\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2+D x^3)}{x^7} \, dx\) [71]

Optimal result

Integrand size = 30, antiderivative size = 172 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {(A b-2 a C) \sqrt {a+b x^2}}{8 a x^4}+\frac {b (A b-2 a C) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac {B \left (a+b x^2\right )^{3/2}}{5 a x^5}+\frac {(2 b B-5 a D) \left (a+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {b^2 (A b-2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}} \] Output:

1/8*(A*b-2*C*a)*(b*x^2+a)^(1/2)/a/x^4+1/16*b*(A*b-2*C*a)*(b*x^2+a)^(1/2)/a 
^2/x^2-1/6*A*(b*x^2+a)^(3/2)/a/x^6-1/5*B*(b*x^2+a)^(3/2)/a/x^5+1/15*(2*B*b 
-5*D*a)*(b*x^2+a)^(3/2)/a^2/x^3-1/16*b^2*(A*b-2*C*a)*arctanh((b*x^2+a)^(1/ 
2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {\sqrt {a+b x^2} \left (b^2 x^4 (15 A+32 B x)-4 a^2 (10 A+x (12 B+5 x (3 C+4 D x)))-2 a b x^2 (5 A+x (8 B+5 x (3 C+8 D x)))\right )}{240 a^2 x^6}+\frac {b^2 (A b-2 a C) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(b^2*x^4*(15*A + 32*B*x) - 4*a^2*(10*A + x*(12*B + 5*x*(3 
*C + 4*D*x))) - 2*a*b*x^2*(5*A + x*(8*B + 5*x*(3*C + 8*D*x)))))/(240*a^2*x 
^6) + (b^2*(A*b - 2*a*C)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/( 
8*a^(5/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2338, 27, 2338, 27, 539, 25, 534, 243, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 539

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/6*(A*(a + b*x^2)^(3/2))/(a*x^6) + ((-2*B*(a + b*x^2)^(3/2))/(5*x^5) + ( 
(5*(A*b - 2*a*C)*(a + b*x^2)^(3/2))/(4*a*x^4) - ((-8*(2*b*B - 5*a*D)*(a + 
b*x^2)^(3/2))/(3*x^3) - (5*b*(A*b - 2*a*C)*(-(Sqrt[a + b*x^2]/x^2) - (b*Ar 
cTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2)/(4*a))/5)/(2*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 = 0.60 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.50

method	result	size
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )}{2 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )+C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )-\frac {D \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}}\)	\(258\)

Input:

int((b*x^2+a)^(1/2)*(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)^(3/2)-1/2*b/a*(-1/4/a/x^4*(b*x^2+a)^(3/2)-1/4*b/a* 
(-1/2/a/x^2*(b*x^2+a)^(3/2)+1/2*b/a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^( 
1/2)*(b*x^2+a)^(1/2))/x)))))+B*(-1/5/a/x^5*(b*x^2+a)^(3/2)+2/15*b/a^2*(b*x 
^2+a)^(3/2)/x^3)+C*(-1/4/a/x^4*(b*x^2+a)^(3/2)-1/4*b/a*(-1/2/a/x^2*(b*x^2+ 
a)^(3/2)+1/2*b/a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2 
))/x))))-1/3*D*(b*x^2+a)^(3/2)/a/x^3
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\left [-\frac {15 \, {\left (2 \, C a b^{2} - A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (16 \, {\left (5 \, D a^{2} b - 2 \, B a b^{2}\right )} x^{5} + 48 \, B a^{3} x + 15 \, {\left (2 \, C a^{2} b - A a b^{2}\right )} x^{4} + 40 \, A a^{3} + 16 \, {\left (5 \, D a^{3} + B a^{2} b\right )} x^{3} + 10 \, {\left (6 \, C a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{480 \, a^{3} x^{6}}, -\frac {15 \, {\left (2 \, C a b^{2} - A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (16 \, {\left (5 \, D a^{2} b - 2 \, B a b^{2}\right )} x^{5} + 48 \, B a^{3} x + 15 \, {\left (2 \, C a^{2} b - A a b^{2}\right )} x^{4} + 40 \, A a^{3} + 16 \, {\left (5 \, D a^{3} + B a^{2} b\right )} x^{3} + 10 \, {\left (6 \, C a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{240 \, a^{3} x^{6}}\right ] \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/x^7,x, algorithm="fricas")
 

