\(\int (c+\frac {d}{x^2})^{3/2} x^6 (a+b x) \, dx\) [14]

Optimal result

Integrand size = 20, antiderivative size = 167 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=-\frac {3 b d^3 \sqrt {c+\frac {d}{x^2}} x^2}{128 c^2}+\frac {b d^2 \sqrt {c+\frac {d}{x^2}} x^4}{64 c}-\frac {2 a d \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{35 c^2}+\frac {3}{16} b d \sqrt {c+\frac {d}{x^2}} x^6+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{7 c}+\frac {1}{8} b c \sqrt {c+\frac {d}{x^2}} x^8+\frac {3 b d^4 \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{128 c^{5/2}} \] Output:

-3/128*b*d^3*(c+d/x^2)^(1/2)*x^2/c^2+1/64*b*d^2*(c+d/x^2)^(1/2)*x^4/c-2/35 
*a*d*(c+d/x^2)^(5/2)*x^5/c^2+3/16*b*d*(c+d/x^2)^(1/2)*x^6+1/7*a*(c+d/x^2)^ 
(5/2)*x^7/c+1/8*b*c*(c+d/x^2)^(1/2)*x^8+3/128*b*d^4*arctanh((c+d/x^2)^(1/2 
)/c^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.81 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (\sqrt {c} \sqrt {d+c x^2} \left (-128 a \left (2 d-5 c x^2\right ) \left (d+c x^2\right )^2+35 b x \left (-3 d^3+2 c d^2 x^2+24 c^2 d x^4+16 c^3 x^6\right )\right )-105 b d^4 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{4480 c^{5/2} \sqrt {d+c x^2}} \] Input:

Integrate[(c + d/x^2)^(3/2)*x^6*(a + b*x),x]
 

Output:

(Sqrt[c + d/x^2]*x*(Sqrt[c]*Sqrt[d + c*x^2]*(-128*a*(2*d - 5*c*x^2)*(d + c 
*x^2)^2 + 35*b*x*(-3*d^3 + 2*c*d^2*x^2 + 24*c^2*d*x^4 + 16*c^3*x^6)) - 105 
*b*d^4*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]]))/(4480*c^(5/2)*Sqrt[d + c*x^2] 
)
 

Rubi [A] (warning: unable to verify)

Time = 0.62 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {1892, 1803, 539, 25, 539, 27, 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.

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

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1803

\(\displaystyle -\int \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {a}{x}+b\right ) x^9d\frac {1}{x}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\int -\left (c+\frac {d}{x^2}\right )^{3/2} \left (8 a c-\frac {3 b d}{x}\right ) x^8d\frac {1}{x}}{8 c}+\frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {\int \left (c+\frac {d}{x^2}\right )^{3/2} \left (8 a c-\frac {3 b d}{x}\right ) x^8d\frac {1}{x}}{8 c}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {\int c d \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {16 a}{x}+21 b\right ) x^7d\frac {1}{x}}{7 c}-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \int \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {16 a}{x}+21 b\right ) x^7d\frac {1}{x}-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (-\frac {\int -3 \left (c+\frac {d}{x^2}\right )^{3/2} \left (32 a c-\frac {7 b d}{x}\right ) x^6d\frac {1}{x}}{6 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {\int \left (c+\frac {d}{x^2}\right )^{3/2} \left (32 a c-\frac {7 b d}{x}\right ) x^6d\frac {1}{x}}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {-7 b d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^5d\frac {1}{x}-\frac {32}{5} a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {-\frac {7}{2} b d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^3d\frac {1}{x^2}-\frac {32}{5} a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {-\frac {7}{2} b d \left (\frac {3}{4} d \int \sqrt {c+\frac {d}{x^2}} x^2d\frac {1}{x^2}-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {32}{5} a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {-\frac {7}{2} b d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}-x \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {32}{5} a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {-\frac {7}{2} b d \left (\frac {3}{4} d \left (\int \frac {1}{\frac {\sqrt {c+\frac {d}{x^2}}}{d}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}-x \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )-\frac {32}{5} a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b x^8 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}-\frac {-\frac {1}{7} d \left (\frac {-\frac {32}{5} a x^5 \left (c+\frac {d}{x^2}\right )^{5/2}-\frac {7}{2} b d \left (\frac {3}{4} d \left (x \left (-\sqrt {c+\frac {d}{x^2}}\right )-\frac {d \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {1}{2} x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{2 c}-\frac {7 b x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}\right )-\frac {8}{7} a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{8 c}\)

