\(\int \frac {x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\) [48]

Optimal result

Integrand size = 40, antiderivative size = 559 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^6}+\frac {\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}+\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16384 c^{11/2} d^{11/2} e^{13/2}} \] Output:

-3/16384*(-a*e^2+c*d^2)^3*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33 
*c^3*d^6)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ 
c^5/d^5/e^6+1/2048*(-a*e^2+c*d^2)*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^ 
4*e^2+33*c^3*d^6)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2 
)^(3/2)/c^4/d^4/e^5+1/112*(5*a/c/d-11*d/e^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(5/2)+1/8*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e-1/4480*(2 
31*c^3*d^6-15*a*c^2*d^4*e^2-95*a^2*c*d^2*e^4-105*a^3*e^6-10*c*d*e*(-15*a^2 
*e^4-10*a*c*d^2*e^2+33*c^2*d^4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2) 
/c^3/d^3/e^4+3/16384*(-a*e^2+c*d^2)^5*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^ 
2*d^4*e^2+33*c^3*d^6)*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^(11/2)/d^(11/2)/e^(13/2)
 

Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (1575 a^7 e^{14}-525 a^6 c d e^{12} (7 d+2 e x)+35 a^5 c^2 d^2 e^{10} \left (29 d^2+68 d e x+24 e^2 x^2\right )+5 a^4 c^3 d^3 e^8 \left (185 d^3-110 d^2 e x-376 d e^2 x^2-144 e^3 x^3\right )+5 a^3 c^4 d^4 e^6 \left (265 d^4-120 d^3 e x+80 d^2 e^2 x^2+320 d e^3 x^3+128 e^4 x^4\right )+a^2 c^5 d^5 e^4 \left (-11193 d^5+7034 d^4 e x-5488 d^3 e^2 x^2+4640 d^2 e^3 x^3+137600 d e^4 x^4+103680 e^5 x^5\right )+a c^6 d^6 e^2 \left (11445 d^6-7476 d^5 e x+5928 d^4 e^2 x^2-5056 d^3 e^3 x^3+4480 d^2 e^4 x^4+212480 d e^5 x^5+168960 e^6 x^6\right )+c^7 d^7 \left (-3465 d^7+2310 d^6 e x-1848 d^5 e^2 x^2+1584 d^4 e^3 x^3-1408 d^3 e^4 x^4+1280 d^2 e^5 x^5+87040 d e^6 x^6+71680 e^7 x^7\right )\right )+\frac {105 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{573440 c^{11/2} d^{11/2} e^{13/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(1575*a^7*e^14 - 5 
25*a^6*c*d*e^12*(7*d + 2*e*x) + 35*a^5*c^2*d^2*e^10*(29*d^2 + 68*d*e*x + 2 
4*e^2*x^2) + 5*a^4*c^3*d^3*e^8*(185*d^3 - 110*d^2*e*x - 376*d*e^2*x^2 - 14 
4*e^3*x^3) + 5*a^3*c^4*d^4*e^6*(265*d^4 - 120*d^3*e*x + 80*d^2*e^2*x^2 + 3 
20*d*e^3*x^3 + 128*e^4*x^4) + a^2*c^5*d^5*e^4*(-11193*d^5 + 7034*d^4*e*x - 
 5488*d^3*e^2*x^2 + 4640*d^2*e^3*x^3 + 137600*d*e^4*x^4 + 103680*e^5*x^5) 
+ a*c^6*d^6*e^2*(11445*d^6 - 7476*d^5*e*x + 5928*d^4*e^2*x^2 - 5056*d^3*e^ 
3*x^3 + 4480*d^2*e^4*x^4 + 212480*d*e^5*x^5 + 168960*e^6*x^6) + c^7*d^7*(- 
3465*d^7 + 2310*d^6*e*x - 1848*d^5*e^2*x^2 + 1584*d^4*e^3*x^3 - 1408*d^3*e 
^4*x^4 + 1280*d^2*e^5*x^5 + 87040*d*e^6*x^6 + 71680*e^7*x^7)) + (105*(c*d^ 
2 - a*e^2)^5*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^ 
6)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/( 
Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(573440*c^(11/2)*d^(11/2)*e^(13/2))
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 543, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1215, 1236, 27, 1236, 27, 1225, 1087, 1087, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {1}{2} c d x^2 \left (6 a d e+\left (11 c d^2-5 a e^2\right ) x\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{8 c d e}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\int x^2 \left (6 a d e+\left (11 c d^2-5 a e^2\right ) x\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{16 e}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {\int -\frac {1}{2} x \left (4 a d e \left (11 c d^2-5 a e^2\right )+3 \left (33 c^2 d^4-10 a c e^2 d^2-15 a^2 e^4\right ) x\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{7 c d e}+\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{16 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}-\frac {\int x \left (4 a d e \left (11 c d^2-5 a e^2\right )+3 \left (33 c^2 d^4-10 a c e^2 d^2-15 a^2 e^4\right ) x\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{14 c d e}}{16 e}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}-\frac {\frac {7 \left (c d^2-a e^2\right ) \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{8 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{20 c^2 d^2 e^2}}{14 c d e}}{16 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}-\frac {\frac {7 \left (c d^2-a e^2\right ) \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{8 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{20 c^2 d^2 e^2}}{14 c d e}}{16 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}-\frac {\frac {7 \left (c d^2-a e^2\right ) \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{8 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{20 c^2 d^2 e^2}}{14 c d e}}{16 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}-\frac {\frac {7 \left (c d^2-a e^2\right ) \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{8 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{20 c^2 d^2 e^2}}{14 c d e}}{16 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}-\frac {\frac {1}{7} x^2 \left (\frac {11 d}{e}-\frac {5 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}-\frac {\frac {7 \left (c d^2-a e^2\right ) \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{8 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{20 c^2 d^2 e^2}}{14 c d e}}{16 e}\)

