\(\int x^5 (\frac {e (a+b x^2)}{c+d x^2})^{3/2} \, dx\) [55]

Optimal result

Integrand size = 26, antiderivative size = 338 \[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\frac {c^2 (b c-a d) e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{d^4}+\frac {\left (29 b^2 c^2-22 a b c d+a^2 d^2\right ) e \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{16 b d^4}-\frac {(19 b c-7 a d) e \left (c+d x^2\right )^2 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{24 d^4}+\frac {b e \left (c+d x^2\right )^3 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{6 d^4}-\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{3/2} d^{9/2}} \] Output:

c^2*(-a*d+b*c)*e*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/d^4+1/16*(a^2*d^2- 
22*a*b*c*d+29*b^2*c^2)*e*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/ 
b/d^4-1/24*(-7*a*d+19*b*c)*e*(d*x^2+c)^2*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^ 
(1/2)/d^4+1/6*b*e*(d*x^2+c)^3*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/d^4-1 
/16*(-a*d+b*c)*(-a^2*d^2-10*a*b*c*d+35*b^2*c^2)*e^(3/2)*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^(3/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 4.70 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.66 \[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (b \sqrt {d} \left (3 a^2 d^2 \left (c+d x^2\right )+2 a b d \left (-50 c^2-19 c d x^2+7 d^2 x^4\right )+b^2 \left (105 c^3+35 c^2 d x^2-14 c d^2 x^4+8 d^3 x^6\right )\right )-\frac {3 (b c-a d)^{3/2} \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {a+b x^2}}\right )}{48 b^2 d^{9/2}} \] Input:

Integrate[x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]
 

Output:

(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(b*Sqrt[d]*(3*a^2*d^2*(c + d*x^2) + 2 
*a*b*d*(-50*c^2 - 19*c*d*x^2 + 7*d^2*x^4) + b^2*(105*c^3 + 35*c^2*d*x^2 - 
14*c*d^2*x^4 + 8*d^3*x^6)) - (3*(b*c - a*d)^(3/2)*(35*b^2*c^2 - 10*a*b*c*d 
 - a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b* 
x^2])/Sqrt[b*c - a*d]])/Sqrt[a + b*x^2]))/(48*b^2*d^(9/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.94 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.89, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2053, 2052, 366, 25, 25, 27, 360, 25, 1471, 27, 299, 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.

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

\(\displaystyle e (b c-a d) \int \frac {x^8 \left (a e-c x^4\right )^2}{\left (b e-d x^4\right )^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\)

\(\Big \downarrow \) 366

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\int -\frac {x^8 \left (-6 b c^2 d e x^4+6 a^2 d^2 e^2-5 (b c e-a d e)^2\right )}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 b d^2 e}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle e (b c-a d) \left (\frac {\int -\frac {e x^8 \left (6 b c^2 d x^4+\left (5 b^2 c^2-10 a b d c-a^2 d^2\right ) e\right )}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 b d^2 e}+\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\int \frac {e x^8 \left (6 b c^2 d x^4+\left (5 b^2 c^2-10 a b d c-a^2 d^2\right ) e\right )}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 b d^2 e}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\int \frac {x^8 \left (6 b c^2 d x^4+\left (5 b^2 c^2-10 a b d c-a^2 d^2\right ) e\right )}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{6 b d^2}\right )\)

\(\Big \downarrow \) 360

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\frac {\int -\frac {24 b c^2 d^3 x^8+4 d^2 (b c-a d) (11 b c+a d) e x^4+b d (b c-a d) (11 b c+a d) e^2}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 d^3}+\frac {b e^2 (b c-a d) (a d+11 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}}{6 b d^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\frac {b e^2 (b c-a d) (a d+11 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\int \frac {24 b c^2 d^3 x^8+4 d^2 (b c-a d) (11 b c+a d) e x^4+b d (b c-a d) (11 b c+a d) e^2}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 d^3}}{6 b d^2}\right )\)

