\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2+D x^3)}{x^{10}} \, dx\) [87]

Optimal result

Integrand size = 30, antiderivative size = 233 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=-\frac {(3 b B+8 a D) \sqrt {a+b x^2}}{48 x^6}-\frac {b (3 b B+56 a D) \sqrt {a+b x^2}}{192 a x^4}+\frac {b^2 (3 b B-8 a D) \sqrt {a+b x^2}}{128 a^2 x^2}-\frac {B \left (a+b x^2\right )^{3/2}}{8 x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac {(4 A b-9 a C) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b (4 A b-9 a C) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}-\frac {b^3 (3 b B-8 a D) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \] Output:

-1/48*(3*B*b+8*D*a)*(b*x^2+a)^(1/2)/x^6-1/192*b*(3*B*b+56*D*a)*(b*x^2+a)^( 
1/2)/a/x^4+1/128*b^2*(3*B*b-8*D*a)*(b*x^2+a)^(1/2)/a^2/x^2-1/8*B*(b*x^2+a) 
^(3/2)/x^8-1/9*A*(b*x^2+a)^(5/2)/a/x^9+1/63*(4*A*b-9*C*a)*(b*x^2+a)^(5/2)/ 
a^2/x^7-2/315*b*(4*A*b-9*C*a)*(b*x^2+a)^(5/2)/a^3/x^5-1/128*b^3*(3*B*b-8*D 
*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=-\frac {\sqrt {a+b x^2} \left (1024 A b^4 x^8+80 a^4 \left (56 A+63 B x+72 C x^2+84 D x^3\right )-a b^3 x^6 (512 A+9 x (105 B+256 C x))+6 a^2 b^2 x^4 \left (64 A+3 x \left (35 B+64 C x+140 D x^2\right )\right )+8 a^3 b x^2 \left (800 A+3 x \left (315 B+384 C x+490 D x^2\right )\right )\right )}{40320 a^3 x^9}+\frac {b^3 (3 b B-8 a D) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{64 a^{5/2}} \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^10,x]
 

Output:

-1/40320*(Sqrt[a + b*x^2]*(1024*A*b^4*x^8 + 80*a^4*(56*A + 63*B*x + 72*C*x 
^2 + 84*D*x^3) - a*b^3*x^6*(512*A + 9*x*(105*B + 256*C*x)) + 6*a^2*b^2*x^4 
*(64*A + 3*x*(35*B + 64*C*x + 140*D*x^2)) + 8*a^3*b*x^2*(800*A + 3*x*(315* 
B + 384*C*x + 490*D*x^2))))/(a^3*x^9) + (b^3*(3*b*B - 8*a*D)*ArcTanh[(Sqrt 
[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(64*a^(5/2))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2338, 25, 2338, 27, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx\)

\(\Big \downarrow \) 2338

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (9 a D x^2-(4 A b-9 a C) x+9 a B\right )}{x^9}dx}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (9 a D x^2-(4 A b-9 a C) x+9 a B\right )}{x^9}dx}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int \frac {a (8 (4 A b-9 a C)+9 (3 b B-8 a D) x) \left (b x^2+a\right )^{3/2}}{x^8}dx}{8 a}-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{8} \int \frac {(8 (4 A b-9 a C)+9 (3 b B-8 a D) x) \left (b x^2+a\right )^{3/2}}{x^8}dx-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\int -\frac {(63 a (3 b B-8 a D)-16 b (4 A b-9 a C) x) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {\int \frac {(63 a (3 b B-8 a D)-16 b (4 A b-9 a C) x) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {\int \frac {3 a b (32 (4 A b-9 a C)+21 (3 b B-8 a D) x) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \int \frac {(32 (4 A b-9 a C)+21 (3 b B-8 a D) x) \left (b x^2+a\right )^{3/2}}{x^6}dx-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \left (21 (3 b B-8 a D) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} (3 b B-8 a D) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} (3 b B-8 a D) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} (3 b B-8 a D) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} (3 b B-8 a D) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{8} \left (\frac {8 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{7 a x^7}-\frac {-\frac {1}{2} b \left (\frac {21}{2} \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right ) (3 b B-8 a D)-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b-9 a C)}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (3 b B-8 a D)}{2 x^6}}{7 a}\right )-\frac {9 B \left (a+b x^2\right )^{5/2}}{8 x^8}}{9 a}-\frac {A \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^10,x]
 

Output:

