\(\int \frac {x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^4} \, dx\) [62]

Optimal result

Integrand size = 40, antiderivative size = 403 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {\left (63 c^2 d^4-14 a c d^2 e^2-a^2 e^4\right ) (a e+c d x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c d e^3 \left (c d^2-a e^2\right )}+\frac {(a e+c d x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d e^2}-\frac {5 \left (63 c^2 d^4-14 a c d^2 e^2-a^2 e^4\right ) \left (3 c d^2-5 a e^2-2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c d e^5}+\frac {2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{e^2 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {5 \left (c d^2-a e^2\right )^2 \left (63 c^2 d^4-14 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{3/2} d^{3/2} e^{11/2}} \] Output:

-1/24*(-a^2*e^4-14*a*c*d^2*e^2+63*c^2*d^4)*(c*d*x+a*e)^2*(a*d*e+(a*e^2+c*d 
^2)*x+c*d*e*x^2)^(1/2)/c/d/e^3/(-a*e^2+c*d^2)+1/4*(c*d*x+a*e)^3*(a*d*e+(a* 
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e^2-5/192*(-a^2*e^4-14*a*c*d^2*e^2+63*c^ 
2*d^4)*(-2*c*d*e*x-5*a*e^2+3*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 
)/c/d/e^5+2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/e^2/(-a*e^2+c*d^2) 
/(e*x+d)^4+5/64*(-a*e^2+c*d^2)^2*(-a^2*e^4-14*a*c*d^2*e^2+63*c^2*d^4)*arct 
anh(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^(11/2)
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {\left (c d^2-a e^2\right )^2 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6 (d+e x)+a^2 c d e^4 \left (-839 d^2-337 d e x+118 e^2 x^2\right )+a c^2 d^2 e^2 \left (1785 d^3+637 d^2 e x-244 d e^2 x^2+136 e^3 x^3\right )-3 c^3 d^3 \left (315 d^4+105 d^3 e x-42 d^2 e^2 x^2+24 d e^3 x^3-16 e^4 x^4\right )\right )}{\left (c d^2-a e^2\right )^2 (a e+c d x)^2 (d+e x)^3}+\frac {15 \left (63 c^2 d^4-14 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{192 c^{3/2} d^{3/2} e^{11/2}} \] Input:

Integrate[(x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^4, 
x]
 

Output:

((c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[ 
e]*(15*a^3*e^6*(d + e*x) + a^2*c*d*e^4*(-839*d^2 - 337*d*e*x + 118*e^2*x^2 
) + a*c^2*d^2*e^2*(1785*d^3 + 637*d^2*e*x - 244*d*e^2*x^2 + 136*e^3*x^3) - 
 3*c^3*d^3*(315*d^4 + 105*d^3*e*x - 42*d^2*e^2*x^2 + 24*d*e^3*x^3 - 16*e^4 
*x^4)))/((c*d^2 - a*e^2)^2*(a*e + c*d*x)^2*(d + e*x)^3) + (15*(63*c^2*d^4 
- 14*a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*S 
qrt[d]*Sqrt[d + e*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(192*c^(3/ 
2)*d^(3/2)*e^(11/2))
 

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1213, 25, 2192, 27, 2192, 27, 2192, 27, 1160, 1092, 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.

\(\displaystyle \int \frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 1213