Input:

Int[(c + d/x^2)^(3/2)*x^6*(a + b*x),x]
 

Output:

(b*(c + d/x^2)^(5/2)*x^8)/(8*c) - ((-8*a*(c + d/x^2)^(5/2)*x^7)/7 - (d*((- 
7*b*(c + d/x^2)^(5/2)*x^6)/(2*c) + ((-32*a*(c + d/x^2)^(5/2)*x^5)/5 - (7*b 
*d*(-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)/(2*c)))/7)/(8*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 = 141, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {\left (560 b \,c^{3} x^{7}+640 a \,c^{3} x^{6}+840 b \,c^{2} d \,x^{5}+1024 a \,c^{2} d \,x^{4}+70 x^{3} b c \,d^{2}+128 a \,d^{2} x^{2} c -105 b \,d^{3} x -256 a \,d^{3}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{4480 c^{2}}+\frac {3 b \,d^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{128 c^{\frac {5}{2}} \sqrt {c \,x^{2}+d}}\)	\(141\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (560 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,x^{3}+640 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,x^{2}-280 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} b d x -256 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} a d +70 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,d^{2} x +105 \sqrt {c}\, \sqrt {c \,x^{2}+d}\, b \,d^{3} x +105 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b \,d^{4}\right )}{4480 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c^{\frac {5}{2}}}\)	\(161\)

Input:

int((c+d/x^2)^(3/2)*x^6*(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/4480*(560*b*c^3*x^7+640*a*c^3*x^6+840*b*c^2*d*x^5+1024*a*c^2*d*x^4+70*b* 
c*d^2*x^3+128*a*c*d^2*x^2-105*b*d^3*x-256*a*d^3)/c^2*x*((c*x^2+d)/x^2)^(1/ 
2)+3/128*b*d^4/c^(5/2)*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 = 0.12 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.73 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=\left [\frac {105 \, b \sqrt {c} d^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (560 \, b c^{4} x^{8} + 640 \, a c^{4} x^{7} + 840 \, b c^{3} d x^{6} + 1024 \, a c^{3} d x^{5} + 70 \, b c^{2} d^{2} x^{4} + 128 \, a c^{2} d^{2} x^{3} - 105 \, b c d^{3} x^{2} - 256 \, a c d^{3} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8960 \, c^{3}}, -\frac {105 \, b \sqrt {-c} d^{4} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (560 \, b c^{4} x^{8} + 640 \, a c^{4} x^{7} + 840 \, b c^{3} d x^{6} + 1024 \, a c^{3} d x^{5} + 70 \, b c^{2} d^{2} x^{4} + 128 \, a c^{2} d^{2} x^{3} - 105 \, b c d^{3} x^{2} - 256 \, a c d^{3} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4480 \, c^{3}}\right ] \] Input:

integrate((c+d/x^2)^(3/2)*x^6*(b*x+a),x, algorithm="fricas")
 

Output:

[1/8960*(105*b*sqrt(c)*d^4*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x 
^2) - d) + 2*(560*b*c^4*x^8 + 640*a*c^4*x^7 + 840*b*c^3*d*x^6 + 1024*a*c^3 
*d*x^5 + 70*b*c^2*d^2*x^4 + 128*a*c^2*d^2*x^3 - 105*b*c*d^3*x^2 - 256*a*c* 
d^3*x)*sqrt((c*x^2 + d)/x^2))/c^3, -1/4480*(105*b*sqrt(-c)*d^4*arctan(sqrt 
(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - (560*b*c^4*x^8 + 640*a*c^4*x 
^7 + 840*b*c^3*d*x^6 + 1024*a*c^3*d*x^5 + 70*b*c^2*d^2*x^4 + 128*a*c^2*d^2 
*x^3 - 105*b*c*d^3*x^2 - 256*a*c*d^3*x)*sqrt((c*x^2 + d)/x^2))/c^3]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (158) = 316\).