Input:

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

Output:

(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(8*e) - ((((11*d)/e - 
(5*a*e)/(c*d))*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/7 - (-1/ 
20*((231*c^3*d^6 - 15*a*c^2*d^4*e^2 - 95*a^2*c*d^2*e^4 - 105*a^3*e^6 - 10* 
c*d*e*(33*c^2*d^4 - 10*a*c*d^2*e^2 - 15*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^ 
2)*x + c*d*e*x^2)^(5/2))/(c^2*d^2*e^2) + (7*(c*d^2 - a*e^2)*(33*c^3*d^6 + 
45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(((c*d^2 + a*e^2 + 2*c*d 
*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*c*d*e) - (3*(c*d^2 
 - a*e^2)^2*(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2])/(4*c*d*e) - ((c*d^2 - a*e^2)^2*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])])/(8*c^(3/2)*d^(3/2)*e^(3/2))))/(16*c*d*e)))/(8*c^2*d^2*e^2))/(14*c*d* 
e))/(16*e)
 

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 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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1215

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

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1894\) vs. \(2(523)=1046\).

Time = 2.57 (sec) , antiderivative size = 1895, normalized size of antiderivative = 3.39

method	result	size
default	\(\text {Expression too large to display}\)	\(1895\)

Input:

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

Output:

d^2/e^3*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5 
/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/8*(2*c*d*e*x+a*e^2 
+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2- 
(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x 
+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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))))+1/e*(1/8*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/ 
d/e/c-9/16*(a*e^2+c*d^2)/d/e/c*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2 
)/d/e/c-1/2*(a*e^2+c*d^2)/d/e/c*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e 
/c*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/ 
d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2 
)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+ 
c*d^2)^2)/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)))))-1/8*a/c*(1/12*(2*c*d*e*x+a 
*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/c/d/e+5/24*(4*a*c*d^2* 
e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^ 
2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/ 
4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/ 
8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e...
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1524, normalized size of antiderivative = 2.73 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2293760*(105*(33*c^8*d^16 - 120*a*c^7*d^14*e^2 + 140*a^2*c^6*d^12*e^4 
- 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^8*e^8 - 8*a^5*c^3*d^6*e^10 - 20*a^6*c 
^2*d^4*e^12 + 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*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*(71680*c^8*d^8*e^8*x^7 - 3465*c^8*d^15*e + 11445*a*c^7*d 
^13*e^3 - 11193*a^2*c^6*d^11*e^5 + 1325*a^3*c^5*d^9*e^7 + 925*a^4*c^4*d^7* 
e^9 + 1015*a^5*c^3*d^5*e^11 - 3675*a^6*c^2*d^3*e^13 + 1575*a^7*c*d*e^15 + 
5120*(17*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 1280*(c^8*d^10*e^6 + 166*a* 
c^7*d^8*e^8 + 81*a^2*c^6*d^6*e^10)*x^5 - 128*(11*c^8*d^11*e^5 - 35*a*c^7*d 
^9*e^7 - 1075*a^2*c^6*d^7*e^9 - 5*a^3*c^5*d^5*e^11)*x^4 + 16*(99*c^8*d^12* 
e^4 - 316*a*c^7*d^10*e^6 + 290*a^2*c^6*d^8*e^8 + 100*a^3*c^5*d^6*e^10 - 45 
*a^4*c^4*d^4*e^12)*x^3 - 8*(231*c^8*d^13*e^3 - 741*a*c^7*d^11*e^5 + 686*a^ 
2*c^6*d^9*e^7 - 50*a^3*c^5*d^7*e^9 + 235*a^4*c^4*d^5*e^11 - 105*a^5*c^3*d^ 
3*e^13)*x^2 + 2*(1155*c^8*d^14*e^2 - 3738*a*c^7*d^12*e^4 + 3517*a^2*c^6*d^ 
10*e^6 - 300*a^3*c^5*d^8*e^8 - 275*a^4*c^4*d^6*e^10 + 1190*a^5*c^3*d^4*e^1 
2 - 525*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/ 
(c^6*d^6*e^7), -1/1146880*(105*(33*c^8*d^16 - 120*a*c^7*d^14*e^2 + 140*a^2 
*c^6*d^12*e^4 - 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^8*e^8 - 8*a^5*c^3*d^6*e 
^10 - 20*a^6*c^2*d^4*e^12 + 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*sqrt(-c*d*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d),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 = 784, normalized size of antiderivative = 1.40 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

