\(\int \frac {x^4 (A+B x)}{(a+b x^2)^{5/2}} \, dx\) [1258]

Optimal result

Integrand size = 20, antiderivative size = 99 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {a (a B-A b x)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac {2 (3 a B-2 A b x)}{3 b^3 \sqrt {a+b x^2}}+\frac {B \sqrt {a+b x^2}}{b^3}+\frac {A \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Output:

-1/3*a*(-A*b*x+B*a)/b^3/(b*x^2+a)^(3/2)+2/3*(-2*A*b*x+3*B*a)/b^3/(b*x^2+a) 
^(1/2)+B*(b*x^2+a)^(1/2)/b^3+A*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {8 a^2 B-3 a b x (A-4 B x)+b^2 x^3 (-4 A+3 B x)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Input:

Integrate[(x^4*(A + B*x))/(a + b*x^2)^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {530, 2345, 27, 455, 224, 219}

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.

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

\(\Big \downarrow \) 530

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

\(\Big \downarrow \) 2345

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

Int[(x^4*(A + B*x))/(a + b*x^2)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

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 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 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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 530

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb 
ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co 
eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po 
lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x 
)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(a 
 + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; 
FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 
1] && IntegerQ[2*p]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[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))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.21

method	result	size
default	\(A \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+B \left (\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{b}\right )\)	\(120\)
risch	\(\frac {B \sqrt {b \,x^{2}+a}}{b^{3}}+\frac {\frac {A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {a \left (A b -B \sqrt {-a b}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{4 b^{2}}-\frac {a \left (A b +B \sqrt {-a b}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{4 b^{2}}-\frac {\left (3 A b -4 B \sqrt {-a b}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{4 b^{2} \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (3 A b +4 B \sqrt {-a b}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{4 b^{2} \left (x -\frac {\sqrt {-a b}}{b}\right )}}{b^{2}}\)	\(472\)

Input:

int(x^4*(B*x+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.61 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (A b^{2} x^{4} + 2 \, A a b x^{2} + A a^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (3 \, B b^{2} x^{4} - 4 \, A b^{2} x^{3} + 12 \, B a b x^{2} - 3 \, A a b x + 8 \, B a^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}}, -\frac {3 \, {\left (A b^{2} x^{4} + 2 \, A a b x^{2} + A a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, B b^{2} x^{4} - 4 \, A b^{2} x^{3} + 12 \, B a b x^{2} - 3 \, A a b x + 8 \, B a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \] Input:

integrate(x^4*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(A*b^2*x^4 + 2*A*a*b*x^2 + A*a^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b* 
x^2 + a)*sqrt(b)*x - a) + 2*(3*B*b^2*x^4 - 4*A*b^2*x^3 + 12*B*a*b*x^2 - 3* 
A*a*b*x + 8*B*a^2)*sqrt(b*x^2 + a))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3), -1/ 
3*(3*(A*b^2*x^4 + 2*A*a*b*x^2 + A*a^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x 
^2 + a)) - (3*B*b^2*x^4 - 4*A*b^2*x^3 + 12*B*a*b*x^2 - 3*A*a*b*x + 8*B*a^2 
)*sqrt(b*x^2 + a))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (92) = 184\).

Time = 5.97 (sec) , antiderivative size = 445, normalized size of antiderivative = 4.49 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=A \left (\frac {3 a^{\frac {39}{2}} b^{11} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{\frac {37}{2}} b^{12} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{19} b^{\frac {23}{2}} x}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {4 a^{18} b^{\frac {25}{2}} x^{3}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} \frac {8 a^{2}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} + \frac {12 a b x^{2}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} + \frac {3 b^{2} x^{4}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate(x**4*(B*x+A)/(b*x**2+a)**(5/2),x)
 

Output:

A*(3*a**(39/2)*b**11*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39 
/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x 
**2/a)) + 3*a**(37/2)*b**12*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a 
))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2* 
sqrt(1 + b*x**2/a)) - 3*a**19*b**(23/2)*x/(3*a**(39/2)*b**(27/2)*sqrt(1 + 
b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) - 4*a**18*b**(2 
5/2)*x**3/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2 
)*x**2*sqrt(1 + b*x**2/a))) + B*Piecewise((8*a**2/(3*a*b**3*sqrt(a + b*x** 
2) + 3*b**4*x**2*sqrt(a + b*x**2)) + 12*a*b*x**2/(3*a*b**3*sqrt(a + b*x**2 
) + 3*b**4*x**2*sqrt(a + b*x**2)) + 3*b**2*x**4/(3*a*b**3*sqrt(a + b*x**2) 
 + 3*b**4*x**2*sqrt(a + b*x**2)), Ne(b, 0)), (x**6/(6*a**(5/2)), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, A x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {B x^{4}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {4 \, B a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {A x}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} + \frac {8 \, B a^{2}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} \] Input:

integrate(x^4*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (\frac {3 \, B x}{b} - \frac {4 \, A}{b}\right )} x + \frac {12 \, B a}{b^{2}}\right )} x - \frac {3 \, A a}{b^{2}}\right )} x + \frac {8 \, B a^{2}}{b^{3}}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {A \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {5}{2}}} \] Input:

integrate(x^4*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

1/3*((((3*B*x/b - 4*A/b)*x + 12*B*a/b^2)*x - 3*A*a/b^2)*x + 8*B*a^2/b^3)/( 
b*x^2 + a)^(3/2) - A*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int((x^4*(A + B*x))/(a + b*x^2)^(5/2),x)
 

Output:

int((x^4*(A + B*x))/(a + b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 6.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.93 \[ \int \frac {x^4 (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a^{2} b x +8 \sqrt {b \,x^{2}+a}\, a^{2} b -4 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{3}+12 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{2}+3 \sqrt {b \,x^{2}+a}\, b^{3} x^{4}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3}+6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b \,x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{4}}{3 b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int(x^4*(B*x+A)/(b*x^2+a)^(5/2),x)
 

Output:

( - 3*sqrt(a + b*x**2)*a**2*b*x + 8*sqrt(a + b*x**2)*a**2*b - 4*sqrt(a + b 
*x**2)*a*b**2*x**3 + 12*sqrt(a + b*x**2)*a*b**2*x**2 + 3*sqrt(a + b*x**2)* 
b**3*x**4 + 3*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3 + 6 
*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*x**2 + 3*sqrt( 
b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*x**4)/(3*b**3*(a**2 
+ 2*a*b*x**2 + b**2*x**4))