Time = 25.26 (sec) , antiderivative size = 585, normalized size of antiderivative = 3.50 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=\frac {15 a c^{6} d^{\frac {9}{2}} x^{10} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {33 a c^{5} d^{\frac {11}{2}} x^{8} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {17 a c^{4} d^{\frac {13}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {3 a c^{3} d^{\frac {15}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {12 a c^{2} d^{\frac {17}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {8 a c d^{\frac {19}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {a d^{\frac {3}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {a d^{\frac {5}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c} - \frac {2 a d^{\frac {7}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c^{2}} + \frac {b c^{2} x^{9}}{8 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {5 b c \sqrt {d} x^{7}}{16 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {13 b d^{\frac {3}{2}} x^{5}}{64 \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b d^{\frac {5}{2}} x^{3}}{128 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {3 b d^{\frac {7}{2}} x}{128 c^{2} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b d^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{128 c^{\frac {5}{2}}} \] Input:

integrate((c+d/x**2)**(3/2)*x**6*(b*x+a),x)
 

Output:

15*a*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*a*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*a*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 + 1 
05*c**3*d**6) + 3*a*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*a*c**2*d**(17/2)*x**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) + 
 8*a*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) + a*d**(3/2)*x**4*sqrt(c*x**2/d + 1)/5 + a*d**(5/2)*x* 
*2*sqrt(c*x**2/d + 1)/(15*c) - 2*a*d**(7/2)*sqrt(c*x**2/d + 1)/(15*c**2) + 
 b*c**2*x**9/(8*sqrt(d)*sqrt(c*x**2/d + 1)) + 5*b*c*sqrt(d)*x**7/(16*sqrt( 
c*x**2/d + 1)) + 13*b*d**(3/2)*x**5/(64*sqrt(c*x**2/d + 1)) - b*d**(5/2)*x 
**3/(128*c*sqrt(c*x**2/d + 1)) - 3*b*d**(7/2)*x/(128*c**2*sqrt(c*x**2/d + 
1)) + 3*b*d**4*asinh(sqrt(c)*x/sqrt(d))/(128*c**(5/2))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.23 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=-\frac {1}{256} \, {\left (\frac {3 \, d^{4} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d^{4} - 11 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c d^{4} - 11 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} d^{4} + 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{3} d^{4}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{4} c^{2} - 4 \, {\left (c + \frac {d}{x^{2}}\right )}^{3} c^{3} + 6 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} c^{4} - 4 \, {\left (c + \frac {d}{x^{2}}\right )} c^{5} + c^{6}}\right )} b + \frac {{\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5}\right )} a}{35 \, c^{2}} \] Input:

integrate((c+d/x^2)^(3/2)*x^6*(b*x+a),x, algorithm="maxima")
 

Output:

-1/256*(3*d^4*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c))) 
/c^(5/2) + 2*(3*(c + d/x^2)^(7/2)*d^4 - 11*(c + d/x^2)^(5/2)*c*d^4 - 11*(c 
 + d/x^2)^(3/2)*c^2*d^4 + 3*sqrt(c + d/x^2)*c^3*d^4)/((c + d/x^2)^4*c^2 - 
4*(c + d/x^2)^3*c^3 + 6*(c + d/x^2)^2*c^4 - 4*(c + d/x^2)*c^5 + c^6))*b + 
1/35*(5*(c + d/x^2)^(7/2)*x^7 - 7*(c + d/x^2)^(5/2)*d*x^5)*a/c^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=-\frac {3 \, b d^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right ) \mathrm {sgn}\left (x\right )}{128 \, c^{\frac {5}{2}}} - \frac {1}{4480} \, \sqrt {c x^{2} + d} {\left (\frac {256 \, a d^{3} \mathrm {sgn}\left (x\right )}{c^{2}} + {\left (\frac {105 \, b d^{3} \mathrm {sgn}\left (x\right )}{c^{2}} - 2 \, {\left (\frac {64 \, a d^{2} \mathrm {sgn}\left (x\right )}{c} + {\left (\frac {35 \, b d^{2} \mathrm {sgn}\left (x\right )}{c} + 4 \, {\left (128 \, a d \mathrm {sgn}\left (x\right ) + 5 \, {\left (21 \, b d \mathrm {sgn}\left (x\right ) + 2 \, {\left (7 \, b c x \mathrm {sgn}\left (x\right ) + 8 \, a c \mathrm {sgn}\left (x\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} + \frac {{\left (105 \, b d^{4} \log \left ({\left | d \right |}\right ) + 512 \, a \sqrt {c} d^{\frac {7}{2}}\right )} \mathrm {sgn}\left (x\right )}{8960 \, c^{\frac {5}{2}}} \] Input:

integrate((c+d/x^2)^(3/2)*x^6*(b*x+a),x, algorithm="giac")
 