1/573440*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*(4*(1 
4*c^2*d^2*e*x + (17*c^9*d^10*e^7 + 33*a*c^8*d^8*e^9)/(c^7*d^7*e^7))*x + (c 
^9*d^11*e^6 + 166*a*c^8*d^9*e^8 + 81*a^2*c^7*d^7*e^10)/(c^7*d^7*e^7))*x - 
(11*c^9*d^12*e^5 - 35*a*c^8*d^10*e^7 - 1075*a^2*c^7*d^8*e^9 - 5*a^3*c^6*d^ 
6*e^11)/(c^7*d^7*e^7))*x + (99*c^9*d^13*e^4 - 316*a*c^8*d^11*e^6 + 290*a^2 
*c^7*d^9*e^8 + 100*a^3*c^6*d^7*e^10 - 45*a^4*c^5*d^5*e^12)/(c^7*d^7*e^7))* 
x - (231*c^9*d^14*e^3 - 741*a*c^8*d^12*e^5 + 686*a^2*c^7*d^10*e^7 - 50*a^3 
*c^6*d^8*e^9 + 235*a^4*c^5*d^6*e^11 - 105*a^5*c^4*d^4*e^13)/(c^7*d^7*e^7)) 
*x + (1155*c^9*d^15*e^2 - 3738*a*c^8*d^13*e^4 + 3517*a^2*c^7*d^11*e^6 - 30 
0*a^3*c^6*d^9*e^8 - 275*a^4*c^5*d^7*e^10 + 1190*a^5*c^4*d^5*e^12 - 525*a^6 
*c^3*d^3*e^14)/(c^7*d^7*e^7))*x - (3465*c^9*d^16*e - 11445*a*c^8*d^14*e^3 
+ 11193*a^2*c^7*d^12*e^5 - 1325*a^3*c^6*d^10*e^7 - 925*a^4*c^5*d^8*e^9 - 1 
015*a^5*c^4*d^6*e^11 + 3675*a^6*c^3*d^4*e^13 - 1575*a^7*c^2*d^2*e^15)/(c^7 
*d^7*e^7)) - 3/32768*(33*c^8*d^16 - 120*a*c^7*d^14*e^2 + 140*a^2*c^6*d^12* 
e^4 - 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^8*e^8 - 8*a^5*c^3*d^6*e^10 - 20*a 
^6*c^2*d^4*e^12 + 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*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))))/(sqrt(c*d*e)*c^5*d^5*e^6)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 1615, normalized size of antiderivative = 2.89 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

(1575*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**7*c*d*e**15 - 3675*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**6*c**2*d**3*e**13 - 1050*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*a**6*c**2*d**2*e**14*x + 1015*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c** 
3*d**5*e**11 + 2380*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**3*d**4*e**12*x 
 + 840*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**3*d**3*e**13*x**2 + 925*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**4*d**7*e**9 - 550*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*a**4*c**4*d**6*e**10*x - 1880*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**4*c**4*d**5*e**11*x**2 - 720*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**4 
*d**4*e**12*x**3 + 1325*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**5*d**9*e** 
7 - 600*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**5*d**8*e**8*x + 400*sqrt(d 
 + e*x)*sqrt(a*e + c*d*x)*a**3*c**5*d**7*e**9*x**2 + 1600*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**3*c**5*d**6*e**10*x**3 + 640*sqrt(d + e*x)*sqrt(a*e + c 
*d*x)*a**3*c**5*d**5*e**11*x**4 - 11193*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
*2*c**6*d**11*e**5 + 7034*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**10* 
e**6*x - 5488*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**9*e**7*x**2 + 4 
640*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**8*e**8*x**3 + 137600*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**7*e**9*x**4 + 103680*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a**2*c**6*d**6*e**10*x**5 + 11445*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a*c**7*d**13*e**3 - 7476*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c** 
7*d**12*e**4*x + 5928*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**7*d**11*e**5...