Output:

[-1/480*(15*(2*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*(5*D*a^2*b - 2*B*a*b^2)*x^5 + 48*B*a^3*x + 1 
5*(2*C*a^2*b - A*a*b^2)*x^4 + 40*A*a^3 + 16*(5*D*a^3 + B*a^2*b)*x^3 + 10*( 
6*C*a^3 + A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a^3*x^6), -1/240*(15*(2*C*a*b^2 
- A*b^3)*sqrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (16*(5*D*a^2*b 
- 2*B*a*b^2)*x^5 + 48*B*a^3*x + 15*(2*C*a^2*b - A*a*b^2)*x^4 + 40*A*a^3 + 
16*(5*D*a^3 + B*a^2*b)*x^3 + 10*(6*C*a^3 + A*a^2*b)*x^2)*sqrt(b*x^2 + a))/ 
(a^3*x^6)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (155) = 310\).

Time = 9.08 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=- \frac {A a}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}}}{48 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {5}{2}}}{16 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {5}{2}}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a x^{2}} + \frac {2 B b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{2}} - \frac {C a}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 C \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {C b^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {C b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} - \frac {D \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {D b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} \] Input:

integrate((b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/x**7,x)
 

Output:

-A*a/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 5*A*sqrt(b)/(24*x**5*sqrt(a/( 
b*x**2) + 1)) + A*b**(3/2)/(48*a*x**3*sqrt(a/(b*x**2) + 1)) + A*b**(5/2)/( 
16*a**2*x*sqrt(a/(b*x**2) + 1)) - A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a* 
*(5/2)) - B*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - B*b**(3/2)*sqrt(a/(b*x 
**2) + 1)/(15*a*x**2) + 2*B*b**(5/2)*sqrt(a/(b*x**2) + 1)/(15*a**2) - C*a/ 
(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*C*sqrt(b)/(8*x**3*sqrt(a/(b*x**2 
) + 1)) - C*b**(3/2)/(8*a*x*sqrt(a/(b*x**2) + 1)) + C*b**2*asinh(sqrt(a)/( 
sqrt(b)*x))/(8*a**(3/2)) - D*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*x**2) - D*b** 
(3/2)*sqrt(a/(b*x**2) + 1)/(3*a)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {C b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {5}{2}}} - \frac {\sqrt {b x^{2} + a} C b^{2}}{8 \, a^{2}} + \frac {\sqrt {b x^{2} + a} A b^{3}}{16 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D}{3 \, a x^{3}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{15 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{8 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{6 \, a x^{6}} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/x^7,x, algorithm="maxima")
 

Output:

1/8*C*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 1/16*A*b^3*arcsinh(a/(sq 
rt(a*b)*abs(x)))/a^(5/2) - 1/8*sqrt(b*x^2 + a)*C*b^2/a^2 + 1/16*sqrt(b*x^2 
 + a)*A*b^3/a^3 + 1/8*(b*x^2 + a)^(3/2)*C*b/(a^2*x^2) - 1/16*(b*x^2 + a)^( 
3/2)*A*b^2/(a^3*x^2) - 1/3*(b*x^2 + a)^(3/2)*D/(a*x^3) + 2/15*(b*x^2 + a)^ 
(3/2)*B*b/(a^2*x^3) - 1/4*(b*x^2 + a)^(3/2)*C/(a*x^4) + 1/8*(b*x^2 + a)^(3 
/2)*A*b/(a^2*x^4) - 1/5*(b*x^2 + a)^(3/2)*B/(a*x^5) - 1/6*(b*x^2 + a)^(3/2 
)*A/(a*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (147) = 294\).