\(\Big \downarrow \) 1471

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\frac {b e^2 (b c-a d) (a d+11 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\frac {d e \left (-5 a^2 d^2-50 a b c d+79 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (b e-d x^4\right )}-\frac {\int \frac {3 b d e \left (16 b c^2 d x^4+\left (19 b^2 c^2-10 a b d c-a^2 d^2\right ) e\right )}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 b e}}{4 d^3}}{6 b d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\frac {b e^2 (b c-a d) (a d+11 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\frac {d e \left (-5 a^2 d^2-50 a b c d+79 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (b e-d x^4\right )}-\frac {3}{2} d \int \frac {16 b c^2 d x^4+\left (19 b^2 c^2-10 a b d c-a^2 d^2\right ) e}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 d^3}}{6 b d^2}\right )\)

\(\Big \downarrow \) 299

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\frac {b e^2 (b c-a d) (a d+11 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\frac {d e \left (-5 a^2 d^2-50 a b c d+79 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (b e-d x^4\right )}-\frac {3}{2} d \left (e \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \int \frac {1}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}-16 b c^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 d^3}}{6 b d^2}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle e (b c-a d) \left (\frac {e x^{10} (b c-a d)^2}{6 b d^2 \left (b e-d x^4\right )^3}-\frac {\frac {b e^2 (b c-a d) (a d+11 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\frac {d e \left (-5 a^2 d^2-50 a b c d+79 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 \left (b e-d x^4\right )}-\frac {3}{2} d \left (\frac {\sqrt {e} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{\sqrt {b} \sqrt {d}}-16 b c^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{4 d^3}}{6 b d^2}\right )\)

Input:

Int[x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]
 

Output:

(b*c - a*d)*e*(((b*c - a*d)^2*e*x^10)/(6*b*d^2*(b*e - d*x^4)^3) - ((b*(b*c 
 - a*d)*(11*b*c + a*d)*e^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*d^2*(b*e 
- d*x^4)^2) - ((d*(79*b^2*c^2 - 50*a*b*c*d - 5*a^2*d^2)*e*Sqrt[(e*(a + b*x 
^2))/(c + d*x^2)])/(2*(b*e - d*x^4)) - (3*d*(-16*b*c^2*Sqrt[(e*(a + b*x^2) 
)/(c + d*x^2)] + ((35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*Sqrt[e]*ArcTanh[(Sqr 
t[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b]*Sqrt[e])])/(Sqrt[b]*Sqrt[ 
d])))/2)/(4*d^3))/(6*b*d^2))
 

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 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 299

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

rule 360

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[(a + b*x^2)^(p + 1)*Expan 
dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 
- 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; 
FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & 
& (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 366

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

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

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 = 0.22 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {\left (8 b^{2} d^{2} x^{4}+14 a b \,d^{2} x^{2}-22 b^{2} c \,x^{2} d +3 a^{2} d^{2}-52 a b c d +57 b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 b \,d^{4}}-\frac {\left (\frac {\left (a^{2} d^{2}+10 a b c d -35 b^{2} c^{2}\right ) \left (a d -b c \right ) \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +b d \,x^{2} e}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right )}{2 \sqrt {b d e}}+\frac {16 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \left (b \,x^{2}+a \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{16 b \,d^{4} \left (b \,x^{2}+a \right )}\)	\(313\)
default	\(\text {Expression too large to display}\)	\(1027\)

Input:

int(x^5*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/48/b*(8*b^2*d^2*x^4+14*a*b*d^2*x^2-22*b^2*c*d*x^2+3*a^2*d^2-52*a*b*c*d+5 
7*b^2*c^2)*(d*x^2+c)/d^4*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16/b/d^4*(1/2*( 
a^2*d^2+10*a*b*c*d-35*b^2*c^2)*(a*d-b*c)*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)+1 
6*c^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*b*(b*x^2+a)/(a*d-b*c)/(b*d*e*x^4+a*d*e*x 
^2+b*c*e*x^2+a*c*e)^(1/2))*e/(b*x^2+a)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x 
^2+c)*(b*x^2+a)*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.64 \[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\left [\frac {3 \, {\left (35 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e \sqrt {\frac {e}{b d}} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e - 4 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d + a b c d^{2} + {\left (3 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{b d}}\right ) + 4 \, {\left (8 \, b^{2} d^{3} e x^{6} - 14 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} e x^{4} + {\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} e x^{2} + {\left (105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, b d^{4}}, \frac {3 \, {\left (35 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e \sqrt {-\frac {e}{b d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{b d}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left (8 \, b^{2} d^{3} e x^{6} - 14 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} e x^{4} + {\left (35 \, b^{2} c^{2} d - 38 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} e x^{2} + {\left (105 \, b^{2} c^{3} - 100 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, b d^{4}}\right ] \] Input:

integrate(x^5*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")
 

