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{16\,c^{3/2}} \] Input:

int(x^5*(c + d/x^2)^(3/2)*(a + b*x),x)
 

Output:

(c + d/x^2)^(1/2)*((b*c*x^7)/7 + (8*b*d*x^5)/35 + (b*d^2*x^3)/(35*c) - (2* 
b*d^3*x)/(35*c^2)) + (a*x^6*(c + d/x^2)^(3/2))/6 + (a*x^6*(c + d/x^2)^(5/2 
))/(16*c) + (a*d^3*atan(((c + d/x^2)^(1/2)*1i)/c^(1/2))*1i)/(16*c^(3/2)) - 
 (a*c*x^6*(c + d/x^2)^(1/2))/16
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5 (a+b x) \, dx=\frac {280 \sqrt {c \,x^{2}+d}\, a \,c^{3} x^{5}+490 \sqrt {c \,x^{2}+d}\, a \,c^{2} d \,x^{3}+105 \sqrt {c \,x^{2}+d}\, a c \,d^{2} x +240 \sqrt {c \,x^{2}+d}\, b \,c^{3} x^{6}+384 \sqrt {c \,x^{2}+d}\, b \,c^{2} d \,x^{4}+48 \sqrt {c \,x^{2}+d}\, b c \,d^{2} x^{2}-96 \sqrt {c \,x^{2}+d}\, b \,d^{3}-105 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a \,d^{3}}{1680 c^{2}} \] Input:

int((c+d/x^2)^(3/2)*x^5*(b*x+a),x)
 

Output:

(280*sqrt(c*x**2 + d)*a*c**3*x**5 + 490*sqrt(c*x**2 + d)*a*c**2*d*x**3 + 1 
05*sqrt(c*x**2 + d)*a*c*d**2*x + 240*sqrt(c*x**2 + d)*b*c**3*x**6 + 384*sq 
rt(c*x**2 + d)*b*c**2*d*x**4 + 48*sqrt(c*x**2 + d)*b*c*d**2*x**2 - 96*sqrt 
(c*x**2 + d)*b*d**3 - 105*sqrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt( 
d))*a*d**3)/(1680*c**2)