\(\int \frac {x^5}{(d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [130]

Optimal result

Integrand size = 40, antiderivative size = 530 \[ \int \frac {x^5}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 a^5 e^5}{c^5 d^5 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \left (c^5 d^{10}+7 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c^5 d^5 e^4 \left (c d^2-a e^2\right )^2 (d+e x)^4}-\frac {2 \left (22 c^5 d^{10}-35 a c^4 d^8 e^2-35 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 e^4 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {2 \left (122 c^5 d^{10}-385 a c^4 d^8 e^2+350 a^2 c^3 d^6 e^4+105 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 e^4 \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac {2 \left (176 c^5 d^{10}-805 a c^4 d^8 e^2+1400 a^2 c^3 d^6 e^4-1050 a^3 c^2 d^4 e^6-105 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^2 d^2 e^4 \left (c d^2-a e^2\right )^5 (d+e x)}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{3/2} d^{3/2} e^{9/2}} \] Output:

2*a^5*e^5/c^5/d^5/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 
2)^(1/2)+2/7*(7*a^5*e^10+c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) 
/c^5/d^5/e^4/(-a*e^2+c*d^2)^2/(e*x+d)^4-2/35*(-35*a^5*e^10-35*a*c^4*d^8*e^ 
2+22*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^4/(-a*e^2 
+c*d^2)^3/(e*x+d)^3+2/105*(105*a^5*e^10+350*a^2*c^3*d^6*e^4-385*a*c^4*d^8* 
e^2+122*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^4/(-a* 
e^2+c*d^2)^4/(e*x+d)^2-2/105*(-105*a^5*e^10-1050*a^3*c^2*d^4*e^6+1400*a^2* 
c^3*d^6*e^4-805*a*c^4*d^8*e^2+176*c^5*d^10)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x 
^2)^(1/2)/c^2/d^2/e^4/(-a*e^2+c*d^2)^5/(e*x+d)+2*arctanh(c^(1/2)*d^(1/2)*( 
e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(3/2)/d^(3/2)/e^ 
(9/2)
 

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) \left (15 c d^6 e^3 (a e+c d x)^4+21 c^2 d^7 e^2 (a e+c d x)^3 (d+e x)-105 a c d^5 e^4 (a e+c d x)^3 (d+e x)+35 c^3 d^8 e (a e+c d x)^2 (d+e x)^2-175 a c^2 d^6 e^3 (a e+c d x)^2 (d+e x)^2+350 a^2 c d^4 e^5 (a e+c d x)^2 (d+e x)^2+105 c^4 d^9 (a e+c d x) (d+e x)^3-525 a c^3 d^7 e^2 (a e+c d x) (d+e x)^3+1050 a^2 c^2 d^5 e^4 (a e+c d x) (d+e x)^3-1050 a^3 c d^3 e^6 (a e+c d x) (d+e x)^3-105 a^5 e^9 (d+e x)^4\right )}{\left (c d^2-a e^2\right )^5 (d+e x)^2}+105 (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )\right )}{105 c^{3/2} d^{3/2} e^{9/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

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

Output:

(2*(-((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(15*c*d^6*e^3*(a*e + c*d*x)^4 
 + 21*c^2*d^7*e^2*(a*e + c*d*x)^3*(d + e*x) - 105*a*c*d^5*e^4*(a*e + c*d*x 
)^3*(d + e*x) + 35*c^3*d^8*e*(a*e + c*d*x)^2*(d + e*x)^2 - 175*a*c^2*d^6*e 
^3*(a*e + c*d*x)^2*(d + e*x)^2 + 350*a^2*c*d^4*e^5*(a*e + c*d*x)^2*(d + e* 
x)^2 + 105*c^4*d^9*(a*e + c*d*x)*(d + e*x)^3 - 525*a*c^3*d^7*e^2*(a*e + c* 
d*x)*(d + e*x)^3 + 1050*a^2*c^2*d^5*e^4*(a*e + c*d*x)*(d + e*x)^3 - 1050*a 
^3*c*d^3*e^6*(a*e + c*d*x)*(d + e*x)^3 - 105*a^5*e^9*(d + e*x)^4))/((c*d^2 
 - a*e^2)^5*(d + e*x)^2)) + 105*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTan 
h[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])]))/(105*c^(3 
/2)*d^(3/2)*e^(9/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1268, 109, 27, 167, 27, 167, 27, 162, 66, 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.

