\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2+D x^3)}{x^3} \, dx\) [80]

Optimal result

Integrand size = 30, antiderivative size = 195 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=\frac {1}{8} (4 (3 A b+2 a C)+3 (4 b B+a D) x) \sqrt {a+b x^2}+\frac {(2 (3 A b+2 a C)+3 (4 b B+a D) x) \left (a+b x^2\right )^{3/2}}{12 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {B \left (a+b x^2\right )^{5/2}}{a x}+\frac {3 a (4 b B+a D) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}-\frac {1}{2} \sqrt {a} (3 A b+2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \] Output:

1/8*(12*A*b+8*C*a+3*(4*B*b+D*a)*x)*(b*x^2+a)^(1/2)+1/12*(6*A*b+4*C*a+3*(4* 
B*b+D*a)*x)*(b*x^2+a)^(3/2)/a-1/2*A*(b*x^2+a)^(5/2)/a/x^2-B*(b*x^2+a)^(5/2 
)/a/x+3/8*a*(4*B*b+D*a)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)-1/2*a^( 
1/2)*(3*A*b+2*C*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=\sqrt {a} (3 A b+2 a C) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {1}{24} \left (\frac {\sqrt {a+b x^2} \left (2 b x^2 \left (12 A+6 B x+4 C x^2+3 D x^3\right )+a \left (-12 A-24 B x+32 C x^2+15 D x^3\right )\right )}{x^2}-\frac {9 a (4 b B+a D) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}\right ) \] Input:

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

Output:

Sqrt[a]*(3*A*b + 2*a*C)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] + ( 
(Sqrt[a + b*x^2]*(2*b*x^2*(12*A + 6*B*x + 4*C*x^2 + 3*D*x^3) + a*(-12*A - 
24*B*x + 32*C*x^2 + 15*D*x^3)))/x^2 - (9*a*(4*b*B + a*D)*Log[-(Sqrt[b]*x) 
+ Sqrt[a + b*x^2]])/Sqrt[b])/24
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.05, 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, 25, 27, 535, 27, 535, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(A*(a + b*x^2)^(5/2))/(a*x^2) + (((2*(3*A*b + 2*a*C) + 3*(4*b*B + a*D 
)*x)*(a + b*x^2)^(3/2))/6 - (2*B*(a + b*x^2)^(5/2))/x + (a*(((4*(3*A*b + 2 
*a*C) + 3*(4*b*B + a*D)*x)*Sqrt[a + b*x^2])/2 + (a*((3*(4*b*B + a*D)*ArcTa 
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (4*(3*A*b + 2*a*C)*ArcTanh[Sqrt 
[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2))/2)/(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 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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 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 [A] (verified)

Time = 0.52 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.35

method	result	size
default	\(D \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )+C \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\)	\(264\)

Input:

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

Output:

D*(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))))+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 
))))+B*(-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)))))+C*(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 = 0.13 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.50 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/48*(9*(D*a^2 + 4*B*a*b)*sqrt(b)*x^2*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sq 
rt(b)*x - a) + 12*(2*C*a*b + 3*A*b^2)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x 
^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(6*D*b^2*x^5 + 8*C*b^2*x^4 - 24*B*a*b*x + 
3*(5*D*a*b + 4*B*b^2)*x^3 - 12*A*a*b + 8*(4*C*a*b + 3*A*b^2)*x^2)*sqrt(b*x 
^2 + a))/(b*x^2), -1/24*(9*(D*a^2 + 4*B*a*b)*sqrt(-b)*x^2*arctan(sqrt(-b)* 
x/sqrt(b*x^2 + a)) - 6*(2*C*a*b + 3*A*b^2)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqr 
t(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - (6*D*b^2*x^5 + 8*C*b^2*x^4 - 24*B*a*b*x 
 + 3*(5*D*a*b + 4*B*b^2)*x^3 - 12*A*a*b + 8*(4*C*a*b + 3*A*b^2)*x^2)*sqrt( 
b*x^2 + a))/(b*x^2), 1/48*(24*(2*C*a*b + 3*A*b^2)*sqrt(-a)*x^2*arctan(sqrt 
(b*x^2 + a)*sqrt(-a)/a) + 9*(D*a^2 + 4*B*a*b)*sqrt(b)*x^2*log(-2*b*x^2 - 2 
*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(6*D*b^2*x^5 + 8*C*b^2*x^4 - 24*B*a*b* 
x + 3*(5*D*a*b + 4*B*b^2)*x^3 - 12*A*a*b + 8*(4*C*a*b + 3*A*b^2)*x^2)*sqrt 
(b*x^2 + a))/(b*x^2), -1/24*(9*(D*a^2 + 4*B*a*b)*sqrt(-b)*x^2*arctan(sqrt( 
-b)*x/sqrt(b*x^2 + a)) - 12*(2*C*a*b + 3*A*b^2)*sqrt(-a)*x^2*arctan(sqrt(b 
*x^2 + a)*sqrt(-a)/a) - (6*D*b^2*x^5 + 8*C*b^2*x^4 - 24*B*a*b*x + 3*(5*D*a 
*b + 4*B*b^2)*x^3 - 12*A*a*b + 8*(4*C*a*b + 3*A*b^2)*x^2)*sqrt(b*x^2 + a)) 
/(b*x^2)]
 

