\(\int \frac {(c+d x)^2 (a+b x^2)^{3/2}}{x^9} \, dx\) [1059]

Optimal result

Integrand size = 22, antiderivative size = 218 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=\frac {\left (3 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{48 x^6}+\frac {7 b \left (3 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{192 a x^4}+\frac {b^2 \left (3 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{128 a^2 x^2}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {2 c d \left (a+b x^2\right )^{5/2}}{7 a x^7}+\frac {4 b c d \left (a+b x^2\right )^{5/2}}{35 a^2 x^5}-\frac {b^3 \left (3 b c^2-8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \] Output:

1/48*(-8*a*d^2+3*b*c^2)*(b*x^2+a)^(1/2)/x^6+7/192*b*(-8*a*d^2+3*b*c^2)*(b* 
x^2+a)^(1/2)/a/x^4+1/128*b^2*(-8*a*d^2+3*b*c^2)*(b*x^2+a)^(1/2)/a^2/x^2-1/ 
8*c^2*(b*x^2+a)^(5/2)/a/x^8-2/7*c*d*(b*x^2+a)^(5/2)/a/x^7+4/35*b*c*d*(b*x^ 
2+a)^(5/2)/a^2/x^5-1/128*b^3*(-8*a*d^2+3*b*c^2)*arctanh((b*x^2+a)^(1/2)/a^ 
(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=-\frac {\sqrt {a+b x^2} \left (-3 b^3 c x^6 (105 c+512 d x)+80 a^3 \left (21 c^2+48 c d x+28 d^2 x^2\right )+6 a b^2 x^4 \left (35 c^2+128 c d x+140 d^2 x^2\right )+8 a^2 b x^2 \left (315 c^2+768 c d x+490 d^2 x^2\right )\right )}{13440 a^2 x^8}+\frac {b^3 \left (3 b c^2-8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{64 a^{5/2}} \] Input:

Integrate[((c + d*x)^2*(a + b*x^2)^(3/2))/x^9,x]
 

Output:

-1/13440*(Sqrt[a + b*x^2]*(-3*b^3*c*x^6*(105*c + 512*d*x) + 80*a^3*(21*c^2 
 + 48*c*d*x + 28*d^2*x^2) + 6*a*b^2*x^4*(35*c^2 + 128*c*d*x + 140*d^2*x^2) 
 + 8*a^2*b*x^2*(315*c^2 + 768*c*d*x + 490*d^2*x^2)))/(a^2*x^8) + (b^3*(3*b 
*c^2 - 8*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(64*a^(5/2 
))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {540, 25, 539, 27, 539, 25, 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.

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 539

\(\displaystyle \frac {-\frac {\int \frac {a \left (7 \left (3 b c^2-8 a d^2\right )+32 b c d x\right ) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{7} \int \frac {\left (7 \left (3 b c^2-8 a d^2\right )+32 b c d x\right ) \left (b x^2+a\right )^{3/2}}{x^7}dx-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{7} \left (\frac {\int -\frac {b \left (192 a c d-7 \left (3 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}+\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {\int \frac {b \left (192 a c d-7 \left (3 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \int \frac {\left (192 a c d-7 \left (3 b c^2-8 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \left (-7 \left (3 b c^2-8 a d^2\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {192 c d \left (a+b x^2\right )^{5/2}}{5 x^5}\right )}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \left (-\frac {7}{2} \left (3 b c^2-8 a d^2\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {192 c d \left (a+b x^2\right )^{5/2}}{5 x^5}\right )}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \left (-\frac {7}{2} \left (3 b c^2-8 a d^2\right ) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {192 c d \left (a+b x^2\right )^{5/2}}{5 x^5}\right )}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \left (-\frac {7}{2} \left (3 b c^2-8 a d^2\right ) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {192 c d \left (a+b x^2\right )^{5/2}}{5 x^5}\right )}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \left (-\frac {7}{2} \left (3 b c^2-8 a d^2\right ) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {192 c d \left (a+b x^2\right )^{5/2}}{5 x^5}\right )}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (a+b x^2\right )^{5/2} \left (3 b c^2-8 a d^2\right )}{6 a x^6}-\frac {b \left (-\frac {7}{2} \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right ) \left (3 b c^2-8 a d^2\right )-\frac {192 c d \left (a+b x^2\right )^{5/2}}{5 x^5}\right )}{6 a}\right )-\frac {16 c d \left (a+b x^2\right )^{5/2}}{7 x^7}}{8 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{8 a x^8}\)

