\(\int \frac {(c+d x) (a+b x^2)^{3/2} (A+B x+C x^2)}{x^5} \, dx\) [55]

Optimal result

Integrand size = 30, antiderivative size = 282 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx=\frac {3 b (A b c+4 a (c C+B d)) \sqrt {a+b x^2}}{8 a}+\frac {b (3 a C d+2 b (B c+A d)) x \sqrt {a+b x^2}}{2 a}-\frac {(A b c+4 a (c C+B d)) \left (a+b x^2\right )^{3/2}}{8 a x^2}-\frac {(3 a C d+2 b (B c+A d)) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A c \left (a+b x^2\right )^{5/2}}{4 a x^4}-\frac {(B c+A d) \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} \sqrt {b} (3 a C d+2 b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {3 b (A b c+4 a (c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}} \] Output:

3/8*b*(A*b*c+4*a*(B*d+C*c))*(b*x^2+a)^(1/2)/a+1/2*b*(3*a*C*d+2*b*(A*d+B*c) 
)*x*(b*x^2+a)^(1/2)/a-1/8*(A*b*c+4*a*(B*d+C*c))*(b*x^2+a)^(3/2)/a/x^2-1/3* 
(3*a*C*d+2*b*(A*d+B*c))*(b*x^2+a)^(3/2)/a/x-1/4*A*c*(b*x^2+a)^(5/2)/a/x^4- 
1/3*(A*d+B*c)*(b*x^2+a)^(5/2)/a/x^3+1/2*b^(1/2)*(3*a*C*d+2*b*(A*d+B*c))*ar 
ctanh(b^(1/2)*x/(b*x^2+a)^(1/2))-3/8*b*(A*b*c+4*a*(B*d+C*c))*arctanh((b*x^ 
2+a)^(1/2)/a^(1/2))/a^(1/2)
 

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.81 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx=-\frac {\sqrt {a+b x^2} \left (b x^2 \left (A (15 c+32 d x)-4 x \left (-8 B c+6 c C x+6 B d x+3 C d x^2\right )\right )+2 a (A (3 c+4 d x)+2 x (3 C x (c+2 d x)+B (2 c+3 d x)))\right )}{24 x^4}+\frac {3 A b^2 c \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}}-3 \sqrt {a} b (c C+B d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {1}{2} \sqrt {b} (3 a C d+2 b (B c+A d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((c + d*x)*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^5,x]
 

Output:

-1/24*(Sqrt[a + b*x^2]*(b*x^2*(A*(15*c + 32*d*x) - 4*x*(-8*B*c + 6*c*C*x + 
 6*B*d*x + 3*C*d*x^2)) + 2*a*(A*(3*c + 4*d*x) + 2*x*(3*C*x*(c + 2*d*x) + B 
*(2*c + 3*d*x)))))/x^4 + (3*A*b^2*c*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/ 
Sqrt[a]])/(4*Sqrt[a]) - 3*Sqrt[a]*b*(c*C + B*d)*ArcTanh[(-(Sqrt[b]*x) + Sq 
rt[a + b*x^2])/Sqrt[a]] - (Sqrt[b]*(3*a*C*d + 2*b*(B*c + A*d))*Log[-(Sqrt[ 
b]*x) + Sqrt[a + b*x^2]])/2
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2338, 25, 2338, 25, 27, 537, 25, 535, 27, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^5,x]
 

Output:

-1/4*(A*c*(a + b*x^2)^(5/2))/(a*x^4) + ((-4*(B*c + A*d)*(a + b*x^2)^(5/2)) 
/(3*x^3) + (-1/2*((3*(A*b*c + 4*a*(c*C + B*d)) + 8*(3*a*C*d + 2*b*(B*c + A 
*d))*x)*(a + b*x^2)^(3/2))/x^2 + (3*b*((3*(A*b*c + 4*a*(c*C + B*d)) + 4*(3 
*a*C*d + 2*b*(B*c + A*d))*x)*Sqrt[a + b*x^2] + a*((4*(3*a*C*d + 2*b*(B*c + 
 A*d))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (3*(A*b*c + 4*a*(c* 
C + B*d))*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])))/2)/3)/(4*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 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 [A] (verified)