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

\(\Big \downarrow \) 1268

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \int \frac {x^5}{(a e+c d x)^{3/2} (d+e x)^{9/2}}dx}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {2 \int \frac {x^3 \left (8 a d e-\left (c d^2-a e^2\right ) x\right )}{2 \sqrt {a e+c d x} (d+e x)^{9/2}}dx}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\int \frac {x^3 \left (8 a d e-\left (c d^2-a e^2\right ) x\right )}{\sqrt {a e+c d x} (d+e x)^{9/2}}dx}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {2 \int \frac {x^2 \left (7 x \left (c d^2-a e^2\right )^2+6 a d e \left (c d^2+7 a e^2\right )\right )}{2 \sqrt {a e+c d x} (d+e x)^{7/2}}dx}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {\int \frac {x^2 \left (7 x \left (c d^2-a e^2\right )^2+6 a d e \left (c d^2+7 a e^2\right )\right )}{\sqrt {a e+c d x} (d+e x)^{7/2}}dx}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {-\frac {2 \int -\frac {x \left (35 x \left (c d^2-a e^2\right )^3+4 a d e \left (7 c^2 d^4-20 a c e^2 d^2-35 a^2 e^4\right )\right )}{2 \sqrt {a e+c d x} (d+e x)^{5/2}}dx}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (-35 a^2 e^4-20 a c d^2 e^2+7 c^2 d^4\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {\frac {\int \frac {x \left (35 x \left (c d^2-a e^2\right )^3+4 a d e \left (7 c^2 d^4-20 a c e^2 d^2-35 a^2 e^4\right )\right )}{\sqrt {a e+c d x} (d+e x)^{5/2}}dx}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (-35 a^2 e^4-20 a c d^2 e^2+7 c^2 d^4\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {\frac {\frac {35 \left (c d^2-a e^2\right )^3 \int \frac {1}{\sqrt {a e+c d x} \sqrt {d+e x}}dx}{e^2}-\frac {2 \sqrt {a e+c d x} \left (-105 a^4 d^2 e^8-790 a^3 c d^4 e^6+896 a^2 c^2 d^6 e^4+2 e x \left (-105 a^4 d e^8-435 a^3 c d^3 e^6+607 a^2 c^2 d^5 e^4-329 a c^3 d^7 e^2+70 c^4 d^9\right )-490 a c^3 d^8 e^2+105 c^4 d^{10}\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (-35 a^2 e^4-20 a c d^2 e^2+7 c^2 d^4\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {\frac {\frac {70 \left (c d^2-a e^2\right )^3 \int \frac {1}{c d-\frac {e (a e+c d x)}{d+e x}}d\frac {\sqrt {a e+c d x}}{\sqrt {d+e x}}}{e^2}-\frac {2 \sqrt {a e+c d x} \left (-105 a^4 d^2 e^8-790 a^3 c d^4 e^6+896 a^2 c^2 d^6 e^4+2 e x \left (-105 a^4 d e^8-435 a^3 c d^3 e^6+607 a^2 c^2 d^5 e^4-329 a c^3 d^7 e^2+70 c^4 d^9\right )-490 a c^3 d^8 e^2+105 c^4 d^{10}\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (-35 a^2 e^4-20 a c d^2 e^2+7 c^2 d^4\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^4}{c d (d+e x)^{7/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^3 \left (7 a e^2+c d^2\right ) \sqrt {a e+c d x}}{7 e (d+e x)^{7/2} \left (c d^2-a e^2\right )}-\frac {\frac {\frac {70 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e^{5/2}}-\frac {2 \sqrt {a e+c d x} \left (-105 a^4 d^2 e^8-790 a^3 c d^4 e^6+896 a^2 c^2 d^6 e^4+2 e x \left (-105 a^4 d e^8-435 a^3 c d^3 e^6+607 a^2 c^2 d^5 e^4-329 a c^3 d^7 e^2+70 c^4 d^9\right )-490 a c^3 d^8 e^2+105 c^4 d^{10}\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}}{5 e \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (-35 a^2 e^4-20 a c d^2 e^2+7 c^2 d^4\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}}{7 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*((2*a*e*x^4)/(c*d*(c*d^2 - a*e^2)*Sqrt[a* 
e + c*d*x]*(d + e*x)^(7/2)) - ((2*d*(c*d^2 + 7*a*e^2)*x^3*Sqrt[a*e + c*d*x 
])/(7*e*(c*d^2 - a*e^2)*(d + e*x)^(7/2)) - ((-2*d*(7*c^2*d^4 - 20*a*c*d^2* 
e^2 - 35*a^2*e^4)*x^2*Sqrt[a*e + c*d*x])/(5*e*(c*d^2 - a*e^2)*(d + e*x)^(5 
/2)) + ((-2*Sqrt[a*e + c*d*x]*(105*c^4*d^10 - 490*a*c^3*d^8*e^2 + 896*a^2* 
c^2*d^6*e^4 - 790*a^3*c*d^4*e^6 - 105*a^4*d^2*e^8 + 2*e*(70*c^4*d^9 - 329* 
a*c^3*d^7*e^2 + 607*a^2*c^2*d^5*e^4 - 435*a^3*c*d^3*e^6 - 105*a^4*d*e^8)*x 
))/(3*e^2*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (70*(c*d^2 - a*e^2)^3*ArcTa 
nh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[c]* 
Sqrt[d]*e^(5/2)))/(5*e*(c*d^2 - a*e^2)))/(7*e*(c*d^2 - a*e^2)))/(c*d*(c*d^ 
2 - a*e^2))))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]
 

