\(\int \frac {(A+B x) (a+b x^2)^{3/2}}{x^5} \, dx\) [1041]

Optimal result

Integrand size = 20, antiderivative size = 111 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=-\frac {b (3 A+8 B x) \sqrt {a+b x^2}}{8 x^2}-\frac {(3 A+4 B x) \left (a+b x^2\right )^{3/2}}{12 x^4}+b^{3/2} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {3 A b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=-\frac {\sqrt {a+b x^2} \left (6 a A+8 a B x+15 A b x^2+32 b B x^3\right )}{24 x^4}+\frac {3 A b^2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}}-b^{3/2} B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {537, 25, 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.

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

\(\Big \downarrow \) 537

\(\displaystyle -\frac {1}{4} b \int -\frac {(3 A+4 B x) \sqrt {b x^2+a}}{x^3}dx-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} b \int \frac {(3 A+4 B x) \sqrt {b x^2+a}}{x^3}dx-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {1}{4} b \left (-\frac {1}{2} b \int -\frac {3 A+8 B x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \int \frac {3 A+8 B x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \left (3 A \int \frac {1}{x \sqrt {b x^2+a}}dx+8 B \int \frac {1}{\sqrt {b x^2+a}}dx\right )-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \left (3 A \int \frac {1}{x \sqrt {b x^2+a}}dx+8 B \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}\right )-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \left (3 A \int \frac {1}{x \sqrt {b x^2+a}}dx+\frac {8 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \left (\frac {3}{2} A \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+\frac {8 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \left (\frac {3 A \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}+\frac {8 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{4} b \left (\frac {1}{2} b \left (\frac {8 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {3 A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )-\frac {\sqrt {a+b x^2} (3 A+8 B x)}{2 x^2}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}\)

Input:

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

Output:

-1/12*((3*A + 4*B*x)*(a + b*x^2)^(3/2))/x^4 + (b*(-1/2*((3*A + 8*B*x)*Sqrt 
[a + b*x^2])/x^2 + (b*((8*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] 
- (3*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2))/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (32 B b \,x^{3}+15 A b \,x^{2}+8 B a x +6 A a \right )}{24 x^{4}}+b^{\frac {3}{2}} B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-\frac {3 b^{2} A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{8 \sqrt {a}}\)	\(93\)
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )\)	\(205\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 480, normalized size of antiderivative = 4.32 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=\left [\frac {24 \, B a b^{\frac {3}{2}} x^{4} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 9 \, A \sqrt {a} b^{2} x^{4} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (32 \, B a b x^{3} + 15 \, A a b x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a x^{4}}, -\frac {48 \, B a \sqrt {-b} b x^{4} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 9 \, A \sqrt {a} b^{2} x^{4} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (32 \, B a b x^{3} + 15 \, A a b x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a x^{4}}, \frac {9 \, A \sqrt {-a} b^{2} x^{4} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + 12 \, B a b^{\frac {3}{2}} x^{4} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (32 \, B a b x^{3} + 15 \, A a b x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{24 \, a x^{4}}, -\frac {24 \, B a \sqrt {-b} b x^{4} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 9 \, A \sqrt {-a} b^{2} x^{4} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (32 \, B a b x^{3} + 15 \, A a b x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{24 \, a x^{4}}\right ] \] Input:

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

Output:

[1/48*(24*B*a*b^(3/2)*x^4*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) 
+ 9*A*sqrt(a)*b^2*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) 
- 2*(32*B*a*b*x^3 + 15*A*a*b*x^2 + 8*B*a^2*x + 6*A*a^2)*sqrt(b*x^2 + a))/( 
a*x^4), -1/48*(48*B*a*sqrt(-b)*b*x^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 
9*A*sqrt(a)*b^2*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 
2*(32*B*a*b*x^3 + 15*A*a*b*x^2 + 8*B*a^2*x + 6*A*a^2)*sqrt(b*x^2 + a))/(a* 
x^4), 1/24*(9*A*sqrt(-a)*b^2*x^4*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 12*B 
*a*b^(3/2)*x^4*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - (32*B*a*b 
*x^3 + 15*A*a*b*x^2 + 8*B*a^2*x + 6*A*a^2)*sqrt(b*x^2 + a))/(a*x^4), -1/24 
*(24*B*a*sqrt(-b)*b*x^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 9*A*sqrt(-a)* 
b^2*x^4*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (32*B*a*b*x^3 + 15*A*a*b*x^2 
+ 8*B*a^2*x + 6*A*a^2)*sqrt(b*x^2 + a))/(a*x^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (102) = 204\).

Time = 4.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.13 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=- \frac {A a^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A a \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {A b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} - \frac {B \sqrt {a} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + B b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

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

Output:

-A*a**2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*A*a*sqrt(b)/(8*x**3*sqrt 
(a/(b*x**2) + 1)) - A*b**(3/2)*sqrt(a/(b*x**2) + 1)/(2*x) - A*b**(3/2)/(8* 
x*sqrt(a/(b*x**2) + 1)) - 3*A*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a)) 
- B*sqrt(a)*b/(x*sqrt(1 + b*x**2/a)) - B*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3 
*x**2) - B*b**(3/2)*sqrt(a/(b*x**2) + 1)/3 + B*b**(3/2)*asinh(sqrt(b)*x/sq 
rt(a)) - B*b**2*x/(sqrt(a)*sqrt(1 + b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {b x^{2} + a} B b^{2} x}{a} + B b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3 \, A b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} A b^{2}}{8 \, a} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{4 \, a x^{4}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (89) = 178\).

Time = 0.14 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.57 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=\frac {3 \, A b^{2} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} - B b^{\frac {3}{2}} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} A b^{2} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a b^{\frac {3}{2}} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A a b^{2} - 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} b^{\frac {3}{2}} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a^{2} b^{2} + 80 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} b^{\frac {3}{2}} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{3} b^{2} - 32 \, B a^{4} b^{\frac {3}{2}}}{12 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{4}} \] Input:

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

Output:

3/4*A*b^2*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - B*b^( 
3/2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) + 1/12*(15*(sqrt(b)*x - sqrt(b 
*x^2 + a))^7*A*b^2 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a*b^(3/2) + 9*(s 
qrt(b)*x - sqrt(b*x^2 + a))^5*A*a*b^2 - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^4 
*B*a^2*b^(3/2) + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^2*b^2 + 80*(sqrt(b) 
*x - sqrt(b*x^2 + a))^2*B*a^3*b^(3/2) + 15*(sqrt(b)*x - sqrt(b*x^2 + a))*A 
*a^3*b^2 - 32*B*a^4*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.56 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^5} \, dx=\frac {-6 \sqrt {b \,x^{2}+a}\, a^{2}-15 \sqrt {b \,x^{2}+a}\, a b \,x^{2}-8 \sqrt {b \,x^{2}+a}\, a b x -32 \sqrt {b \,x^{2}+a}\, b^{2} x^{3}+9 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}-9 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}+8 \sqrt {b}\, b^{2} x^{4}}{24 x^{4}} \] Input:

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

Output:

( - 6*sqrt(a + b*x**2)*a**2 - 15*sqrt(a + b*x**2)*a*b*x**2 - 8*sqrt(a + b* 
x**2)*a*b*x - 32*sqrt(a + b*x**2)*b**2*x**3 + 9*sqrt(a)*log((sqrt(a + b*x* 
*2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4 - 9*sqrt(a)*log((sqrt(a + b* 
x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4 + 24*sqrt(b)*log((sqrt(a + 
 b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*x**4 + 8*sqrt(b)*b**2*x**4)/(24*x**4)