\(\int \frac {(c+d x) (A+B x+C x^2)}{x (a+b x^2)^{3/2}} \, dx\) [136]

Optimal result

Integrand size = 30, antiderivative size = 121 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b c-a (c C+B d)}{a b \sqrt {a+b x^2}}+\frac {(b B c+A b d-a C d) x}{a b \sqrt {a+b x^2}}+\frac {C d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {A c \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \] Output:

(A*b*c-a*(B*d+C*c))/a/b/(b*x^2+a)^(1/2)+(A*b*d+B*b*c-C*a*d)*x/a/b/(b*x^2+a 
)^(1/2)+C*d*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)-A*c*arctanh((b*x^2+ 
a)^(1/2)/a^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {b B c x+A b (c+d x)-a (c C+B d+C d x)}{a b \sqrt {a+b x^2}}+\frac {2 A c \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {C d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}} \] Input:

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

Output:

(b*B*c*x + A*b*(c + d*x) - a*(c*C + B*d + C*d*x))/(a*b*Sqrt[a + b*x^2]) + 
(2*A*c*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(3/2) - (C*d*Log[ 
-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(3/2)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2336, 25, 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.

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

\(\Big \downarrow \) 2336

\(\displaystyle \frac {x (-a C d+A b d+b B c)-a B d-a c C+A b c}{a b \sqrt {a+b x^2}}-\frac {\int -\frac {A b c+a C d x}{b x \sqrt {b x^2+a}}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {A b c+a C d x}{b x \sqrt {b x^2+a}}dx}{a}+\frac {x (-a C d+A b d+b B c)-a B d-a c C+A b c}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {A b c+a C d x}{x \sqrt {b x^2+a}}dx}{a b}+\frac {x (-a C d+A b d+b B c)-a B d-a c C+A b c}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {A b c \int \frac {1}{x \sqrt {b x^2+a}}dx+a C d \int \frac {1}{\sqrt {b x^2+a}}dx}{a b}+\frac {x (-a C d+A b d+b B c)-a B d-a c C+A b c}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {A b c \int \frac {1}{x \sqrt {b x^2+a}}dx+a C d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{a b}+\frac {x (-a C d+A b d+b B c)-a B d-a c C+A b c}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(A*b*c - a*c*C - a*B*d + (b*B*c + A*b*d - a*C*d)*x)/(a*b*Sqrt[a + b*x^2]) 
+ ((a*C*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (A*b*c*ArcTanh[S 
qrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/(a*b)
 

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 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 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {A d x}{a \sqrt {b \,x^{2}+a}}+\frac {B c x}{a \sqrt {b \,x^{2}+a}}+A c \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {B d}{b \sqrt {b \,x^{2}+a}}-\frac {C c}{b \sqrt {b \,x^{2}+a}}+d C \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\)	\(150\)

Input:

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

Output:

A*d*x/a/(b*x^2+a)^(1/2)+B*c*x/a/(b*x^2+a)^(1/2)+A*c*(1/a/(b*x^2+a)^(1/2)-1 
/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))-B*d/b/(b*x^2+a)^(1/2)-C*c/ 
b/(b*x^2+a)^(1/2)+d*C*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+ 
a)^(1/2)))
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 706, normalized size of antiderivative = 5.83 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (C a^{2} b d x^{2} + C a^{3} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + {\left (A b^{3} c x^{2} + A a b^{2} c\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (B a^{2} b d + {\left (C a^{2} b - A a b^{2}\right )} c - {\left (B a b^{2} c - {\left (C a^{2} b - A a b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, -\frac {2 \, {\left (C a^{2} b d x^{2} + C a^{3} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (A b^{3} c x^{2} + A a b^{2} c\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (B a^{2} b d + {\left (C a^{2} b - A a b^{2}\right )} c - {\left (B a b^{2} c - {\left (C a^{2} b - A a b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac {2 \, {\left (A b^{3} c x^{2} + A a b^{2} c\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (C a^{2} b d x^{2} + C a^{3} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (B a^{2} b d + {\left (C a^{2} b - A a b^{2}\right )} c - {\left (B a b^{2} c - {\left (C a^{2} b - A a b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, -\frac {{\left (C a^{2} b d x^{2} + C a^{3} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (A b^{3} c x^{2} + A a b^{2} c\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (B a^{2} b d + {\left (C a^{2} b - A a b^{2}\right )} c - {\left (B a b^{2} c - {\left (C a^{2} b - A a b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{a^{2} b^{3} x^{2} + a^{3} b^{2}}\right ] \] Input:

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

Output:

[1/2*((C*a^2*b*d*x^2 + C*a^3*d)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*s 
qrt(b)*x - a) + (A*b^3*c*x^2 + A*a*b^2*c)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x 
^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(B*a^2*b*d + (C*a^2*b - A*a*b^2)*c - (B*a* 
b^2*c - (C*a^2*b - A*a*b^2)*d)*x)*sqrt(b*x^2 + a))/(a^2*b^3*x^2 + a^3*b^2) 
, -1/2*(2*(C*a^2*b*d*x^2 + C*a^3*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 
+ a)) - (A*b^3*c*x^2 + A*a*b^2*c)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)* 
sqrt(a) + 2*a)/x^2) + 2*(B*a^2*b*d + (C*a^2*b - A*a*b^2)*c - (B*a*b^2*c - 
(C*a^2*b - A*a*b^2)*d)*x)*sqrt(b*x^2 + a))/(a^2*b^3*x^2 + a^3*b^2), 1/2*(2 
*(A*b^3*c*x^2 + A*a*b^2*c)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + ( 
C*a^2*b*d*x^2 + C*a^3*d)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)* 
x - a) - 2*(B*a^2*b*d + (C*a^2*b - A*a*b^2)*c - (B*a*b^2*c - (C*a^2*b - A* 
a*b^2)*d)*x)*sqrt(b*x^2 + a))/(a^2*b^3*x^2 + a^3*b^2), -((C*a^2*b*d*x^2 + 
C*a^3*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (A*b^3*c*x^2 + A*a* 
b^2*c)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (B*a^2*b*d + (C*a^2*b 
 - A*a*b^2)*c - (B*a*b^2*c - (C*a^2*b - A*a*b^2)*d)*x)*sqrt(b*x^2 + a))/(a 
^2*b^3*x^2 + a^3*b^2)]
 

Sympy [A] (verification not implemented)

Time = 10.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.74 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=A c \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}}\right ) + \frac {A d x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + B d \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {B c x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + C c \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + C d \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:

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

Output:

A*c*(2*a**3*sqrt(1 + b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**3*log 
(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**3*log(sqrt(1 + b*x**2/a 
) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**2*b*x**2*log(b*x**2/a)/(2*a** 
(9/2) + 2*a**(7/2)*b*x**2) - 2*a**2*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(2* 
a**(9/2) + 2*a**(7/2)*b*x**2)) + A*d*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + B*d 
*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)) 
 + B*c*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + C*c*Piecewise((-1/(b*sqrt(a + b*x 
**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)) + C*d*(asinh(sqrt(b)*x/sqrt(a 
))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a)))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {B c x}{\sqrt {b x^{2} + a} a} + \frac {A d x}{\sqrt {b x^{2} + a} a} - \frac {C d x}{\sqrt {b x^{2} + a} b} + \frac {C d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {A c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {A c}{\sqrt {b x^{2} + a} a} - \frac {C c}{\sqrt {b x^{2} + a} b} - \frac {B d}{\sqrt {b x^{2} + a} b} \] Input:

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

Output:

B*c*x/(sqrt(b*x^2 + a)*a) + A*d*x/(sqrt(b*x^2 + a)*a) - C*d*x/(sqrt(b*x^2 
+ a)*b) + C*d*arcsinh(b*x/sqrt(a*b))/b^(3/2) - A*c*arcsinh(a/(sqrt(a*b)*ab 
s(x)))/a^(3/2) + A*c/(sqrt(b*x^2 + a)*a) - C*c/(sqrt(b*x^2 + a)*b) - B*d/( 
sqrt(b*x^2 + a)*b)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.84 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x \left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b \,x^{2}+a}\, a \,b^{2} c +\sqrt {b \,x^{2}+a}\, a \,b^{2} d x -\sqrt {b \,x^{2}+a}\, a \,b^{2} d -\sqrt {b \,x^{2}+a}\, a b \,c^{2}-\sqrt {b \,x^{2}+a}\, a b c d x +\sqrt {b \,x^{2}+a}\, b^{3} c x +\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c +\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c -\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{2}+\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} c d +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{2}+\sqrt {b}\, a^{2} b d -\sqrt {b}\, a^{2} c d +\sqrt {b}\, a \,b^{2} c +\sqrt {b}\, a \,b^{2} d \,x^{2}-\sqrt {b}\, a b c d \,x^{2}+\sqrt {b}\, b^{3} c \,x^{2}}{a \,b^{2} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(sqrt(a + b*x**2)*a*b**2*c + sqrt(a + b*x**2)*a*b**2*d*x - sqrt(a + b*x**2 
)*a*b**2*d - sqrt(a + b*x**2)*a*b*c**2 - sqrt(a + b*x**2)*a*b*c*d*x + sqrt 
(a + b*x**2)*b**3*c*x + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)* 
x)/sqrt(a))*a*b**2*c + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x 
)/sqrt(a))*b**3*c*x**2 - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b) 
*x)/sqrt(a))*a*b**2*c - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)* 
x)/sqrt(a))*b**3*c*x**2 + sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt( 
a))*a**2*c*d + sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c*d 
*x**2 + sqrt(b)*a**2*b*d - sqrt(b)*a**2*c*d + sqrt(b)*a*b**2*c + sqrt(b)*a 
*b**2*d*x**2 - sqrt(b)*a*b*c*d*x**2 + sqrt(b)*b**3*c*x**2)/(a*b**2*(a + b* 
x**2))