Output:

-3/128*b*d^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))*sgn(x)/c^(5/2) - 1/448 
0*sqrt(c*x^2 + d)*(256*a*d^3*sgn(x)/c^2 + (105*b*d^3*sgn(x)/c^2 - 2*(64*a* 
d^2*sgn(x)/c + (35*b*d^2*sgn(x)/c + 4*(128*a*d*sgn(x) + 5*(21*b*d*sgn(x) + 
 2*(7*b*c*x*sgn(x) + 8*a*c*sgn(x))*x)*x)*x)*x)*x)*x) + 1/8960*(105*b*d^4*l 
og(abs(d)) + 512*a*sqrt(c)*d^(7/2))*sgn(x)/c^(5/2)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 7.65 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=\sqrt {c+\frac {d}{x^2}}\,\left (\frac {a\,c\,x^7}{7}+\frac {8\,a\,d\,x^5}{35}+\frac {a\,d^2\,x^3}{35\,c}-\frac {2\,a\,d^3\,x}{35\,c^2}\right )+\frac {11\,b\,x^8\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{128}+\frac {11\,b\,x^8\,{\left (c+\frac {d}{x^2}\right )}^{5/2}}{128\,c}-\frac {3\,b\,x^8\,{\left (c+\frac {d}{x^2}\right )}^{7/2}}{128\,c^2}-\frac {3\,b\,c\,x^8\,\sqrt {c+\frac {d}{x^2}}}{128}-\frac {b\,d^4\,\mathrm {atan}\left (\frac {\sqrt {c+\frac {d}{x^2}}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,3{}\mathrm {i}}{128\,c^{5/2}} \] Input:

int(x^6*(c + d/x^2)^(3/2)*(a + b*x),x)
 

Output:

(c + d/x^2)^(1/2)*((a*c*x^7)/7 + (8*a*d*x^5)/35 + (a*d^2*x^3)/(35*c) - (2* 
a*d^3*x)/(35*c^2)) + (11*b*x^8*(c + d/x^2)^(3/2))/128 + (11*b*x^8*(c + d/x 
^2)^(5/2))/(128*c) - (3*b*x^8*(c + d/x^2)^(7/2))/(128*c^2) - (b*d^4*atan(( 
(c + d/x^2)^(1/2)*1i)/c^(1/2))*3i)/(128*c^(5/2)) - (3*b*c*x^8*(c + d/x^2)^ 
(1/2))/128
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.04 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 (a+b x) \, dx=\frac {640 \sqrt {c \,x^{2}+d}\, a \,c^{4} x^{6}+1024 \sqrt {c \,x^{2}+d}\, a \,c^{3} d \,x^{4}+128 \sqrt {c \,x^{2}+d}\, a \,c^{2} d^{2} x^{2}-256 \sqrt {c \,x^{2}+d}\, a c \,d^{3}+560 \sqrt {c \,x^{2}+d}\, b \,c^{4} x^{7}+840 \sqrt {c \,x^{2}+d}\, b \,c^{3} d \,x^{5}+70 \sqrt {c \,x^{2}+d}\, b \,c^{2} d^{2} x^{3}-105 \sqrt {c \,x^{2}+d}\, b c \,d^{3} x +105 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b \,d^{4}}{4480 c^{3}} \] Input:

int((c+d/x^2)^(3/2)*x^6*(b*x+a),x)
 

Output:

(640*sqrt(c*x**2 + d)*a*c**4*x**6 + 1024*sqrt(c*x**2 + d)*a*c**3*d*x**4 + 
128*sqrt(c*x**2 + d)*a*c**2*d**2*x**2 - 256*sqrt(c*x**2 + d)*a*c*d**3 + 56 
0*sqrt(c*x**2 + d)*b*c**4*x**7 + 840*sqrt(c*x**2 + d)*b*c**3*d*x**5 + 70*s 
qrt(c*x**2 + d)*b*c**2*d**2*x**3 - 105*sqrt(c*x**2 + d)*b*c*d**3*x + 105*s 
qrt(c)*log((sqrt(c*x**2 + d) + sqrt(c)*x)/sqrt(d))*b*d**4)/(4480*c**3)