\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2+D x^3)}{x^7} \, dx\) [84]

Optimal result

Integrand size = 30, antiderivative size = 178 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {b (A b-6 a C-16 a D x) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a C-8 a D x) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {b^2 (A b-6 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:

1/16*b*(-16*D*a*x+A*b-6*C*a)*(b*x^2+a)^(1/2)/a/x^2+1/24*(-8*D*a*x+A*b-6*C* 
a)*(b*x^2+a)^(3/2)/a/x^4-1/6*A*(b*x^2+a)^(5/2)/a/x^6-1/5*B*(b*x^2+a)^(5/2) 
/a/x^5+b^(3/2)*D*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))+1/16*b^2*(A*b-6*C*a)*a 
rctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=-\frac {\sqrt {a+b x^2} \left (3 b^2 x^4 (5 A+16 B x)+a^2 \left (40 A+48 B x+60 C x^2+80 D x^3\right )+2 a b x^2 \left (35 A+48 B x+75 C x^2+160 D x^3\right )\right )}{240 a x^6}+\frac {b^2 (-A b+6 a C) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-b^{3/2} D \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^7,x]
 

Output:

-1/240*(Sqrt[a + b*x^2]*(3*b^2*x^4*(5*A + 16*B*x) + a^2*(40*A + 48*B*x + 6 
0*C*x^2 + 80*D*x^3) + 2*a*b*x^2*(35*A + 48*B*x + 75*C*x^2 + 160*D*x^3)))/( 
a*x^6) + (b^2*(-(A*b) + 6*a*C)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[ 
a]])/(8*a^(3/2)) - b^(3/2)*D*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.04, 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, 537, 27, 537, 25, 538, 224, 219, 243, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

\(\displaystyle \frac {-\frac {3}{4} b \left (-\frac {1}{2} b \int -\frac {A b-6 a C-16 a D x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \int \frac {A b-6 a C-16 a D x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left ((A b-6 a C) \int \frac {1}{x \sqrt {b x^2+a}}dx-16 a D \int \frac {1}{\sqrt {b x^2+a}}dx\right )-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left ((A b-6 a C) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {16 a D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {1}{2} (A b-6 a C) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {16 a D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {(A b-6 a C) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {16 a D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (-\frac {(A b-6 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {16 a D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (-6 a C-16 a D x+A b)}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} (-6 a C-8 a D x+A b)}{4 x^4}-\frac {6 B \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^7,x]
 

Output:

-1/6*(A*(a + b*x^2)^(5/2))/(a*x^6) + (((A*b - 6*a*C - 8*a*D*x)*(a + b*x^2) 
^(3/2))/(4*x^4) - (6*B*(a + b*x^2)^(5/2))/(5*x^5) - (3*b*(-1/2*((A*b - 6*a 
*C - 16*a*D*x)*Sqrt[a + b*x^2])/x^2 + (b*((-16*a*D*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/Sqrt[b] - ((A*b - 6*a*C)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/ 
Sqrt[a]))/2))/4)/(6*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 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 537

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

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

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. \(348\) vs. \(2(148)=296\).

Time = 0.69 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.96

method	result	size
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )-\frac {B \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}+C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )+D \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )\)	\(349\)

Input:

int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^7,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

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))))))-1/5*B*(b*x^2+a)^(5/2) 
/a/x^5+C*(-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)))))+D*(-1/3/a/x^3*(b*x^2+a)^(5/2)+2/3*b/a*(-1/a/x*(b*x^ 
2+a)^(5/2)+4*b/a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a 
/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 772, normalized size of antiderivative = 4.34 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^7,x, algorithm="fricas")
 

Output:

[1/480*(240*D*a^2*b^(3/2)*x^6*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - 
 a) - 15*(6*C*a*b^2 - A*b^3)*sqrt(a)*x^6*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*s 
qrt(a) + 2*a)/x^2) - 2*(16*(20*D*a^2*b + 3*B*a*b^2)*x^5 + 48*B*a^3*x + 15* 
(10*C*a^2*b + A*a*b^2)*x^4 + 40*A*a^3 + 16*(5*D*a^3 + 6*B*a^2*b)*x^3 + 10* 
(6*C*a^3 + 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6), -1/480*(480*D*a^2*s 
qrt(-b)*b*x^6*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + 15*(6*C*a*b^2 - A*b^3)* 
sqrt(a)*x^6*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(16*(2 
0*D*a^2*b + 3*B*a*b^2)*x^5 + 48*B*a^3*x + 15*(10*C*a^2*b + A*a*b^2)*x^4 + 
40*A*a^3 + 16*(5*D*a^3 + 6*B*a^2*b)*x^3 + 10*(6*C*a^3 + 7*A*a^2*b)*x^2)*sq 
rt(b*x^2 + a))/(a^2*x^6), 1/240*(120*D*a^2*b^(3/2)*x^6*log(-2*b*x^2 - 2*sq 
rt(b*x^2 + a)*sqrt(b)*x - a) + 15*(6*C*a*b^2 - A*b^3)*sqrt(-a)*x^6*arctan( 
sqrt(b*x^2 + a)*sqrt(-a)/a) - (16*(20*D*a^2*b + 3*B*a*b^2)*x^5 + 48*B*a^3* 
x + 15*(10*C*a^2*b + A*a*b^2)*x^4 + 40*A*a^3 + 16*(5*D*a^3 + 6*B*a^2*b)*x^ 
3 + 10*(6*C*a^3 + 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6), -1/240*(240* 
D*a^2*sqrt(-b)*b*x^6*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 15*(6*C*a*b^2 - 
A*b^3)*sqrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (16*(20*D*a^2*b + 
 3*B*a*b^2)*x^5 + 48*B*a^3*x + 15*(10*C*a^2*b + A*a*b^2)*x^4 + 40*A*a^3 + 
16*(5*D*a^3 + 6*B*a^2*b)*x^3 + 10*(6*C*a^3 + 7*A*a^2*b)*x^2)*sqrt(b*x^2 + 
a))/(a^2*x^6)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (163) = 326\).

Time = 14.39 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=- \frac {A a^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 A a \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} - \frac {B a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {2 B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {B b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {C a^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 C a \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {C b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {C b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 C b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} - \frac {D \sqrt {a} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {D a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {D b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + D b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {D b^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

integrate((b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/x**7,x)
 

Output:

-A*a**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 11*A*a*sqrt(b)/(24*x**5*sq 
rt(a/(b*x**2) + 1)) - 17*A*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - A*b** 
(5/2)/(16*a*x*sqrt(a/(b*x**2) + 1)) + A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(1 
6*a**(3/2)) - B*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 2*B*b**(3/2)*sqr 
t(a/(b*x**2) + 1)/(5*x**2) - B*b**(5/2)*sqrt(a/(b*x**2) + 1)/(5*a) - C*a** 
2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*C*a*sqrt(b)/(8*x**3*sqrt(a/(b* 
x**2) + 1)) - C*b**(3/2)*sqrt(a/(b*x**2) + 1)/(2*x) - C*b**(3/2)/(8*x*sqrt 
(a/(b*x**2) + 1)) - 3*C*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a)) - D*sq 
rt(a)*b/(x*sqrt(1 + b*x**2/a)) - D*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*x**2) 
 - D*b**(3/2)*sqrt(a/(b*x**2) + 1)/3 + D*b**(3/2)*asinh(sqrt(b)*x/sqrt(a)) 
 - D*b**2*x/(sqrt(a)*sqrt(1 + b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {\sqrt {b x^{2} + a} D b^{2} x}{a} + D b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3 \, C b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} C b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} A b^{3}}{16 \, a^{2}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C b}{8 \, a^{2} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{6 \, a x^{6}} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^7,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (151) = 302\).