-1/9*(A*(a + b*x^2)^(5/2))/(a*x^9) + ((-9*B*(a + b*x^2)^(5/2))/(8*x^8) + ( 
(8*(4*A*b - 9*a*C)*(a + b*x^2)^(5/2))/(7*a*x^7) - ((-21*(3*b*B - 8*a*D)*(a 
 + b*x^2)^(5/2))/(2*x^6) - (b*((-32*(4*A*b - 9*a*C)*(a + b*x^2)^(5/2))/(5* 
a*x^5) + (21*(3*b*B - 8*a*D)*(-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))/2)/ 
(7*a))/8)/(9*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 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.62

method	result	size
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )+D \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )\)	\(378\)

Input:

int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^10,x,method=_RETURNVERBOSE)
 

Output:

A*(-1/9/a/x^9*(b*x^2+a)^(5/2)-4/9*b/a*(-1/7/a/x^7*(b*x^2+a)^(5/2)+2/35*b/a 
^2/x^5*(b*x^2+a)^(5/2)))+B*(-1/8/a/x^8*(b*x^2+a)^(5/2)-3/8*b/a*(-1/6/a/x^6 
*(b*x^2+a)^(5/2)-1/6*b/a*(-1/4/a/x^4*(b*x^2+a)^(5/2)+1/4*b/a*(-1/2/a/x^2*( 
b*x^2+a)^(5/2)+3/2*b/a*(1/3*(b*x^2+a)^(3/2)+a*((b*x^2+a)^(1/2)-a^(1/2)*ln( 
(2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)))))))+C*(-1/7/a/x^7*(b*x^2+a)^(5/2)+2/3 
5*b/a^2/x^5*(b*x^2+a)^(5/2))+D*(-1/6/a/x^6*(b*x^2+a)^(5/2)-1/6*b/a*(-1/4/a 
/x^4*(b*x^2+a)^(5/2)+1/4*b/a*(-1/2/a/x^2*(b*x^2+a)^(5/2)+3/2*b/a*(1/3*(b*x 
^2+a)^(3/2)+a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/ 
x))))))
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=\left [-\frac {315 \, {\left (8 \, D a b^{3} - 3 \, B b^{4}\right )} \sqrt {a} x^{9} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (256 \, {\left (9 \, C a b^{3} - 4 \, A b^{4}\right )} x^{8} - 315 \, {\left (8 \, D a^{2} b^{2} - 3 \, B a b^{3}\right )} x^{7} - 128 \, {\left (9 \, C a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 5040 \, B a^{4} x - 210 \, {\left (56 \, D a^{3} b + 3 \, B a^{2} b^{2}\right )} x^{5} - 4480 \, A a^{4} - 384 \, {\left (24 \, C a^{3} b + A a^{2} b^{2}\right )} x^{4} - 840 \, {\left (8 \, D a^{4} + 9 \, B a^{3} b\right )} x^{3} - 640 \, {\left (9 \, C a^{4} + 10 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{80640 \, a^{3} x^{9}}, -\frac {315 \, {\left (8 \, D a b^{3} - 3 \, B b^{4}\right )} \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (256 \, {\left (9 \, C a b^{3} - 4 \, A b^{4}\right )} x^{8} - 315 \, {\left (8 \, D a^{2} b^{2} - 3 \, B a b^{3}\right )} x^{7} - 128 \, {\left (9 \, C a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 5040 \, B a^{4} x - 210 \, {\left (56 \, D a^{3} b + 3 \, B a^{2} b^{2}\right )} x^{5} - 4480 \, A a^{4} - 384 \, {\left (24 \, C a^{3} b + A a^{2} b^{2}\right )} x^{4} - 840 \, {\left (8 \, D a^{4} + 9 \, B a^{3} b\right )} x^{3} - 640 \, {\left (9 \, C a^{4} + 10 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{40320 \, a^{3} x^{9}}\right ] \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^10,x, algorithm="fricas")
 

Output:

[-1/80640*(315*(8*D*a*b^3 - 3*B*b^4)*sqrt(a)*x^9*log(-(b*x^2 - 2*sqrt(b*x^ 
2 + a)*sqrt(a) + 2*a)/x^2) - 2*(256*(9*C*a*b^3 - 4*A*b^4)*x^8 - 315*(8*D*a 
^2*b^2 - 3*B*a*b^3)*x^7 - 128*(9*C*a^2*b^2 - 4*A*a*b^3)*x^6 - 5040*B*a^4*x 
 - 210*(56*D*a^3*b + 3*B*a^2*b^2)*x^5 - 4480*A*a^4 - 384*(24*C*a^3*b + A*a 
^2*b^2)*x^4 - 840*(8*D*a^4 + 9*B*a^3*b)*x^3 - 640*(9*C*a^4 + 10*A*a^3*b)*x 
^2)*sqrt(b*x^2 + a))/(a^3*x^9), -1/40320*(315*(8*D*a*b^3 - 3*B*b^4)*sqrt(- 
a)*x^9*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (256*(9*C*a*b^3 - 4*A*b^4)*x^8 
 - 315*(8*D*a^2*b^2 - 3*B*a*b^3)*x^7 - 128*(9*C*a^2*b^2 - 4*A*a*b^3)*x^6 - 
 5040*B*a^4*x - 210*(56*D*a^3*b + 3*B*a^2*b^2)*x^5 - 4480*A*a^4 - 384*(24* 
C*a^3*b + A*a^2*b^2)*x^4 - 840*(8*D*a^4 + 9*B*a^3*b)*x^3 - 640*(9*C*a^4 + 
10*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^3*x^9)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (219) = 438\).

Time = 39.19 (sec) , antiderivative size = 1697, normalized size of antiderivative = 7.28 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/x**10,x)
 

Output:

-35*A*a**8*b**(19/2)*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b 
**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 110*A*a**7*b** 
(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**1 
0 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*A*a**6*b**(23/2)*x* 
*4*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a 
**5*b**11*x**12 + 315*a**4*b**12*x**14) - 40*A*a**5*b**(25/2)*x**6*sqrt(a/ 
(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11* 
x**12 + 315*a**4*b**12*x**14) - 15*A*a**5*b**(11/2)*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) + 5*A*a**4* 
b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x 
**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 33*A*a**4*b**(13/2)* 
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) + 30*A*a**3*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(315*a**7 
*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12* 
x**14) - 17*A*a**3*b**(15/2)*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) + 40*A*a**2*b**(31/2)*x**12*s 
qrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5* 
b**11*x**12 + 315*a**4*b**12*x**14) - 3*A*a**2*b**(17/2)*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) + 
 16*A*a*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=\frac {D b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {3 \, B b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} D b^{3}}{16 \, a^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{4}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} B b^{4}}{128 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{64 \, a^{3} x^{4}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{315 \, a^{3} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D}{6 \, a x^{6}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C}{7 \, a x^{7}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{8 \, a x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{9 \, a x^{9}} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^10,x, algorithm="maxima")
 

Output:

1/16*D*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/128*B*b^4*arcsinh(a/( 
sqrt(a*b)*abs(x)))/a^(5/2) - 1/48*(b*x^2 + a)^(3/2)*D*b^3/a^3 - 1/16*sqrt( 
b*x^2 + a)*D*b^3/a^2 + 1/128*(b*x^2 + a)^(3/2)*B*b^4/a^4 + 3/128*sqrt(b*x^ 
2 + a)*B*b^4/a^3 + 1/48*(b*x^2 + a)^(5/2)*D*b^2/(a^3*x^2) - 1/128*(b*x^2 + 
 a)^(5/2)*B*b^3/(a^4*x^2) + 1/24*(b*x^2 + a)^(5/2)*D*b/(a^2*x^4) - 1/64*(b 
*x^2 + a)^(5/2)*B*b^2/(a^3*x^4) + 2/35*(b*x^2 + a)^(5/2)*C*b/(a^2*x^5) - 8 
/315*(b*x^2 + a)^(5/2)*A*b^2/(a^3*x^5) - 1/6*(b*x^2 + a)^(5/2)*D/(a*x^6) + 
 1/16*(b*x^2 + a)^(5/2)*B*b/(a^2*x^6) - 1/7*(b*x^2 + a)^(5/2)*C/(a*x^7) + 
4/63*(b*x^2 + a)^(5/2)*A*b/(a^2*x^7) - 1/8*(b*x^2 + a)^(5/2)*B/(a*x^8) - 1 
/9*(b*x^2 + a)^(5/2)*A/(a*x^9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (197) = 394\).