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

rule 167

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*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 1268

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d 
 + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f 
 + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1179\) vs. \(2(498)=996\).

Time = 3.24 (sec) , antiderivative size = 1180, normalized size of antiderivative = 2.23

method	result	size
default	\(\text {Expression too large to display}\)	\(1180\)

Input:

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

Output:

1/e^3*(-x/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/ 
d/e/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/ 
c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c* 
d^2)*x+c*d*x^2*e)^(1/2))+1/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^ 
(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))+12*d^2/e^5*( 
2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2 
)*x+c*d*x^2*e)^(1/2)-3*d/e^4*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^( 
1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d 
^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))-10/e^6*d^3*(-2/3/(a*e^2-c* 
d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e*c/(a*e^ 
2-c*d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x 
+d/e))^(1/2))+5/e^7*d^4*(-2/5/(a*e^2-c*d^2)/(x+d/e)^2/(d*e*c*(x+d/e)^2+(a* 
e^2-c*d^2)*(x+d/e))^(1/2)-6/5*d*e*c/(a*e^2-c*d^2)*(-2/3/(a*e^2-c*d^2)/(x+d 
/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e*c/(a*e^2-c*d^2)^ 
3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1 
/2)))-d^5/e^8*(-2/7/(a*e^2-c*d^2)/(x+d/e)^3/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2) 
*(x+d/e))^(1/2)-8/7*d*e*c/(a*e^2-c*d^2)*(-2/5/(a*e^2-c*d^2)/(x+d/e)^2/(d*e 
*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-6/5*d*e*c/(a*e^2-c*d^2)*(-2/3/(a 
*e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e* 
c/(a*e^2-c*d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1586 vs. \(2 (498) = 996\).

Time = 12.95 (sec) , antiderivative size = 3186, normalized size of antiderivative = 6.01 \[ \int \frac {x^5}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^5/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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