\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2+D x^3)}{x^8} \, dx\) [85]

Optimal result

Integrand size = 30, antiderivative size = 169 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=\frac {(b B-6 a D) \sqrt {a+b x^2}}{24 x^4}+\frac {5 b (b B-6 a D) \sqrt {a+b x^2}}{48 a x^2}-\frac {A \left (a+b x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac {(2 A b-7 a C) \left (a+b x^2\right )^{5/2}}{35 a^2 x^5}+\frac {b^2 (b B-6 a D) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:

1/24*(B*b-6*D*a)*(b*x^2+a)^(1/2)/x^4+5/48*b*(B*b-6*D*a)*(b*x^2+a)^(1/2)/a/ 
x^2-1/7*A*(b*x^2+a)^(5/2)/a/x^7-1/6*B*(b*x^2+a)^(5/2)/a/x^6+1/35*(2*A*b-7* 
C*a)*(b*x^2+a)^(5/2)/a^2/x^5+1/16*b^2*(B*b-6*D*a)*arctanh((b*x^2+a)^(1/2)/ 
a^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=-\frac {\frac {\sqrt {a+b x^2} \left (-96 A b^3 x^6+3 a b^2 x^4 (16 A+7 x (5 B+16 C x))+2 a^2 b x^2 \left (192 A+7 x \left (35 B+48 C x+75 D x^2\right )\right )+4 a^3 (60 A+7 x (10 B+3 x (4 C+5 D x)))\right )}{x^7}+105 \sqrt {a} b^2 (-b B+6 a D) \log (x)-105 \sqrt {a} b^2 (-b B+6 a D) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{1680 a^2} \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^8,x]
 

Output:

-1/1680*((Sqrt[a + b*x^2]*(-96*A*b^3*x^6 + 3*a*b^2*x^4*(16*A + 7*x*(5*B + 
16*C*x)) + 2*a^2*b*x^2*(192*A + 7*x*(35*B + 48*C*x + 75*D*x^2)) + 4*a^3*(6 
0*A + 7*x*(10*B + 3*x*(4*C + 5*D*x)))))/x^7 + 105*Sqrt[a]*b^2*(-(b*B) + 6* 
a*D)*Log[x] - 105*Sqrt[a]*b^2*(-(b*B) + 6*a*D)*Log[-Sqrt[a] + Sqrt[a + b*x 
^2]])/a^2
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2338, 25, 2338, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^8,x]
 

Output:

-1/7*(A*(a + b*x^2)^(5/2))/(a*x^7) + ((-7*B*(a + b*x^2)^(5/2))/(6*x^6) + ( 
(6*(2*A*b - 7*a*C)*(a + b*x^2)^(5/2))/(5*a*x^5) - (7*(b*B - 6*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)/6)/(7*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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(285\) vs. \(2(141)=282\).

Time = 0.62 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.69

method	result	size
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )-\frac {C \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}+D \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )\)	\(286\)

Input:

int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^8,x,method=_RETURNVERBOSE)
 

Output:

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*(-1/6/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))))))-1/5*C/a/x^5*(b*x^2+a)^(5/2)+D*(-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 = 0.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=\left [-\frac {105 \, {\left (6 \, D a b^{2} - B b^{3}\right )} \sqrt {a} x^{7} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (48 \, {\left (7 \, C a b^{2} - 2 \, A b^{3}\right )} x^{6} + 105 \, {\left (10 \, D a^{2} b + B a b^{2}\right )} x^{5} + 280 \, B a^{3} x + 48 \, {\left (14 \, C a^{2} b + A a b^{2}\right )} x^{4} + 240 \, A a^{3} + 70 \, {\left (6 \, D a^{3} + 7 \, B a^{2} b\right )} x^{3} + 48 \, {\left (7 \, C a^{3} + 8 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3360 \, a^{2} x^{7}}, \frac {105 \, {\left (6 \, D a b^{2} - B b^{3}\right )} \sqrt {-a} x^{7} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (48 \, {\left (7 \, C a b^{2} - 2 \, A b^{3}\right )} x^{6} + 105 \, {\left (10 \, D a^{2} b + B a b^{2}\right )} x^{5} + 280 \, B a^{3} x + 48 \, {\left (14 \, C a^{2} b + A a b^{2}\right )} x^{4} + 240 \, A a^{3} + 70 \, {\left (6 \, D a^{3} + 7 \, B a^{2} b\right )} x^{3} + 48 \, {\left (7 \, C a^{3} + 8 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{1680 \, a^{2} x^{7}}\right ] \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^8,x, algorithm="fricas")
 