Time = 0.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (32 A b d \,x^{3}+32 B b c \,x^{3}+24 C a d \,x^{3}+15 A b c \,x^{2}+12 B a d \,x^{2}+12 C a c \,x^{2}+8 A a d x +8 B a c x +6 A a c \right )}{24 x^{4}}+\frac {b \left (-\frac {\left (3 A b c +12 B a d +12 C a c \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+8 A \sqrt {b}\, d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+8 B \sqrt {b}\, c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {16 a C d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+8 B d \sqrt {b \,x^{2}+a}+8 C c \sqrt {b \,x^{2}+a}+8 C b d \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\right )}{8}\)	\(269\)
default	\(\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )+A c \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )+d C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )\)	\(374\)

Input:

int((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/24*(b*x^2+a)^(1/2)*(32*A*b*d*x^3+32*B*b*c*x^3+24*C*a*d*x^3+15*A*b*c*x^2 
+12*B*a*d*x^2+12*C*a*c*x^2+8*A*a*d*x+8*B*a*c*x+6*A*a*c)/x^4+1/8*b*(-(3*A*b 
*c+12*B*a*d+12*C*a*c)/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+8*A*b^ 
(1/2)*d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+8*B*b^(1/2)*c*ln(b^(1/2)*x+(b*x^2+a) 
^(1/2))+16*a*C*d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+8*B*d*(b*x^2+a)^(1/ 
2)+8*C*c*(b*x^2+a)^(1/2)+8*C*b*d*(1/2*x/b*(b*x^2+a)^(1/2)-1/2*a/b^(3/2)*ln 
(b^(1/2)*x+(b*x^2+a)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 906, normalized size of antiderivative = 3.21 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5,x, algorithm="fricas")
 

Output:

[1/48*(12*(2*B*a*b*c + (3*C*a^2 + 2*A*a*b)*d)*sqrt(b)*x^4*log(-2*b*x^2 - 2 
*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 9*(4*B*a*b*d + (4*C*a*b + A*b^2)*c)*sqrt 
(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(12*C*a*b* 
d*x^5 + 24*(C*a*b*c + B*a*b*d)*x^4 - 6*A*a^2*c - 8*(4*B*a*b*c + (3*C*a^2 + 
 4*A*a*b)*d)*x^3 - 3*(4*B*a^2*d + (4*C*a^2 + 5*A*a*b)*c)*x^2 - 8*(B*a^2*c 
+ A*a^2*d)*x)*sqrt(b*x^2 + a))/(a*x^4), -1/48*(24*(2*B*a*b*c + (3*C*a^2 + 
2*A*a*b)*d)*sqrt(-b)*x^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 9*(4*B*a*b*d 
 + (4*C*a*b + A*b^2)*c)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a 
) + 2*a)/x^2) - 2*(12*C*a*b*d*x^5 + 24*(C*a*b*c + B*a*b*d)*x^4 - 6*A*a^2*c 
 - 8*(4*B*a*b*c + (3*C*a^2 + 4*A*a*b)*d)*x^3 - 3*(4*B*a^2*d + (4*C*a^2 + 5 
*A*a*b)*c)*x^2 - 8*(B*a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + a))/(a*x^4), 1/24*( 
9*(4*B*a*b*d + (4*C*a*b + A*b^2)*c)*sqrt(-a)*x^4*arctan(sqrt(b*x^2 + a)*sq 
rt(-a)/a) + 6*(2*B*a*b*c + (3*C*a^2 + 2*A*a*b)*d)*sqrt(b)*x^4*log(-2*b*x^2 
 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (12*C*a*b*d*x^5 + 24*(C*a*b*c + B*a* 
b*d)*x^4 - 6*A*a^2*c - 8*(4*B*a*b*c + (3*C*a^2 + 4*A*a*b)*d)*x^3 - 3*(4*B* 
a^2*d + (4*C*a^2 + 5*A*a*b)*c)*x^2 - 8*(B*a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + 
 a))/(a*x^4), -1/24*(12*(2*B*a*b*c + (3*C*a^2 + 2*A*a*b)*d)*sqrt(-b)*x^4*a 
rctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 9*(4*B*a*b*d + (4*C*a*b + A*b^2)*c)*sq 
rt(-a)*x^4*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (12*C*a*b*d*x^5 + 24*(C*a* 
b*c + B*a*b*d)*x^4 - 6*A*a^2*c - 8*(4*B*a*b*c + (3*C*a^2 + 4*A*a*b)*d)*...
 

