\(\int \frac {x^4 (A+B x^2)}{(a+b x^2)^{3/2}} \, dx\) [267]

Optimal result

Integrand size = 22, antiderivative size = 114 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {a (A b-a B) x}{b^3 \sqrt {a+b x^2}}+\frac {(4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^3}+\frac {B x^3 \sqrt {a+b x^2}}{4 b^2}-\frac {3 a (4 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}} \] Output:

a*(A*b-B*a)*x/b^3/(b*x^2+a)^(1/2)+1/8*(4*A*b-7*B*a)*x*(b*x^2+a)^(1/2)/b^3+ 
1/4*B*x^3*(b*x^2+a)^(1/2)/b^2-3/8*a*(4*A*b-5*B*a)*arctanh(b^(1/2)*x/(b*x^2 
+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {12 a A b x-15 a^2 B x+4 A b^2 x^3-5 a b B x^3+2 b^2 B x^5}{8 b^3 \sqrt {a+b x^2}}+\frac {3 a (-4 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{4 b^{7/2}} \] Input:

Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x]
 

Output:

(12*a*A*b*x - 15*a^2*B*x + 4*A*b^2*x^3 - 5*a*b*B*x^3 + 2*b^2*B*x^5)/(8*b^3 
*Sqrt[a + b*x^2]) + (3*a*(-4*A*b + 5*a*B)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + 
Sqrt[a + b*x^2])])/(4*b^(7/2))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {363, 252, 262, 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.

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

Int[(x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {-3 \sqrt {b \,x^{2}+a}\, a \left (A b -\frac {5 B a}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (3 \left (-\frac {5 x^{2} B}{12}+A \right ) a \,b^{\frac {3}{2}}+x^{2} \left (\frac {x^{2} B}{2}+A \right ) b^{\frac {5}{2}}-\frac {15 B \,a^{2} \sqrt {b}}{4}\right ) x}{2 \sqrt {b \,x^{2}+a}\, b^{\frac {7}{2}}}\)	\(94\)
risch	\(\frac {x \left (2 b B \,x^{2}+4 A b -7 B a \right ) \sqrt {b \,x^{2}+a}}{8 b^{3}}-\frac {a \left (3 b \left (4 A b -5 B a \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )-\frac {7 a B x}{\sqrt {b \,x^{2}+a}}+\frac {4 b A x}{\sqrt {b \,x^{2}+a}}\right )}{8 b^{3}}\)	\(116\)
default	\(A \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+B \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )\)	\(150\)

Input:

int(x^4*(B*x^2+A)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/2*(-3*(b*x^2+a)^(1/2)*a*(A*b-5/4*B*a)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2)) 
+(3*(-5/12*x^2*B+A)*a*b^(3/2)+x^2*(1/2*x^2*B+A)*b^(5/2)-15/4*B*a^2*b^(1/2) 
)*x)/(b*x^2+a)^(1/2)/b^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.40 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2 \, B b^{3} x^{5} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, -\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b^{3} x^{5} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \] Input:

integrate(x^4*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/16*(3*(5*B*a^3 - 4*A*a^2*b + (5*B*a^2*b - 4*A*a*b^2)*x^2)*sqrt(b)*log( 
-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(2*B*b^3*x^5 - (5*B*a*b^2 
- 4*A*b^3)*x^3 - 3*(5*B*a^2*b - 4*A*a*b^2)*x)*sqrt(b*x^2 + a))/(b^5*x^2 + 
a*b^4), -1/8*(3*(5*B*a^3 - 4*A*a^2*b + (5*B*a^2*b - 4*A*a*b^2)*x^2)*sqrt(- 
b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2*B*b^3*x^5 - (5*B*a*b^2 - 4*A*b^ 
3)*x^3 - 3*(5*B*a^2*b - 4*A*a*b^2)*x)*sqrt(b*x^2 + a))/(b^5*x^2 + a*b^4)]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 6.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=A \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (- \frac {15 a^{\frac {3}{2}} x}{8 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 \sqrt {a} x^{3}}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{5}}{4 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:

integrate(x**4*(B*x**2+A)/(b*x**2+a)**(3/2),x)
 

Output:

A*(3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/ 
(2*b**(5/2)) + x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a))) + B*(-15*a**(3/2)*x/ 
(8*b**3*sqrt(1 + b*x**2/a)) - 5*sqrt(a)*x**3/(8*b**2*sqrt(1 + b*x**2/a)) + 
 15*a**2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(7/2)) + x**5/(4*sqrt(a)*b*sqrt(1 
+ b*x**2/a)))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B x^{5}}{4 \, \sqrt {b x^{2} + a} b} - \frac {5 \, B a x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {15 \, B a^{2} x}{8 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, A a x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {15 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {3 \, A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} \] Input:

integrate(x^4*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

1/4*B*x^5/(sqrt(b*x^2 + a)*b) - 5/8*B*a*x^3/(sqrt(b*x^2 + a)*b^2) + 1/2*A* 
x^3/(sqrt(b*x^2 + a)*b) - 15/8*B*a^2*x/(sqrt(b*x^2 + a)*b^3) + 3/2*A*a*x/( 
sqrt(b*x^2 + a)*b^2) + 15/8*B*a^2*arcsinh(b*x/sqrt(a*b))/b^(7/2) - 3/2*A*a 
*arcsinh(b*x/sqrt(a*b))/b^(5/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, B x^{2}}{b} - \frac {5 \, B a b^{3} - 4 \, A b^{4}}{b^{5}}\right )} x^{2} - \frac {3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )}}{b^{5}}\right )} x}{8 \, \sqrt {b x^{2} + a}} - \frac {3 \, {\left (5 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \] Input:

integrate(x^4*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

1/8*((2*B*x^2/b - (5*B*a*b^3 - 4*A*b^4)/b^5)*x^2 - 3*(5*B*a^2*b^2 - 4*A*a* 
b^3)/b^5)*x/sqrt(b*x^2 + a) - 3/8*(5*B*a^2 - 4*A*a*b)*log(abs(-sqrt(b)*x + 
 sqrt(b*x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int((x^4*(A + B*x^2))/(a + b*x^2)^(3/2),x)
 

Output:

int((x^4*(A + B*x^2))/(a + b*x^2)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a b x +2 \sqrt {b \,x^{2}+a}\, b^{2} x^{3}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2}}{8 b^{3}} \] Input:

int(x^4*(B*x^2+A)/(b*x^2+a)^(3/2),x)
 

Output:

( - 3*sqrt(a + b*x**2)*a*b*x + 2*sqrt(a + b*x**2)*b**2*x**3 + 3*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2)/(8*b**3)