Output:

[-1/3360*(105*(6*D*a*b^2 - B*b^3)*sqrt(a)*x^7*log(-(b*x^2 + 2*sqrt(b*x^2 + 
 a)*sqrt(a) + 2*a)/x^2) + 2*(48*(7*C*a*b^2 - 2*A*b^3)*x^6 + 105*(10*D*a^2* 
b + B*a*b^2)*x^5 + 280*B*a^3*x + 48*(14*C*a^2*b + A*a*b^2)*x^4 + 240*A*a^3 
 + 70*(6*D*a^3 + 7*B*a^2*b)*x^3 + 48*(7*C*a^3 + 8*A*a^2*b)*x^2)*sqrt(b*x^2 
 + a))/(a^2*x^7), 1/1680*(105*(6*D*a*b^2 - B*b^3)*sqrt(-a)*x^7*arctan(sqrt 
(b*x^2 + a)*sqrt(-a)/a) - (48*(7*C*a*b^2 - 2*A*b^3)*x^6 + 105*(10*D*a^2*b 
+ B*a*b^2)*x^5 + 280*B*a^3*x + 48*(14*C*a^2*b + A*a*b^2)*x^4 + 240*A*a^3 + 
 70*(6*D*a^3 + 7*B*a^2*b)*x^3 + 48*(7*C*a^3 + 8*A*a^2*b)*x^2)*sqrt(b*x^2 + 
 a))/(a^2*x^7)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (153) = 306\).

Time = 14.98 (sec) , antiderivative size = 774, normalized size of antiderivative = 4.58 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=- \frac {15 A a^{6} b^{\frac {9}{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 {33 A a^{5} b^{\frac {11}{2}} 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 {17 A a^{4} b^{\frac {13}{2}} 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 {3 A a^{3} b^{\frac {15}{2}} 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 {12 A a^{2} b^{\frac {17}{2}} 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 {8 A a b^{\frac {19}{2}} 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 {A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a x^{2}} + \frac {2 A b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{2}} - \frac {B a^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 B a \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 B b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} - \frac {C a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {2 C b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {C b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {D a^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 D a \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {D b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {D b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 D b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} \] Input:

integrate((b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/x**8,x)
 

Output:

-15*A*a**6*b**(9/2)*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) - 33*A*a**5*b**(11/2)*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) - 17* 
A*a**4*b**(13/2)*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) - 3*A*a**3*b**(15/2)*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) - 12 
*A*a**2*b**(17/2)*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) - 8*A*a*b**(19/2)*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) - A*b 
**(3/2)*sqrt(a/(b*x**2) + 1)/(5*x**4) - A*b**(5/2)*sqrt(a/(b*x**2) + 1)/(1 
5*a*x**2) + 2*A*b**(7/2)*sqrt(a/(b*x**2) + 1)/(15*a**2) - B*a**2/(6*sqrt(b 
)*x**7*sqrt(a/(b*x**2) + 1)) - 11*B*a*sqrt(b)/(24*x**5*sqrt(a/(b*x**2) + 1 
)) - 17*B*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - B*b**(5/2)/(16*a*x*sqr 
t(a/(b*x**2) + 1)) + B*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(3/2)) - C*a 
*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 2*C*b**(3/2)*sqrt(a/(b*x**2) + 1) 
/(5*x**2) - C*b**(5/2)*sqrt(a/(b*x**2) + 1)/(5*a) - D*a**2/(4*sqrt(b)*x**5 
*sqrt(a/(b*x**2) + 1)) - 3*D*a*sqrt(b)/(8*x**3*sqrt(a/(b*x**2) + 1)) - D*b 
**(3/2)*sqrt(a/(b*x**2) + 1)/(2*x) - D*b**(3/2)/(8*x*sqrt(a/(b*x**2) + 1)) 
 - 3*D*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=-\frac {3 \, D b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {B b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} D b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} B b^{3}}{16 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D b}{8 \, a^{2} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C}{5 \, a x^{5}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{35 \, a^{2} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{6 \, a x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{7 \, a x^{7}} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^8,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (144) = 288\).