Sympy [A] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.52 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**5,x)
 

Output:

-A*sqrt(a)*b*d/(x*sqrt(1 + b*x**2/a)) - A*a**2*c/(4*sqrt(b)*x**5*sqrt(a/(b 
*x**2) + 1)) - 3*A*a*sqrt(b)*c/(8*x**3*sqrt(a/(b*x**2) + 1)) - A*a*sqrt(b) 
*d*sqrt(a/(b*x**2) + 1)/(3*x**2) - A*b**(3/2)*c*sqrt(a/(b*x**2) + 1)/(2*x) 
 - A*b**(3/2)*c/(8*x*sqrt(a/(b*x**2) + 1)) - A*b**(3/2)*d*sqrt(a/(b*x**2) 
+ 1)/3 + A*b**(3/2)*d*asinh(sqrt(b)*x/sqrt(a)) - 3*A*b**2*c*asinh(sqrt(a)/ 
(sqrt(b)*x))/(8*sqrt(a)) - A*b**2*d*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - B*sqr 
t(a)*b*c/(x*sqrt(1 + b*x**2/a)) - 3*B*sqrt(a)*b*d*asinh(sqrt(a)/(sqrt(b)*x 
))/2 - B*a*sqrt(b)*c*sqrt(a/(b*x**2) + 1)/(3*x**2) - B*a*sqrt(b)*d*sqrt(a/ 
(b*x**2) + 1)/(2*x) + B*a*sqrt(b)*d/(x*sqrt(a/(b*x**2) + 1)) - B*b**(3/2)* 
c*sqrt(a/(b*x**2) + 1)/3 + B*b**(3/2)*c*asinh(sqrt(b)*x/sqrt(a)) + B*b**(3 
/2)*d*x/sqrt(a/(b*x**2) + 1) - B*b**2*c*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - C 
*a**(3/2)*d/(x*sqrt(1 + b*x**2/a)) - 3*C*sqrt(a)*b*c*asinh(sqrt(a)/(sqrt(b 
)*x))/2 - C*sqrt(a)*b*d*x/sqrt(1 + b*x**2/a) - C*a*sqrt(b)*c*sqrt(a/(b*x** 
2) + 1)/(2*x) + C*a*sqrt(b)*c/(x*sqrt(a/(b*x**2) + 1)) + C*a*sqrt(b)*d*asi 
nh(sqrt(b)*x/sqrt(a)) + C*b**(3/2)*c*x/sqrt(a/(b*x**2) + 1) + C*b*d*Piecew 
ise((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)), (s 
qrt(a)*x, True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx=\frac {3}{2} \, \sqrt {b x^{2} + a} C b d x + \frac {3}{2} \, C a \sqrt {b} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3 \, A b^{2} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} c}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} A b^{2} c}{8 \, a} + \frac {\sqrt {b x^{2} + a} {\left (B c + A d\right )} b^{2} x}{a} + {\left (B c + A d\right )} b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3}{2} \, {\left (C c + B d\right )} \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b x^{2} + a} {\left (C c + B d\right )} b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )} b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C d}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b c}{8 \, a^{2} x^{2}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )}}{2 \, a x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c}{4 \, a x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )}}{3 \, a x^{3}} \] Input:

integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (242) = 484\).

Time = 0.20 (sec) , antiderivative size = 754, normalized size of antiderivative = 2.67 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5,x, algorithm="giac")
 

Output:

-1/2*(2*B*b^(3/2)*c + 3*C*a*sqrt(b)*d + 2*A*b^(3/2)*d)*log(abs(-sqrt(b)*x 
+ sqrt(b*x^2 + a))) + 1/2*(C*b*d*x + 2*C*b*c + 2*B*b*d)*sqrt(b*x^2 + a) + 
3/4*(4*C*a*b*c + A*b^2*c + 4*B*a*b*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a) 
)/sqrt(-a))/sqrt(-a) + 1/12*(12*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a*b*c + 
15*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*b^2*c + 12*(sqrt(b)*x - sqrt(b*x^2 + 
a))^7*B*a*b*d + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a*b^(3/2)*c + 24*(sqr 
t(b)*x - sqrt(b*x^2 + a))^6*C*a^2*sqrt(b)*d + 48*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^6*A*a*b^(3/2)*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^2*b*c + 9*(s 
qrt(b)*x - sqrt(b*x^2 + a))^5*A*a*b^2*c - 12*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^5*B*a^2*b*d - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*b^(3/2)*c - 72*(sq 
rt(b)*x - sqrt(b*x^2 + a))^4*C*a^3*sqrt(b)*d - 96*(sqrt(b)*x - sqrt(b*x^2 
+ a))^4*A*a^2*b^(3/2)*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*a^3*b*c + 9 
*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^2*b^2*c - 12*(sqrt(b)*x - sqrt(b*x^2 
+ a))^3*B*a^3*b*d + 80*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^3*b^(3/2)*c + 7 
2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^4*sqrt(b)*d + 80*(sqrt(b)*x - sqrt(b 
*x^2 + a))^2*A*a^3*b^(3/2)*d + 12*(sqrt(b)*x - sqrt(b*x^2 + a))*C*a^4*b*c 
+ 15*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^3*b^2*c + 12*(sqrt(b)*x - sqrt(b*x^ 
2 + a))*B*a^4*b*d - 32*B*a^4*b^(3/2)*c - 24*C*a^5*sqrt(b)*d - 32*A*a^4*b^( 
3/2)*d)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^5} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2))/x^5,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2))/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.82 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5} \, dx=\frac {-24 \sqrt {b \,x^{2}+a}\, a^{2} c -32 \sqrt {b \,x^{2}+a}\, a^{2} d x -60 \sqrt {b \,x^{2}+a}\, a b c \,x^{2}-32 \sqrt {b \,x^{2}+a}\, a b c x -128 \sqrt {b \,x^{2}+a}\, a b d \,x^{3}-48 \sqrt {b \,x^{2}+a}\, a b d \,x^{2}-48 \sqrt {b \,x^{2}+a}\, a \,c^{2} x^{2}-96 \sqrt {b \,x^{2}+a}\, a c d \,x^{3}-128 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{3}+96 \sqrt {b \,x^{2}+a}\, b^{2} d \,x^{4}+96 \sqrt {b \,x^{2}+a}\, b \,c^{2} x^{4}+48 \sqrt {b \,x^{2}+a}\, b c d \,x^{5}+36 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{4}+144 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d \,x^{4}+144 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} x^{4}-36 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{4}-144 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d \,x^{4}-144 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} x^{4}+96 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b d \,x^{4}+144 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a c d \,x^{4}+96 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{4}+32 \sqrt {b}\, a b d \,x^{4}+63 \sqrt {b}\, a c d \,x^{4}+32 \sqrt {b}\, b^{2} c \,x^{4}}{96 x^{4}} \] Input:

int((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5,x)
 

Output:

( - 24*sqrt(a + b*x**2)*a**2*c - 32*sqrt(a + b*x**2)*a**2*d*x - 60*sqrt(a 
+ b*x**2)*a*b*c*x**2 - 32*sqrt(a + b*x**2)*a*b*c*x - 128*sqrt(a + b*x**2)* 
a*b*d*x**3 - 48*sqrt(a + b*x**2)*a*b*d*x**2 - 48*sqrt(a + b*x**2)*a*c**2*x 
**2 - 96*sqrt(a + b*x**2)*a*c*d*x**3 - 128*sqrt(a + b*x**2)*b**2*c*x**3 + 
96*sqrt(a + b*x**2)*b**2*d*x**4 + 96*sqrt(a + b*x**2)*b*c**2*x**4 + 48*sqr 
t(a + b*x**2)*b*c*d*x**5 + 36*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sq 
rt(b)*x)/sqrt(a))*b**2*c*x**4 + 144*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a 
) + sqrt(b)*x)/sqrt(a))*b**2*d*x**4 + 144*sqrt(a)*log((sqrt(a + b*x**2) - 
sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**2*x**4 - 36*sqrt(a)*log((sqrt(a + b*x** 
2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*x**4 - 144*sqrt(a)*log((sqrt(a + 
 b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*d*x**4 - 144*sqrt(a)*log((sq 
rt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**2*x**4 + 96*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*d*x**4 + 144*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*c*d*x**4 + 96*sqrt(b)*log((sqrt(a + 
 b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*c*x**4 + 32*sqrt(b)*a*b*d*x**4 + 63*sq 
rt(b)*a*c*d*x**4 + 32*sqrt(b)*b**2*c*x**4)/(96*x**4)