Time = 0.16 (sec) , antiderivative size = 726, normalized size of antiderivative = 4.08 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^7,x, algorithm="giac")
 

Output:

-D*b^(3/2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) + 1/8*(6*C*a*b^2 - A*b^3 
)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/120*(15 
0*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a*b^2 + 15*(sqrt(b)*x - sqrt(b*x^2 + 
a))^11*A*b^3 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^2*b^(3/2) + 240*(s 
qrt(b)*x - sqrt(b*x^2 + a))^10*B*a*b^(5/2) - 210*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^9*C*a^2*b^2 + 235*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a*b^3 - 1920*(sqr 
t(b)*x - sqrt(b*x^2 + a))^8*D*a^3*b^(3/2) - 240*(sqrt(b)*x - sqrt(b*x^2 + 
a))^8*B*a^2*b^(5/2) + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^3*b^2 + 390*( 
sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a^2*b^3 + 3200*(sqrt(b)*x - sqrt(b*x^2 + 
a))^6*D*a^4*b^(3/2) + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^3*b^(5/2) + 
60*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^4*b^2 + 390*(sqrt(b)*x - sqrt(b*x^2 
 + a))^5*A*a^3*b^3 - 2880*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a^5*b^(3/2) - 
480*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^4*b^(5/2) - 210*(sqrt(b)*x - sqrt( 
b*x^2 + a))^3*C*a^5*b^2 + 235*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^4*b^3 + 
1440*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a^6*b^(3/2) + 48*(sqrt(b)*x - sqrt( 
b*x^2 + a))^2*B*a^5*b^(5/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))*C*a^6*b^2 
+ 15*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^5*b^3 - 320*D*a^7*b^(3/2) - 48*B*a^ 
6*b^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^6*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{x^7} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^7,x)
 

Output:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {-40 \sqrt {b \,x^{2}+a}\, a^{3}-70 \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}-48 \sqrt {b \,x^{2}+a}\, a^{2} b x -60 \sqrt {b \,x^{2}+a}\, a^{2} c \,x^{2}-80 \sqrt {b \,x^{2}+a}\, a^{2} d \,x^{3}-15 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}-96 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{3}-150 \sqrt {b \,x^{2}+a}\, a b c \,x^{4}-320 \sqrt {b \,x^{2}+a}\, a b d \,x^{5}-48 \sqrt {b \,x^{2}+a}\, b^{3} x^{5}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}+90 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{6}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}-90 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{6}+240 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b d \,x^{6}+160 \sqrt {b}\, a b d \,x^{6}-32 \sqrt {b}\, b^{3} x^{6}}{240 a \,x^{6}} \] Input:

int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^7,x)
 

Output:

( - 40*sqrt(a + b*x**2)*a**3 - 70*sqrt(a + b*x**2)*a**2*b*x**2 - 48*sqrt(a 
 + b*x**2)*a**2*b*x - 60*sqrt(a + b*x**2)*a**2*c*x**2 - 80*sqrt(a + b*x**2 
)*a**2*d*x**3 - 15*sqrt(a + b*x**2)*a*b**2*x**4 - 96*sqrt(a + b*x**2)*a*b* 
*2*x**3 - 150*sqrt(a + b*x**2)*a*b*c*x**4 - 320*sqrt(a + b*x**2)*a*b*d*x** 
5 - 48*sqrt(a + b*x**2)*b**3*x**5 - 15*sqrt(a)*log((sqrt(a + b*x**2) - sqr 
t(a) + sqrt(b)*x)/sqrt(a))*b**3*x**6 + 90*sqrt(a)*log((sqrt(a + b*x**2) - 
sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*x**6 + 15*sqrt(a)*log((sqrt(a + b*x** 
2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*x**6 - 90*sqrt(a)*log((sqrt(a + b* 
x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*x**6 + 240*sqrt(b)*log((sqrt( 
a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*d*x**6 + 160*sqrt(b)*a*b*d*x**6 - 32 
*sqrt(b)*b**3*x**6)/(240*a*x**6)