\(\int \frac {x^3}{\sqrt {\frac {e (a+b x^2)}{c+d x^2}}} \, dx\) [76]

Optimal result

Integrand size = 26, antiderivative size = 205 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {(b c+3 a d) \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{8 b^2 d e}+\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{4 b d e}-\frac {(b c-a d) (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {b} \sqrt {e}}\right )}{8 b^{5/2} d^{3/2} \sqrt {e}} \] Output:

-1/8*(3*a*d+b*c)*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/b^2/d/e+ 
1/4*(d*x^2+c)^2*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/b/d/e-1/8*(-a*d+b*c 
)*(3*a*d+b*c)*arctanh(d^(1/2)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/b^(1/ 
2)/e^(1/2))/b^(5/2)/d^(3/2)/e^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.79 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {b} \sqrt {d} \left (a+b x^2\right ) \sqrt {c+d x^2} \left (-3 a d+b \left (c+2 d x^2\right )\right )-\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{5/2} d^{3/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \] Input:

Integrate[x^3/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]
 

Output:

(Sqrt[b]*Sqrt[d]*(a + b*x^2)*Sqrt[c + d*x^2]*(-3*a*d + b*(c + 2*d*x^2)) - 
(b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[d]*Sqrt[a 
+ b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(8*b^(5/2)*d^(3/2)*Sqrt[(e*(a + b*x^ 
2))/(c + d*x^2)]*Sqrt[c + d*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2053, 2052, 25, 298, 215, 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.

\(\displaystyle \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx\)

\(\Big \downarrow \) 2053

\(\displaystyle \frac {1}{2} \int \frac {x^2}{\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}dx^2\)

\(\Big \downarrow \) 2052

\(\displaystyle e (b c-a d) \int -\frac {a e-c x^4}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\left (e (b c-a d) \int \frac {a e-c x^4}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\right )\)

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 221

\(\displaystyle e (b c-a d) \left (\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b d \left (b e-d x^4\right )^2}-\frac {(3 a d+b c) \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 b^{3/2} \sqrt {d} e^{3/2}}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b d}\right )\)

Input:

Int[x^3/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]
 

Output:

(b*c - a*d)*e*(((b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*b*d*(b*e 
 - d*x^4)^2) - ((b*c + 3*a*d)*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/(2*b*e*(b 
*e - d*x^4)) + ArcTanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b 
]*Sqrt[e])]/(2*b^(3/2)*Sqrt[d]*e^(3/2))))/(4*b*d))
 

Defintions of rubi rules used

rule 25

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

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 298

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

rule 2052

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S 
ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d)   Subst[Int[x^(q* 
(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* 
x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] 
&& IntegerQ[m]
 

rule 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( 
(a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, 
 x] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {\left (-2 b d \,x^{2}+3 a d -b c \right ) \left (b \,x^{2}+a \right )}{8 b^{2} d \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +b d \,x^{2} e}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{16 b^{2} d \sqrt {b d e}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(191\)
default	\(-\frac {\left (b \,x^{2}+a \right ) \left (-4 \sqrt {b d}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b d \,x^{2}-3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{2}+2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c d +\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2}+6 \sqrt {b d}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a d -2 \sqrt {b d}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b c \right )}{16 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} d \sqrt {b d}}\)	\(341\)

Input:

int(x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(-2*b*d*x^2+3*a*d-b*c)*(b*x^2+a)/b^2/d/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+ 
1/16*(3*a^2*d^2-2*a*b*c*d-b^2*c^2)/b^2/d*ln((1/2*a*d*e+1/2*b*c*e+b*d*x^2*e 
)/(b*d*e)^(1/2)+(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^(1/2))/(b*d*e)^(1/2)/( 
e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*(b*x^2+a)*e)^(1/2)/(d*x^2+c)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.01 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\left [-\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {b d e} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e + 4 \, {\left (2 \, b d^{2} x^{4} + b c^{2} + a c d + {\left (3 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}\right ) - 4 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d - 3 \, a b c d^{2} + 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, b^{3} d^{2} e}, \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-b d e} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {-b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (b^{2} d e x^{2} + a b d e\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d - 3 \, a b c d^{2} + 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, b^{3} d^{2} e}\right ] \] Input:

integrate(x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="fricas")
 

Output:

[-1/32*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*sqrt(b*d*e)*log(8*b^2*d^2*e*x^4 
+ 8*(b^2*c*d + a*b*d^2)*e*x^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e + 4*(2*b 
*d^2*x^4 + b*c^2 + a*c*d + (3*b*c*d + a*d^2)*x^2)*sqrt(b*d*e)*sqrt((b*e*x^ 
2 + a*e)/(d*x^2 + c))) - 4*(2*b^2*d^3*x^4 + b^2*c^2*d - 3*a*b*c*d^2 + 3*(b 
^2*c*d^2 - a*b*d^3)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^3*d^2*e), 1 
/16*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*sqrt(-b*d*e)*arctan(1/2*(2*b*d*x^2 
+ b*c + a*d)*sqrt(-b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(b^2*d*e*x^2 + 
 a*b*d*e)) + 2*(2*b^2*d^3*x^4 + b^2*c^2*d - 3*a*b*c*d^2 + 3*(b^2*c*d^2 - a 
*b*d^3)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^3*d^2*e)]
 

Sympy [F(-1)]

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

integrate(x**3/(e*(b*x**2+a)/(d*x**2+c))**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {2 \, \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e} {\left (\frac {2 \, x^{2}}{b e} + \frac {b c e - 3 \, a d e}{b^{2} d e^{2}}\right )} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left ({\left | -b c e - a d e - 2 \, \sqrt {b d e} {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} \right |}\right )}{\sqrt {b d e} b^{2} d}}{16 \, \mathrm {sgn}\left (d x^{2} + c\right )} \] Input:

integrate(x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="giac")
 

Output:

1/16*(2*sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e)*(2*x^2/(b*e) + (b* 
c*e - 3*a*d*e)/(b^2*d*e^2)) + (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*log(abs(-b 
*c*e - a*d*e - 2*sqrt(b*d*e)*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 
 + a*d*e*x^2 + a*c*e))))/(sqrt(b*d*e)*b^2*d))/sgn(d*x^2 + c)
 

Mupad [F(-1)]

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

int(x^3/((e*(a + b*x^2))/(c + d*x^2))^(1/2),x)
 

Output:

int(x^3/((e*(a + b*x^2))/(c + d*x^2))^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {e}\, \left (-3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2}+\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d +2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{2}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} d^{2}-2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a b c d -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{2} c^{2}\right )}{8 b^{3} d^{2} e} \] Input:

int(x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x)
 

Output:

(sqrt(e)*( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2 + sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*b**2*c*d + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d* 
*2*x**2 + 3*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d - sqrt(d)*sq 
rt(c + d*x**2)*b)*a**2*d**2 - 2*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b* 
x**2)*d - sqrt(d)*sqrt(c + d*x**2)*b)*a*b*c*d - sqrt(d)*sqrt(b)*log( - sqr 
t(b)*sqrt(a + b*x**2)*d - sqrt(d)*sqrt(c + d*x**2)*b)*b**2*c**2))/(8*b**3* 
d**2*e)