Time = 0.14 (sec) , antiderivative size = 900, normalized size of antiderivative = 3.86 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^10,x, algorithm="giac")
 

Output:

-1/64*(8*D*a*b^3 - 3*B*b^4)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a) 
)/(sqrt(-a)*a^2) + 1/20160*(2520*(sqrt(b)*x - sqrt(b*x^2 + a))^17*D*a*b^3 
- 945*(sqrt(b)*x - sqrt(b*x^2 + a))^17*B*b^4 + 31920*(sqrt(b)*x - sqrt(b*x 
^2 + a))^15*D*a^2*b^3 + 8190*(sqrt(b)*x - sqrt(b*x^2 + a))^15*B*a*b^4 + 80 
640*(sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a^2*b^(7/2) - 45360*(sqrt(b)*x - sq 
rt(b*x^2 + a))^13*D*a^3*b^3 + 97650*(sqrt(b)*x - sqrt(b*x^2 + a))^13*B*a^2 
*b^4 - 80640*(sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^3*b^(7/2) + 215040*(sqrt 
(b)*x - sqrt(b*x^2 + a))^12*A*a^2*b^(9/2) - 15120*(sqrt(b)*x - sqrt(b*x^2 
+ a))^11*D*a^4*b^3 + 106470*(sqrt(b)*x - sqrt(b*x^2 + a))^11*B*a^3*b^4 + 8 
0640*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^4*b^(7/2) + 322560*(sqrt(b)*x - 
sqrt(b*x^2 + a))^10*A*a^3*b^(9/2) - 209664*(sqrt(b)*x - sqrt(b*x^2 + a))^8 
*C*a^5*b^(7/2) + 451584*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^4*b^(9/2) + 15 
120*(sqrt(b)*x - sqrt(b*x^2 + a))^7*D*a^6*b^3 - 106470*(sqrt(b)*x - sqrt(b 
*x^2 + a))^7*B*a^5*b^4 + 112896*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^6*b^(7 
/2) + 129024*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^5*b^(9/2) + 45360*(sqrt(b 
)*x - sqrt(b*x^2 + a))^5*D*a^7*b^3 - 97650*(sqrt(b)*x - sqrt(b*x^2 + a))^5 
*B*a^6*b^4 - 2304*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^7*b^(7/2) + 36864*(s 
qrt(b)*x - sqrt(b*x^2 + a))^4*A*a^6*b^(9/2) - 31920*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^3*D*a^8*b^3 - 8190*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a^7*b^4 + 207 
36*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^8*b^(7/2) - 9216*(sqrt(b)*x - sq...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{x^{10}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^10,x)
 

Output:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^10, x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{10}} \, dx=\frac {-4480 \sqrt {b \,x^{2}+a}\, a^{5}-6400 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-5040 \sqrt {b \,x^{2}+a}\, a^{4} b x -5760 \sqrt {b \,x^{2}+a}\, a^{4} c \,x^{2}-6720 \sqrt {b \,x^{2}+a}\, a^{4} d \,x^{3}-384 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-7560 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{3}-9216 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{4}-11760 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{5}+512 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}-630 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{5}-1152 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{6}-2520 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{7}-1024 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}+945 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{7}+2304 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{8}-2520 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d \,x^{9}+945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{9}+2520 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d \,x^{9}-945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{9}+1024 \sqrt {b}\, a \,b^{4} x^{9}-2304 \sqrt {b}\, a \,b^{3} c \,x^{9}}{40320 a^{3} x^{9}} \] Input:

int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^10,x)
 

Output:

( - 4480*sqrt(a + b*x**2)*a**5 - 6400*sqrt(a + b*x**2)*a**4*b*x**2 - 5040* 
sqrt(a + b*x**2)*a**4*b*x - 5760*sqrt(a + b*x**2)*a**4*c*x**2 - 6720*sqrt( 
a + b*x**2)*a**4*d*x**3 - 384*sqrt(a + b*x**2)*a**3*b**2*x**4 - 7560*sqrt( 
a + b*x**2)*a**3*b**2*x**3 - 9216*sqrt(a + b*x**2)*a**3*b*c*x**4 - 11760*s 
qrt(a + b*x**2)*a**3*b*d*x**5 + 512*sqrt(a + b*x**2)*a**2*b**3*x**6 - 630* 
sqrt(a + b*x**2)*a**2*b**3*x**5 - 1152*sqrt(a + b*x**2)*a**2*b**2*c*x**6 - 
 2520*sqrt(a + b*x**2)*a**2*b**2*d*x**7 - 1024*sqrt(a + b*x**2)*a*b**4*x** 
8 + 945*sqrt(a + b*x**2)*a*b**4*x**7 + 2304*sqrt(a + b*x**2)*a*b**3*c*x**8 
 - 2520*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b* 
*3*d*x**9 + 945*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt( 
a))*b**5*x**9 + 2520*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/ 
sqrt(a))*a*b**3*d*x**9 - 945*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqr 
t(b)*x)/sqrt(a))*b**5*x**9 + 1024*sqrt(b)*a*b**4*x**9 - 2304*sqrt(b)*a*b** 
3*c*x**9)/(40320*a**3*x**9)