Time = 0.15 (sec) , antiderivative size = 728, normalized size of antiderivative = 4.31 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^8,x, algorithm="giac")
 

Output:

1/8*(6*D*a*b^2 - B*b^3)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(s 
qrt(-a)*a) + 1/840*(1050*(sqrt(b)*x - sqrt(b*x^2 + a))^13*D*a*b^2 + 105*(s 
qrt(b)*x - sqrt(b*x^2 + a))^13*B*b^3 + 1680*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
12*C*a*b^(5/2) - 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^11*D*a^2*b^2 + 1540*(s 
qrt(b)*x - sqrt(b*x^2 + a))^11*B*a*b^3 - 3360*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^10*C*a^2*b^(5/2) + 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a*b^(7/2) + 1 
890*(sqrt(b)*x - sqrt(b*x^2 + a))^9*D*a^3*b^2 + 1085*(sqrt(b)*x - sqrt(b*x 
^2 + a))^9*B*a^2*b^3 + 5040*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^3*b^(5/2) 
+ 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(7/2) - 6720*(sqrt(b)*x - s 
qrt(b*x^2 + a))^6*C*a^4*b^(5/2) + 6720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a 
^3*b^(7/2) - 1890*(sqrt(b)*x - sqrt(b*x^2 + a))^5*D*a^5*b^2 - 1085*(sqrt(b 
)*x - sqrt(b*x^2 + a))^5*B*a^4*b^3 + 3696*(sqrt(b)*x - sqrt(b*x^2 + a))^4* 
C*a^5*b^(5/2) + 1344*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^4*b^(7/2) + 2520* 
(sqrt(b)*x - sqrt(b*x^2 + a))^3*D*a^6*b^2 - 1540*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^3*B*a^5*b^3 - 672*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^6*b^(5/2) + 672 
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^5*b^(7/2) - 1050*(sqrt(b)*x - sqrt(b* 
x^2 + a))*D*a^7*b^2 - 105*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^6*b^3 + 336*C* 
a^7*b^(5/2) - 96*A*a^6*b^(7/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7*a 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{x^8} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^8,x)
 

Output:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^8} \, dx=\frac {-240 \sqrt {b \,x^{2}+a}\, a^{4}-384 \sqrt {b \,x^{2}+a}\, a^{3} b \,x^{2}-280 \sqrt {b \,x^{2}+a}\, a^{3} b x -336 \sqrt {b \,x^{2}+a}\, a^{3} c \,x^{2}-420 \sqrt {b \,x^{2}+a}\, a^{3} d \,x^{3}-48 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} x^{4}-490 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} x^{3}-672 \sqrt {b \,x^{2}+a}\, a^{2} b c \,x^{4}-1050 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{5}+96 \sqrt {b \,x^{2}+a}\, a \,b^{3} x^{6}-105 \sqrt {b \,x^{2}+a}\, a \,b^{3} x^{5}-336 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{6}+630 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{7}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} x^{7}-630 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{7}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} x^{7}-96 \sqrt {b}\, a \,b^{3} x^{7}-144 \sqrt {b}\, a \,b^{2} c \,x^{7}}{1680 a^{2} x^{7}} \] Input:

int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^8,x)
 

Output:

( - 240*sqrt(a + b*x**2)*a**4 - 384*sqrt(a + b*x**2)*a**3*b*x**2 - 280*sqr 
t(a + b*x**2)*a**3*b*x - 336*sqrt(a + b*x**2)*a**3*c*x**2 - 420*sqrt(a + b 
*x**2)*a**3*d*x**3 - 48*sqrt(a + b*x**2)*a**2*b**2*x**4 - 490*sqrt(a + b*x 
**2)*a**2*b**2*x**3 - 672*sqrt(a + b*x**2)*a**2*b*c*x**4 - 1050*sqrt(a + b 
*x**2)*a**2*b*d*x**5 + 96*sqrt(a + b*x**2)*a*b**3*x**6 - 105*sqrt(a + b*x* 
*2)*a*b**3*x**5 - 336*sqrt(a + b*x**2)*a*b**2*c*x**6 + 630*sqrt(a)*log((sq 
rt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*d*x**7 - 105*sqrt(a) 
*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*x**7 - 630*sqr 
t(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*d*x**7 + 
 105*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*x* 
*7 - 96*sqrt(b)*a*b**3*x**7 - 144*sqrt(b)*a*b**2*c*x**7)/(1680*a**2*x**7)