Time = 0.13 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.77 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=-\frac {{\left (2 \, C a b^{2} - A b^{3}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{2}} + \frac {30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} C a b^{2} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} A b^{3} + 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} D a^{2} b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} C a^{2} b^{2} + 85 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} A a b^{3} - 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} D a^{3} b^{\frac {3}{2}} + 480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{2} b^{\frac {5}{2}} - 180 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} C a^{3} b^{2} + 570 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} A a^{2} b^{3} + 800 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} D a^{4} b^{\frac {3}{2}} - 320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{3} b^{\frac {5}{2}} - 180 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} C a^{4} b^{2} + 570 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A a^{3} b^{3} - 480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} D a^{5} b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} C a^{5} b^{2} + 85 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a^{4} b^{3} + 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} D a^{6} b^{\frac {3}{2}} - 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{5} b^{\frac {5}{2}} + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} C a^{6} b^{2} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{5} b^{3} - 80 \, D a^{7} b^{\frac {3}{2}} + 32 \, B a^{6} b^{\frac {5}{2}}}{120 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{6} a^{2}} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/x^7,x, algorithm="giac")
 

Output:

-1/8*(2*C*a*b^2 - A*b^3)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/( 
sqrt(-a)*a^2) + 1/120*(30*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a*b^2 - 15*(s 
qrt(b)*x - sqrt(b*x^2 + a))^11*A*b^3 + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^1 
0*D*a^2*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^9*C*a^2*b^2 + 85*(sqrt 
(b)*x - sqrt(b*x^2 + a))^9*A*a*b^3 - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D 
*a^3*b^(3/2) + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^2*b^(5/2) - 180*(sq 
rt(b)*x - sqrt(b*x^2 + a))^7*C*a^3*b^2 + 570*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^7*A*a^2*b^3 + 800*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^4*b^(3/2) - 320*(sq 
rt(b)*x - sqrt(b*x^2 + a))^6*B*a^3*b^(5/2) - 180*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^5*C*a^4*b^2 + 570*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*a^3*b^3 - 480*(sq 
rt(b)*x - sqrt(b*x^2 + a))^4*D*a^5*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^3*C*a^5*b^2 + 85*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^4*b^3 + 240*(sqr 
t(b)*x - sqrt(b*x^2 + a))^2*D*a^6*b^(3/2) - 192*(sqrt(b)*x - sqrt(b*x^2 + 
a))^2*B*a^5*b^(5/2) + 30*(sqrt(b)*x - sqrt(b*x^2 + a))*C*a^6*b^2 - 15*(sqr 
t(b)*x - sqrt(b*x^2 + a))*A*a^5*b^3 - 80*D*a^7*b^(3/2) + 32*B*a^6*b^(5/2)) 
/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^6*a^2)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/x^7,x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.80 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{x^7} \, dx=\frac {-40 \sqrt {b \,x^{2}+a}\, a^{3}-10 \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}-48 \sqrt {b \,x^{2}+a}\, a^{2} b x -60 \sqrt {b \,x^{2}+a}\, a^{2} c \,x^{2}-80 \sqrt {b \,x^{2}+a}\, a^{2} d \,x^{3}+15 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}-16 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{3}-30 \sqrt {b \,x^{2}+a}\, a b c \,x^{4}-80 \sqrt {b \,x^{2}+a}\, a b d \,x^{5}+32 \sqrt {b \,x^{2}+a}\, b^{3} x^{5}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}-30 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{6}+30 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{6}-32 \sqrt {b}\, b^{3} x^{6}}{240 a^{2} x^{6}} \] Input:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/x^7,x)
 

Output:

( - 40*sqrt(a + b*x**2)*a**3 - 10*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 - 16*sqrt(a + b*x**2)*a*b* 
*2*x**3 - 30*sqrt(a + b*x**2)*a*b*c*x**4 - 80*sqrt(a + b*x**2)*a*b*d*x**5 
+ 32*sqrt(a + b*x**2)*b**3*x**5 + 15*sqrt(a)*log((sqrt(a + b*x**2) - sqrt( 
a) + sqrt(b)*x)/sqrt(a))*b**3*x**6 - 30*sqrt(a)*log((sqrt(a + b*x**2) - sq 
rt(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 + 30*sqrt(a)*log((sqrt(a + b*x* 
*2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*x**6 - 32*sqrt(b)*b**3*x**6)/(2 
40*a**2*x**6)