Input:

Int[((c + d*x)^2*(a + b*x^2)^(3/2))/x^9,x]
 

Output:

-1/8*(c^2*(a + b*x^2)^(5/2))/(a*x^8) + ((-16*c*d*(a + b*x^2)^(5/2))/(7*x^7 
) + ((7*(3*b*c^2 - 8*a*d^2)*(a + b*x^2)^(5/2))/(6*a*x^6) - (b*((-192*c*d*( 
a + b*x^2)^(5/2))/(5*x^5) - (7*(3*b*c^2 - 8*a*d^2)*(-1/2*(a + b*x^2)^(3/2) 
/x^4 + (3*b*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]) 
/Sqrt[a]))/4))/2))/(6*a))/7)/(8*a)
 

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 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*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*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-1536 d \,x^{7} c \,b^{3}+840 a \,b^{2} d^{2} x^{6}-315 b^{3} c^{2} x^{6}+768 a \,b^{2} c d \,x^{5}+3920 a^{2} b \,d^{2} x^{4}+210 a \,b^{2} c^{2} x^{4}+6144 a^{2} b c d \,x^{3}+2240 a^{3} d^{2} x^{2}+2520 a^{2} b \,c^{2} x^{2}+3840 a^{3} c d x +1680 c^{2} a^{3}\right )}{13440 x^{8} a^{2}}+\frac {\left (8 a \,d^{2}-3 b \,c^{2}\right ) b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {5}{2}}}\)	\(182\)
default	\(c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+2 c d \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )\)	\(322\)

Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)/x^9,x,method=_RETURNVERBOSE)
 

Output:

-1/13440*(b*x^2+a)^(1/2)*(-1536*b^3*c*d*x^7+840*a*b^2*d^2*x^6-315*b^3*c^2* 
x^6+768*a*b^2*c*d*x^5+3920*a^2*b*d^2*x^4+210*a*b^2*c^2*x^4+6144*a^2*b*c*d* 
x^3+2240*a^3*d^2*x^2+2520*a^2*b*c^2*x^2+3840*a^3*c*d*x+1680*a^3*c^2)/x^8/a 
^2+1/128*(8*a*d^2-3*b*c^2)*b^3/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/ 
x)
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.82 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=\left [-\frac {105 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{3} d^{2}\right )} \sqrt {a} x^{8} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (1536 \, a b^{3} c d x^{7} - 768 \, a^{2} b^{2} c d x^{5} - 6144 \, a^{3} b c d x^{3} - 3840 \, a^{4} c d x + 105 \, {\left (3 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} d^{2}\right )} x^{6} - 1680 \, a^{4} c^{2} - 70 \, {\left (3 \, a^{2} b^{2} c^{2} + 56 \, a^{3} b d^{2}\right )} x^{4} - 280 \, {\left (9 \, a^{3} b c^{2} + 8 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{26880 \, a^{3} x^{8}}, \frac {105 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{3} d^{2}\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (1536 \, a b^{3} c d x^{7} - 768 \, a^{2} b^{2} c d x^{5} - 6144 \, a^{3} b c d x^{3} - 3840 \, a^{4} c d x + 105 \, {\left (3 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} d^{2}\right )} x^{6} - 1680 \, a^{4} c^{2} - 70 \, {\left (3 \, a^{2} b^{2} c^{2} + 56 \, a^{3} b d^{2}\right )} x^{4} - 280 \, {\left (9 \, a^{3} b c^{2} + 8 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{13440 \, a^{3} x^{8}}\right ] \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^9,x, algorithm="fricas")
 

Output:

[-1/26880*(105*(3*b^4*c^2 - 8*a*b^3*d^2)*sqrt(a)*x^8*log(-(b*x^2 + 2*sqrt( 
b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(1536*a*b^3*c*d*x^7 - 768*a^2*b^2*c*d*x 
^5 - 6144*a^3*b*c*d*x^3 - 3840*a^4*c*d*x + 105*(3*a*b^3*c^2 - 8*a^2*b^2*d^ 
2)*x^6 - 1680*a^4*c^2 - 70*(3*a^2*b^2*c^2 + 56*a^3*b*d^2)*x^4 - 280*(9*a^3 
*b*c^2 + 8*a^4*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3*x^8), 1/13440*(105*(3*b^4*c 
^2 - 8*a*b^3*d^2)*sqrt(-a)*x^8*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (1536* 
a*b^3*c*d*x^7 - 768*a^2*b^2*c*d*x^5 - 6144*a^3*b*c*d*x^3 - 3840*a^4*c*d*x 
+ 105*(3*a*b^3*c^2 - 8*a^2*b^2*d^2)*x^6 - 1680*a^4*c^2 - 70*(3*a^2*b^2*c^2 
 + 56*a^3*b*d^2)*x^4 - 280*(9*a^3*b*c^2 + 8*a^4*d^2)*x^2)*sqrt(b*x^2 + a)) 
/(a^3*x^8)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (204) = 408\).

Time = 36.00 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.54 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=- \frac {30 a^{6} b^{\frac {9}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {66 a^{5} b^{\frac {11}{2}} c d x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {34 a^{4} b^{\frac {13}{2}} c d x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {6 a^{3} b^{\frac {15}{2}} c d x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {24 a^{2} b^{\frac {17}{2}} c d x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {a^{2} c^{2}}{8 \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a^{2} d^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {16 a b^{\frac {19}{2}} c d x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {5 a \sqrt {b} c^{2}}{16 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 a \sqrt {b} d^{2}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {13 b^{\frac {3}{2}} c^{2}}{64 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 b^{\frac {3}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {17 b^{\frac {3}{2}} d^{2}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {5}{2}} c^{2}}{128 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 b^{\frac {5}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{15 a x^{2}} - \frac {b^{\frac {5}{2}} d^{2}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {7}{2}} c^{2}}{128 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {4 b^{\frac {7}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{2}} + \frac {b^{3} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} - \frac {3 b^{4} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{128 a^{\frac {5}{2}}} \] Input:

integrate((d*x+c)**2*(b*x**2+a)**(3/2)/x**9,x)
 

Output:

-30*a**6*b**(9/2)*c*d*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4* 
b**5*x**8 + 105*a**3*b**6*x**10) - 66*a**5*b**(11/2)*c*d*x**2*sqrt(a/(b*x* 
*2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 
 34*a**4*b**(13/2)*c*d*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210 
*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 6*a**3*b**(15/2)*c*d*x**6*sqrt(a/ 
(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x** 
10) - 24*a**2*b**(17/2)*c*d*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 
+ 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - a**2*c**2/(8*sqrt(b)*x**9*sq 
rt(a/(b*x**2) + 1)) - a**2*d**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 16 
*a*b**(19/2)*c*d*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4 
*b**5*x**8 + 105*a**3*b**6*x**10) - 5*a*sqrt(b)*c**2/(16*x**7*sqrt(a/(b*x* 
*2) + 1)) - 11*a*sqrt(b)*d**2/(24*x**5*sqrt(a/(b*x**2) + 1)) - 13*b**(3/2) 
*c**2/(64*x**5*sqrt(a/(b*x**2) + 1)) - 2*b**(3/2)*c*d*sqrt(a/(b*x**2) + 1) 
/(5*x**4) - 17*b**(3/2)*d**2/(48*x**3*sqrt(a/(b*x**2) + 1)) + b**(5/2)*c** 
2/(128*a*x**3*sqrt(a/(b*x**2) + 1)) - 2*b**(5/2)*c*d*sqrt(a/(b*x**2) + 1)/ 
(15*a*x**2) - b**(5/2)*d**2/(16*a*x*sqrt(a/(b*x**2) + 1)) + 3*b**(7/2)*c** 
2/(128*a**2*x*sqrt(a/(b*x**2) + 1)) + 4*b**(7/2)*c*d*sqrt(a/(b*x**2) + 1)/ 
(15*a**2) + b**3*d**2*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(3/2)) - 3*b**4*c* 
*2*asinh(sqrt(a)/(sqrt(b)*x))/(128*a**(5/2))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=-\frac {3 \, b^{4} c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} + \frac {b^{3} d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4} c^{2}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} b^{4} c^{2}}{128 \, a^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} d^{2}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} b^{3} d^{2}}{16 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} c^{2}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} d^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} c^{2}}{64 \, a^{3} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b d^{2}}{24 \, a^{2} x^{4}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c d}{35 \, a^{2} x^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b c^{2}}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{2}}{6 \, a x^{6}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d}{7 \, a x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2}}{8 \, a x^{8}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^9,x, algorithm="maxima")
 