\(\displaystyle -\frac {\int -\frac {c^3 d^3 x^4 e^7-c^2 d^2 \left (c d^2-3 a e^2\right ) x^3 e^6+c d \left (c^2 d^4-3 a c e^2 d^2+3 a^2 e^4\right ) x^2 e^5-\left (c d^2-a e^2\right )^3 x e^4+d \left (c d^2-a e^2\right )^3 e^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {c^3 d^3 x^4 e^7-c^2 d^2 \left (c d^2-3 a e^2\right ) x^3 e^6+c d \left (c^2 d^4-3 a c e^2 d^2+3 a^2 e^4\right ) x^2 e^5-\left (c d^2-a e^2\right )^3 x e^4+d \left (c d^2-a e^2\right )^3 e^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {-c^3 d^3 \left (15 c d^2-17 a e^2\right ) x^3 e^7+2 c^2 d^2 \left (4 c^2 d^4-15 a c e^2 d^2+12 a^2 e^4\right ) x^2 e^6-8 c d \left (c d^2-a e^2\right )^3 x e^5+8 c d^2 \left (c d^2-a e^2\right )^3 e^4}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-c^3 d^3 \left (15 c d^2-17 a e^2\right ) x^3 e^7+2 c^2 d^2 \left (4 c^2 d^4-15 a c e^2 d^2+12 a^2 e^4\right ) x^2 e^6-8 c d \left (c d^2-a e^2\right )^3 x e^5+8 c d^2 \left (c d^2-a e^2\right )^3 e^4}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {c^3 d^3 \left (123 c^2 d^4-190 a c e^2 d^2+59 a^2 e^4\right ) x^2 e^7-4 c^2 d^2 \left (12 c^3 d^6-51 a c^2 e^2 d^4+53 a^2 c e^4 d^2-12 a^3 e^6\right ) x e^6+48 c^2 d^3 \left (c d^2-a e^2\right )^3 e^5}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {c^3 d^3 \left (123 c^2 d^4-190 a c e^2 d^2+59 a^2 e^4\right ) x^2 e^7-4 c^2 d^2 \left (12 c^3 d^6-51 a c^2 e^2 d^4+53 a^2 c e^4 d^2-12 a^3 e^6\right ) x e^6+48 c^2 d^3 \left (c d^2-a e^2\right )^3 e^5}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {c^3 d^3 e^6 \left (2 d \left (96 c^3 d^6-411 a c^2 e^2 d^4+478 a^2 c e^4 d^2-155 a^3 e^6\right )-e \left (561 c^3 d^6-1017 a c^2 e^2 d^4+455 a^2 c e^4 d^2-15 a^3 e^6\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c^2 d^2 e^6 x \left (59 a^2 e^4-190 a c d^2 e^2+123 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {1}{4} c^2 d^2 e^5 \int \frac {2 d \left (96 c^3 d^6-411 a c^2 e^2 d^4+478 a^2 c e^4 d^2-155 a^3 e^6\right )-e \left (561 c^3 d^6-1017 a c^2 e^2 d^4+455 a^2 c e^4 d^2-15 a^3 e^6\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c^2 d^2 e^6 x \left (59 a^2 e^4-190 a c d^2 e^2+123 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {\frac {1}{4} c^2 d^2 e^5 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-a^2 e^4-14 a c d^2 e^2+63 c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}-\frac {\left (-15 a^3 e^6+455 a^2 c d^2 e^4-1017 a c^2 d^4 e^2+561 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c^2 d^2 e^6 x \left (59 a^2 e^4-190 a c d^2 e^2+123 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\frac {1}{4} c^2 d^2 e^5 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-a^2 e^4-14 a c d^2 e^2+63 c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{c d}-\frac {\left (-15 a^3 e^6+455 a^2 c d^2 e^4-1017 a c^2 d^4 e^2+561 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c^2 d^2 e^6 x \left (59 a^2 e^4-190 a c d^2 e^2+123 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c^2 d^2 e^6 x \left (59 a^2 e^4-190 a c d^2 e^2+123 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {1}{4} c^2 d^2 e^5 \left (\frac {15 \left (c d^2-a e^2\right )^2 \left (-a^2 e^4-14 a c d^2 e^2+63 c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt {e}}-\frac {\left (-15 a^3 e^6+455 a^2 c d^2 e^4-1017 a c^2 d^4 e^2+561 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )}{6 c d e}-\frac {1}{3} c^2 d^2 e^6 x^2 \left (15 c d^2-17 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^2 d^2 e^6 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^8}-\frac {2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

Input:

Int[(x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^4,x]
 

Output:

(-2*d^2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^ 
5*(d + e*x)) + ((c^2*d^2*e^6*x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2])/4 + (-1/3*(c^2*d^2*e^6*(15*c*d^2 - 17*a*e^2)*x^2*Sqrt[a*d*e + (c*d^2 + 
 a*e^2)*x + c*d*e*x^2]) + ((c^2*d^2*e^6*(123*c^2*d^4 - 190*a*c*d^2*e^2 + 5 
9*a^2*e^4)*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 + (c^2*d^2*e^5 
*(-(((561*c^3*d^6 - 1017*a*c^2*d^4*e^2 + 455*a^2*c*d^2*e^4 - 15*a^3*e^6)*S 
qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d)) + (15*(c*d^2 - a*e^2)^2 
*(63*c^2*d^4 - 14*a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e* 
x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) 
])/(2*c^(3/2)*d^(3/2)*Sqrt[e])))/4)/(6*c*d*e))/(8*c*d*e))/e^8
 

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 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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

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

rule 1213

Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 3)*(Sqrt[a + b*x + c*x^2] 
/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m - n + 2)   Int[Expan 
dToSum[((-d)^n*(-2*c*d + b*e)^(-m - 1) - e^n*x^n*((-c)*d + b*e + c*e*x)^(-m 
 - 1))/(d + e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e} 
, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IGtQ[n, 0] && EqQ[m 
+ p, -3/2]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1470\) vs. \(2(373)=746\).

Time = 3.05 (sec) , antiderivative size = 1471, normalized size of antiderivative = 3.65

method	result	size
default	\(\text {Expression too large to display}\)	\(1471\)

Input:

int(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^4,x,method=_RETURN 
VERBOSE)
 

Output:

1/e^4*(2/3/(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)) 
^(7/2)-10/3*d*e*c/(a*e^2-c*d^2)*(1/5*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e 
))^(5/2)+1/2*(a*e^2-c*d^2)*(1/8*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c 
*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/d/e/c*(1/4*(2 
*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^ 
(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d* 
e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))) 
)+d^2/e^6*(-2/(a*e^2-c*d^2)/(x+d/e)^4*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/ 
e))^(7/2)+6*d*e*c/(a*e^2-c*d^2)*(2/(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))^(7/2)-8*d*e*c/(a*e^2-c*d^2)*(2/3/(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))^(7/2)-10/3*d*e*c/(a*e^2-c* 
d^2)*(1/5*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+1/2*(a*e^2-c*d^2)* 
(1/8*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x 
+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/d/e/c*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2) 
/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d 
/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2 
+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))))))-2*d/e^5*(2/(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))^(7/2)-8*d*e*c/(a*e^2-c 
*d^2)*(2/3/(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)) 
^(7/2)-10/3*d*e*c/(a*e^2-c*d^2)*(1/5*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+...
 

Fricas [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.32 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^4,x, algorit 
hm="fricas")
 

Output:

[-1/768*(15*(63*c^4*d^9 - 140*a*c^3*d^7*e^2 + 90*a^2*c^2*d^5*e^4 - 12*a^3* 
c*d^3*e^6 - a^4*d*e^8 + (63*c^4*d^8*e - 140*a*c^3*d^6*e^3 + 90*a^2*c^2*d^4 
*e^5 - 12*a^3*c*d^2*e^7 - a^4*e^9)*x)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + 
c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a* 
e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3 
)*x) - 4*(48*c^4*d^4*e^5*x^4 - 945*c^4*d^8*e + 1785*a*c^3*d^6*e^3 - 839*a^ 
2*c^2*d^4*e^5 + 15*a^3*c*d^2*e^7 - 8*(9*c^4*d^5*e^4 - 17*a*c^3*d^3*e^6)*x^ 
3 + 2*(63*c^4*d^6*e^3 - 122*a*c^3*d^4*e^5 + 59*a^2*c^2*d^2*e^7)*x^2 - (315 
*c^4*d^7*e^2 - 637*a*c^3*d^5*e^4 + 337*a^2*c^2*d^3*e^6 - 15*a^3*c*d*e^8)*x 
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^7*x + c^2*d^3*e^ 
6), -1/384*(15*(63*c^4*d^9 - 140*a*c^3*d^7*e^2 + 90*a^2*c^2*d^5*e^4 - 12*a 
^3*c*d^3*e^6 - a^4*d*e^8 + (63*c^4*d^8*e - 140*a*c^3*d^6*e^3 + 90*a^2*c^2* 
d^4*e^5 - 12*a^3*c*d^2*e^7 - a^4*e^9)*x)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d* 
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e 
)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*(48*c^4 
*d^4*e^5*x^4 - 945*c^4*d^8*e + 1785*a*c^3*d^6*e^3 - 839*a^2*c^2*d^4*e^5 + 
15*a^3*c*d^2*e^7 - 8*(9*c^4*d^5*e^4 - 17*a*c^3*d^3*e^6)*x^3 + 2*(63*c^4*d^ 
6*e^3 - 122*a*c^3*d^4*e^5 + 59*a^2*c^2*d^2*e^7)*x^2 - (315*c^4*d^7*e^2 - 6 
37*a*c^3*d^5*e^4 + 337*a^2*c^2*d^3*e^6 - 15*a^3*c*d*e^8)*x)*sqrt(c*d*e*x^2 
 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^7*x + c^2*d^3*e^6)]
 