Sympy [A] (verification not implemented)

Time = 5.50 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=- \frac {3 A \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {A a \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + B a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + B b \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) - C a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {C a^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {C a \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + C b \left (\begin {cases} \frac {a \sqrt {a + b x^{2}}}{3 b} + \frac {x^{2} \sqrt {a + b x^{2}}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + D a \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + D b \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a x \sqrt {a + b x^{2}}}{8 b} + \frac {x^{3} \sqrt {a + b x^{2}}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-3*A*sqrt(a)*b*asinh(sqrt(a)/(sqrt(b)*x))/2 - A*a*sqrt(b)*sqrt(a/(b*x**2) 
+ 1)/(2*x) + A*a*sqrt(b)/(x*sqrt(a/(b*x**2) + 1)) + A*b**(3/2)*x/sqrt(a/(b 
*x**2) + 1) - B*a**(3/2)/(x*sqrt(1 + b*x**2/a)) - B*sqrt(a)*b*x/sqrt(1 + b 
*x**2/a) + B*a*sqrt(b)*asinh(sqrt(b)*x/sqrt(a)) + B*b*Piecewise((a*Piecewi 
se((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/ 
sqrt(b*x**2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True 
)) - C*a**(3/2)*asinh(sqrt(a)/(sqrt(b)*x)) + C*a**2/(sqrt(b)*x*sqrt(a/(b*x 
**2) + 1)) + C*a*sqrt(b)*x/sqrt(a/(b*x**2) + 1) + C*b*Piecewise((a*sqrt(a 
+ b*x**2)/(3*b) + x**2*sqrt(a + b*x**2)/3, Ne(b, 0)), (sqrt(a)*x**2/2, Tru 
e)) + D*a*Piecewise((a*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/ 
sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/2 + x*sqrt(a + b*x**2)/ 
2, Ne(b, 0)), (sqrt(a)*x, True)) + D*b*Piecewise((-a**2*Piecewise((log(2*s 
qrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2 
), True))/(8*b) + a*x*sqrt(a + b*x**2)/(8*b) + x**3*sqrt(a + b*x**2)/4, Ne 
(b, 0)), (sqrt(a)*x**3/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=\frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D x + \frac {3}{8} \, \sqrt {b x^{2} + a} D a x + \frac {3}{2} \, \sqrt {b x^{2} + a} B b x + \frac {3 \, D a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {3}{2} \, B a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - C a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {3}{2} \, A \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C + \sqrt {b x^{2} + a} C a + \frac {3}{2} \, \sqrt {b x^{2} + a} A b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{2 \, a x^{2}} \] Input:

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

Output:

1/4*(b*x^2 + a)^(3/2)*D*x + 3/8*sqrt(b*x^2 + a)*D*a*x + 3/2*sqrt(b*x^2 + a 
)*B*b*x + 3/8*D*a^2*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 3/2*B*a*sqrt(b)*arcsi 
nh(b*x/sqrt(a*b)) - C*a^(3/2)*arcsinh(a/(sqrt(a*b)*abs(x))) - 3/2*A*sqrt(a 
)*b*arcsinh(a/(sqrt(a*b)*abs(x))) + 1/3*(b*x^2 + a)^(3/2)*C + sqrt(b*x^2 + 
 a)*C*a + 3/2*sqrt(b*x^2 + a)*A*b + 1/2*(b*x^2 + a)^(3/2)*A*b/a - (b*x^2 + 
 a)^(3/2)*B/x - 1/2*(b*x^2 + a)^(5/2)*A/(a*x^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=\frac {1}{24} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left (3 \, D b x + 4 \, C b\right )} x + \frac {3 \, {\left (5 \, D a b^{2} + 4 \, B b^{3}\right )}}{b^{2}}\right )} x + \frac {8 \, {\left (4 \, C a b^{2} + 3 \, A b^{3}\right )}}{b^{2}}\right )} + \frac {{\left (2 \, C a^{2} + 3 \, A a b\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3 \, {\left (D a^{2} + 4 \, B a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, \sqrt {b}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b - 2 \, B a^{3} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \] Input:

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

Output:

1/24*sqrt(b*x^2 + a)*((2*(3*D*b*x + 4*C*b)*x + 3*(5*D*a*b^2 + 4*B*b^3)/b^2 
)*x + 8*(4*C*a*b^2 + 3*A*b^3)/b^2) + (2*C*a^2 + 3*A*a*b)*arctan(-(sqrt(b)* 
x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - 3/8*(D*a^2 + 4*B*a*b)*log(abs(-s 
qrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + ((sqrt(b)*x - sqrt(b*x^2 + a))^3*A* 
a*b + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b) + (sqrt(b)*x - sqrt( 
b*x^2 + a))*A*a^2*b - 2*B*a^3*sqrt(b))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - 
a)^2
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 3.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=\frac {C\,{\left (b\,x^2+a\right )}^{3/2}}{3}-C\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+A\,b\,\sqrt {b\,x^2+a}+C\,a\,\sqrt {b\,x^2+a}-\frac {A\,a\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {3\,A\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2}-\frac {B\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}}+\frac {x\,{\left (b\,x^2+a\right )}^{3/2}\,D\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \] Input:

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

Output:

(C*(a + b*x^2)^(3/2))/3 - C*a^(3/2)*atanh((a + b*x^2)^(1/2)/a^(1/2)) + A*b 
*(a + b*x^2)^(1/2) + C*a*(a + b*x^2)^(1/2) - (A*a*(a + b*x^2)^(1/2))/(2*x^ 
2) - (3*A*a^(1/2)*b*atanh((a + b*x^2)^(1/2)/a^(1/2)))/2 - (B*(a + b*x^2)^( 
3/2)*hypergeom([-3/2, -1/2], 1/2, -(b*x^2)/a))/(x*((b*x^2)/a + 1)^(3/2)) + 
 (x*(a + b*x^2)^(3/2)*D*hypergeom([-3/2, 1/2], 3/2, -(b*x^2)/a))/((b*x^2)/ 
a + 1)^(3/2)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^3} \, dx=\frac {-12 \sqrt {b \,x^{2}+a}\, a^{2} b +24 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{2}-24 \sqrt {b \,x^{2}+a}\, a \,b^{2} x +32 \sqrt {b \,x^{2}+a}\, a b c \,x^{2}+15 \sqrt {b \,x^{2}+a}\, a b d \,x^{3}+12 \sqrt {b \,x^{2}+a}\, b^{3} x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{4}+6 \sqrt {b \,x^{2}+a}\, b^{2} d \,x^{5}+36 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{2}+24 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,x^{2}-36 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{2}-24 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,x^{2}+9 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d \,x^{2}+36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{2}}{24 b \,x^{2}} \] Input:

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

Output:

( - 12*sqrt(a + b*x**2)*a**2*b + 24*sqrt(a + b*x**2)*a*b**2*x**2 - 24*sqrt 
(a + b*x**2)*a*b**2*x + 32*sqrt(a + b*x**2)*a*b*c*x**2 + 15*sqrt(a + b*x** 
2)*a*b*d*x**3 + 12*sqrt(a + b*x**2)*b**3*x**3 + 8*sqrt(a + b*x**2)*b**2*c* 
x**4 + 6*sqrt(a + b*x**2)*b**2*d*x**5 + 36*sqrt(a)*log((sqrt(a + b*x**2) - 
 sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*x**2 + 24*sqrt(a)*log((sqrt(a + b*x* 
*2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*x**2 - 36*sqrt(a)*log((sqrt(a + 
b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*x**2 - 24*sqrt(a)*log((sqrt 
(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*x**2 + 9*sqrt(b)*log((s 
qrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*d*x**2 + 36*sqrt(b)*log((sqrt(a 
 + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*x**2)/(24*b*x**2)