Output:

-3/128*b^4*c^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/16*b^3*d^2*arcsin 
h(a/(sqrt(a*b)*abs(x)))/a^(3/2) + 1/128*(b*x^2 + a)^(3/2)*b^4*c^2/a^4 + 3/ 
128*sqrt(b*x^2 + a)*b^4*c^2/a^3 - 1/48*(b*x^2 + a)^(3/2)*b^3*d^2/a^3 - 1/1 
6*sqrt(b*x^2 + a)*b^3*d^2/a^2 - 1/128*(b*x^2 + a)^(5/2)*b^3*c^2/(a^4*x^2) 
+ 1/48*(b*x^2 + a)^(5/2)*b^2*d^2/(a^3*x^2) - 1/64*(b*x^2 + a)^(5/2)*b^2*c^ 
2/(a^3*x^4) + 1/24*(b*x^2 + a)^(5/2)*b*d^2/(a^2*x^4) + 4/35*(b*x^2 + a)^(5 
/2)*b*c*d/(a^2*x^5) + 1/16*(b*x^2 + a)^(5/2)*b*c^2/(a^2*x^6) - 1/6*(b*x^2 
+ a)^(5/2)*d^2/(a*x^6) - 2/7*(b*x^2 + a)^(5/2)*c*d/(a*x^7) - 1/8*(b*x^2 + 
a)^(5/2)*c^2/(a*x^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (186) = 372\).

Time = 0.14 (sec) , antiderivative size = 709, normalized size of antiderivative = 3.25 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=\frac {{\left (3 \, b^{4} c^{2} - 8 \, a b^{3} d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{2}} - \frac {315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{15} b^{4} c^{2} - 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{15} a b^{3} d^{2} - 2415 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{13} a b^{4} c^{2} - 11480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{13} a^{2} b^{3} d^{2} - 53760 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{2} b^{\frac {7}{2}} c d - 34965 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} a^{2} b^{4} c^{2} + 3640 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} a^{3} b^{3} d^{2} - 70455 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} a^{3} b^{4} c^{2} + 8680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} a^{4} b^{3} d^{2} - 53760 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} b^{\frac {7}{2}} c d - 70455 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a^{4} b^{4} c^{2} + 8680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a^{5} b^{3} d^{2} + 86016 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} b^{\frac {7}{2}} c d - 34965 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{5} b^{4} c^{2} + 3640 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{6} b^{3} d^{2} + 10752 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} b^{\frac {7}{2}} c d - 2415 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{6} b^{4} c^{2} - 11480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{7} b^{3} d^{2} + 12288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} b^{\frac {7}{2}} c d + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{7} b^{4} c^{2} - 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{8} b^{3} d^{2} - 1536 \, a^{8} b^{\frac {7}{2}} c d}{6720 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{8} a^{2}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^9,x, algorithm="giac")
 

Output:

1/64*(3*b^4*c^2 - 8*a*b^3*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt( 
-a))/(sqrt(-a)*a^2) - 1/6720*(315*(sqrt(b)*x - sqrt(b*x^2 + a))^15*b^4*c^2 
 - 840*(sqrt(b)*x - sqrt(b*x^2 + a))^15*a*b^3*d^2 - 2415*(sqrt(b)*x - sqrt 
(b*x^2 + a))^13*a*b^4*c^2 - 11480*(sqrt(b)*x - sqrt(b*x^2 + a))^13*a^2*b^3 
*d^2 - 53760*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(7/2)*c*d - 34965*(sqr 
t(b)*x - sqrt(b*x^2 + a))^11*a^2*b^4*c^2 + 3640*(sqrt(b)*x - sqrt(b*x^2 + 
a))^11*a^3*b^3*d^2 - 70455*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a^3*b^4*c^2 + 8 
680*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a^4*b^3*d^2 - 53760*(sqrt(b)*x - sqrt( 
b*x^2 + a))^8*a^4*b^(7/2)*c*d - 70455*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^4* 
b^4*c^2 + 8680*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^5*b^3*d^2 + 86016*(sqrt(b 
)*x - sqrt(b*x^2 + a))^6*a^5*b^(7/2)*c*d - 34965*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^5*a^5*b^4*c^2 + 3640*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a^6*b^3*d^2 + 10 
752*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^6*b^(7/2)*c*d - 2415*(sqrt(b)*x - sq 
rt(b*x^2 + a))^3*a^6*b^4*c^2 - 11480*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^7*b 
^3*d^2 + 12288*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(7/2)*c*d + 315*(sqrt 
(b)*x - sqrt(b*x^2 + a))*a^7*b^4*c^2 - 840*(sqrt(b)*x - sqrt(b*x^2 + a))*a 
^8*b^3*d^2 - 1536*a^8*b^(7/2)*c*d)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^ 
8*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2}{x^9} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)^2)/x^9,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)^2)/x^9, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.73 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^9} \, dx=\frac {-1680 \sqrt {b \,x^{2}+a}\, a^{4} c^{2}-3840 \sqrt {b \,x^{2}+a}\, a^{4} c d x -2240 \sqrt {b \,x^{2}+a}\, a^{4} d^{2} x^{2}-2520 \sqrt {b \,x^{2}+a}\, a^{3} b \,c^{2} x^{2}-6144 \sqrt {b \,x^{2}+a}\, a^{3} b c d \,x^{3}-3920 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{2} x^{4}-210 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} x^{4}-768 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c d \,x^{5}-840 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{2} x^{6}+315 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} x^{6}+1536 \sqrt {b \,x^{2}+a}\, a \,b^{3} c d \,x^{7}-840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d^{2} x^{8}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c^{2} x^{8}+840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d^{2} x^{8}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c^{2} x^{8}-1536 \sqrt {b}\, a \,b^{3} c d \,x^{8}}{13440 a^{3} x^{8}} \] Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)/x^9,x)
 

Output:

( - 1680*sqrt(a + b*x**2)*a**4*c**2 - 3840*sqrt(a + b*x**2)*a**4*c*d*x - 2 
240*sqrt(a + b*x**2)*a**4*d**2*x**2 - 2520*sqrt(a + b*x**2)*a**3*b*c**2*x* 
*2 - 6144*sqrt(a + b*x**2)*a**3*b*c*d*x**3 - 3920*sqrt(a + b*x**2)*a**3*b* 
d**2*x**4 - 210*sqrt(a + b*x**2)*a**2*b**2*c**2*x**4 - 768*sqrt(a + b*x**2 
)*a**2*b**2*c*d*x**5 - 840*sqrt(a + b*x**2)*a**2*b**2*d**2*x**6 + 315*sqrt 
(a + b*x**2)*a*b**3*c**2*x**6 + 1536*sqrt(a + b*x**2)*a*b**3*c*d*x**7 - 84 
0*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*d** 
2*x**8 + 315*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a)) 
*b**4*c**2*x**8 + 840*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x) 
/sqrt(a))*a*b**3*d**2*x**8 - 315*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + 
 sqrt(b)*x)/sqrt(a))*b**4*c**2*x**8 - 1536*sqrt(b)*a*b**3*c*d*x**8)/(13440 
*a**3*x**8)