Sympy [F(-1)]

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

integrate(x**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^4,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 [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (\frac {6 \, c^{2} d^{2} x}{e^{2}} - \frac {15 \, c^{5} d^{6} e^{16} - 17 \, a c^{4} d^{4} e^{18}}{c^{3} d^{3} e^{19}}\right )} x + \frac {123 \, c^{5} d^{7} e^{15} - 190 \, a c^{4} d^{5} e^{17} + 59 \, a^{2} c^{3} d^{3} e^{19}}{c^{3} d^{3} e^{19}}\right )} x - \frac {561 \, c^{5} d^{8} e^{14} - 1017 \, a c^{4} d^{6} e^{16} + 455 \, a^{2} c^{3} d^{4} e^{18} - 15 \, a^{3} c^{2} d^{2} e^{20}}{c^{3} d^{3} e^{19}}\right )} - \frac {2 \, {\left (c^{3} d^{8} - 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} - a^{3} d^{2} e^{6}\right )}}{{\left ({\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} e + \sqrt {c d e} d\right )} e^{5}} - \frac {5 \, {\left (63 \, c^{4} d^{8} - 140 \, a c^{3} d^{6} e^{2} + 90 \, a^{2} c^{2} d^{4} e^{4} - 12 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} \log \left ({\left | c d^{2} + a e^{2} + 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c d e^{5}} \] Input:

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^4,x, algorit 
hm="giac")
 

Output:

1/192*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*c^2*d^2*x/e^2 - 
 (15*c^5*d^6*e^16 - 17*a*c^4*d^4*e^18)/(c^3*d^3*e^19))*x + (123*c^5*d^7*e^ 
15 - 190*a*c^4*d^5*e^17 + 59*a^2*c^3*d^3*e^19)/(c^3*d^3*e^19))*x - (561*c^ 
5*d^8*e^14 - 1017*a*c^4*d^6*e^16 + 455*a^2*c^3*d^4*e^18 - 15*a^3*c^2*d^2*e 
^20)/(c^3*d^3*e^19)) - 2*(c^3*d^8 - 3*a*c^2*d^6*e^2 + 3*a^2*c*d^4*e^4 - a^ 
3*d^2*e^6)/(((sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)) 
*e + sqrt(c*d*e)*d)*e^5) - 5/128*(63*c^4*d^8 - 140*a*c^3*d^6*e^2 + 90*a^2* 
c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 - a^4*e^8)*log(abs(c*d^2 + a*e^2 + 2*sqrt(c 
*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqr 
t(c*d*e)*c*d*e^5)
 

Mupad [F(-1)]

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

int((x^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^4,x)
 

Output:

int((x^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^4, x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {x^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{\left (e x +d \right )^{4}}d x \] Input:

int(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^4,x)
 

Output:

int(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^4,x)