Output:

[1/192*(3*(35*b^3*c^3 - 45*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e*sqrt(e 
/(b*d))*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^2*d^3*x^4 + b^2*c^2*d + 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))*sqrt(e/(b*d))) + 4*( 
8*b^2*d^3*e*x^6 - 14*(b^2*c*d^2 - a*b*d^3)*e*x^4 + (35*b^2*c^2*d - 38*a*b* 
c*d^2 + 3*a^2*d^3)*e*x^2 + (105*b^2*c^3 - 100*a*b*c^2*d + 3*a^2*c*d^2)*e)* 
sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b*d^4), 1/96*(3*(35*b^3*c^3 - 45*a*b^2 
*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e*sqrt(-e/(b*d))*arctan(1/2*(2*b*d*x^2 + 
 b*c + a*d)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(-e/(b*d))/(b*e*x^2 + a* 
e)) + 2*(8*b^2*d^3*e*x^6 - 14*(b^2*c*d^2 - a*b*d^3)*e*x^4 + (35*b^2*c^2*d 
- 38*a*b*c*d^2 + 3*a^2*d^3)*e*x^2 + (105*b^2*c^3 - 100*a*b*c^2*d + 3*a^2*c 
*d^2)*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b*d^4)]
 

Sympy [F(-1)]

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

integrate(x**5*(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^5*(e*(b*x^2+a)/(d*x^2+c))^(3/2),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 [F(-2)]

Exception generated. \[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{2,[0,5,0]%%%},[2,0,0,0]%%%}+%%%{%%{[%%%{-4,[0,4,0]%%%} 
,0]:[1,0,
 

Mupad [F(-1)]

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

int(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x)
 

Output:

int(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.80 \[ \int x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {e}\, e \left (3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b c \,d^{3}+3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{4} x^{2}-100 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} d^{2}-38 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,d^{3} x^{2}+14 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} d^{4} x^{4}+105 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} c^{3} d +35 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} c^{2} d^{2} x^{2}-14 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} c \,d^{3} x^{4}+8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} d^{4} x^{6}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} c \,d^{3}+3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} d^{4} x^{2}+27 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} b \,c^{2} d^{2}+27 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} b c \,d^{3} x^{2}-135 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a \,b^{2} c^{3} d -135 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a \,b^{2} c^{2} d^{2} x^{2}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{3} c^{4}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d +\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{3} c^{3} d \,x^{2}\right )}{48 b^{2} d^{5} \left (d \,x^{2}+c \right )} \] Input:

int(x^5*(e*(b*x^2+a)/(d*x^2+c))^(3/2),x)
 

Output:

(sqrt(e)*e*(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c*d**3 + 3*sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*a**2*b*d**4*x**2 - 100*sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*a*b**2*c**2*d**2 - 38*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c* 
d**3*x**2 + 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**4*x**4 + 105*sq 
rt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*d + 35*sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*b**3*c**2*d**2*x**2 - 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c 
*d**3*x**4 + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d**4*x**6 + 3*sqrt(d 
)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)* 
a**3*c*d**3 + 3*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d 
)*sqrt(c + d*x**2)*b)*a**3*d**4*x**2 + 27*sqrt(d)*sqrt(b)*log( - sqrt(b)*s 
qrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a**2*b*c**2*d**2 + 27*sqrt 
(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b 
)*a**2*b*c*d**3*x**2 - 135*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2) 
*d + sqrt(d)*sqrt(c + d*x**2)*b)*a*b**2*c**3*d - 135*sqrt(d)*sqrt(b)*log( 
- sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*a*b**2*c**2*d** 
2*x**2 + 105*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + b*x**2)*d + sqrt(d)*s 
qrt(c + d*x**2)*b)*b**3*c**4 + 105*sqrt(d)*sqrt(b)*log( - sqrt(b)*sqrt(a + 
 b*x**2)*d + sqrt(d)*sqrt(c + d*x**2)*b)*b**3*c**3*d*x**2))/(48*b**2*d**5* 
(c + d*x**2))