\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^7 (d+e x)} \, dx\) [38]

Optimal result

Integrand size = 40, antiderivative size = 481 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 a^{9/2} d^{11/2} e^{9/2}} \] Output:

1/512*(-21*a^4*e^8+6*a^2*c^2*d^4*e^4+8*a*c^3*d^6*e^2+7*c^4*d^8)*(2*a*d*e+( 
a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^4/d^5/e^4/x^2-1/ 
6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^6-1/20*(c/a/e-3*e/d^2)*(a*d* 
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^5+1/160*(-21*a^2*e^4+6*a*c*d^2*e^2+7* 
c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^2/d^3/e^2/x^4-1/960*(-1 
05*a^3*e^6+21*a^2*c*d^2*e^4+33*a*c^2*d^4*e^2+35*c^3*d^6)*(a*d*e+(a*e^2+c*d 
^2)*x+c*d*e*x^2)^(3/2)/a^3/d^4/e^3/x^3-1/512*(-a*e^2+c*d^2)^3*(21*a^3*e^6+ 
21*a^2*c*d^2*e^4+15*a*c^2*d^4*e^2+7*c^3*d^6)*arctanh(a^(1/2)*e^(1/2)*(e*x+ 
d)/d^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/a^(9/2)/d^(11/2)/e^(9/ 
2)
 

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-105 c^5 d^{10} x^5+5 a c^4 d^8 e x^4 (14 d+11 e x)-2 a^2 c^3 d^6 e^2 x^3 \left (28 d^2+16 d e x-27 e^2 x^2\right )+6 a^3 c^2 d^4 e^3 x^2 \left (8 d^3+4 d^2 e x-6 d e^2 x^2+13 e^3 x^3\right )+a^4 c d^2 e^4 x \left (1664 d^4+224 d^3 e x-264 d^2 e^2 x^2+336 d e^3 x^3-525 e^4 x^4\right )+a^5 e^5 \left (1280 d^5+128 d^4 e x-144 d^3 e^2 x^2+168 d^2 e^3 x^3-210 d e^4 x^4+315 e^5 x^5\right )\right )}{x^6}-\frac {15 \left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 a^{9/2} d^{11/2} e^{9/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-105*c^5*d^10* 
x^5 + 5*a*c^4*d^8*e*x^4*(14*d + 11*e*x) - 2*a^2*c^3*d^6*e^2*x^3*(28*d^2 + 
16*d*e*x - 27*e^2*x^2) + 6*a^3*c^2*d^4*e^3*x^2*(8*d^3 + 4*d^2*e*x - 6*d*e^ 
2*x^2 + 13*e^3*x^3) + a^4*c*d^2*e^4*x*(1664*d^4 + 224*d^3*e*x - 264*d^2*e^ 
2*x^2 + 336*d*e^3*x^3 - 525*e^4*x^4) + a^5*e^5*(1280*d^5 + 128*d^4*e*x - 1 
44*d^3*e^2*x^2 + 168*d^2*e^3*x^3 - 210*d*e^4*x^4 + 315*e^5*x^5)))/x^6) - ( 
15*(c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21 
*a^3*e^6)*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d* 
x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(7680*a^(9/2)*d^(11/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1215, 1237, 27, 1237, 27, 1237, 27, 1228, 1152, 1154, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {\int \frac {\left (7 c^2 d^4+6 a c e^2 d^2+4 c e \left (c d^2-3 a e^2\right ) x d-21 a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 x^5}dx}{5 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\left (7 c^2 d^4+6 a c e^2 d^2+4 c e \left (c d^2-3 a e^2\right ) x d-21 a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^5}dx}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\left (35 c^3 d^6+33 a c^2 e^2 d^4+21 a^2 c e^4 d^2+2 c e \left (7 c^2 d^4+6 a c e^2 d^2-21 a^2 e^4\right ) x d-105 a^3 e^6\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 x^4}dx}{4 a d e}-\frac {\left (\frac {7 c^2 d^4}{a}-21 a e^4+6 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d e x^4}}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\left (35 c^3 d^6+33 a c^2 e^2 d^4+21 a^2 c e^4 d^2+2 c e \left (7 c^2 d^4+6 a c e^2 d^2-21 a^2 e^4\right ) x d-105 a^3 e^6\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^4}dx}{8 a d e}-\frac {\left (\frac {7 c^2 d^4}{a}-21 a e^4+6 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d e x^4}}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {-\frac {-\frac {-\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^3}dx}{2 a d e}-\frac {\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 a d e x^3}}{8 a d e}-\frac {\left (\frac {7 c^2 d^4}{a}-21 a e^4+6 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d e x^4}}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {-\frac {-\frac {-\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 a d e}-\frac {\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 a d e x^3}}{8 a d e}-\frac {\left (\frac {7 c^2 d^4}{a}-21 a e^4+6 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d e x^4}}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {-\frac {-\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 a d e}-\frac {\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 a d e x^3}}{8 a d e}-\frac {\left (\frac {7 c^2 d^4}{a}-21 a e^4+6 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d e x^4}}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {-\frac {-\frac {\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 a d e x^3}-\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 a d e}}{8 a d e}-\frac {\left (\frac {7 c^2 d^4}{a}-21 a e^4+6 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d e x^4}}{10 a d e}-\frac {\left (\frac {c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 x^5}}{4 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]
 

Output:

-1/6*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d*x^6) + (-1/5*(((c*d) 
/(a*e) - (3*e)/d)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/x^5 - (-1 
/4*(((7*c^2*d^4)/a + 6*c*d^2*e^2 - 21*a*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2)^(3/2))/(d*e*x^4) - (-1/3*((35*c^3*d^6 + 33*a*c^2*d^4*e^2 + 21*a 
^2*c*d^2*e^4 - 105*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) 
/(a*d*e*x^3) - (5*(7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^ 
4*e^8)*(-1/4*((2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
 + c*d*e*x^2])/(a*d*e*x^2) + ((c*d^2 - a*e^2)^2*ArcTanh[(2*a*d*e + (c*d^2 
+ a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2])])/(8*a^(3/2)*d^(3/2)*e^(3/2))))/(2*a*d*e))/(8*a*d*e))/(10*a*d*e) 
)/(4*d)
 

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, 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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(16882\) vs. \(2(445)=890\).

Time = 4.73 (sec) , antiderivative size = 16883, normalized size of antiderivative = 35.10

method	result	size
default	\(\text {Expression too large to display}\)	\(16883\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/x^7/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 30.08 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm 
="fricas")
 

Output:

[-1/30720*(15*(7*c^6*d^12 - 6*a*c^5*d^10*e^2 - 3*a^2*c^4*d^8*e^4 - 4*a^3*c 
^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 42*a^5*c*d^2*e^10 - 21*a^6*e^12)*sqrt(a* 
d*e)*x^6*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 4* 
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)* 
sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*(1280*a^6*d^6*e^6 - (1 
05*a*c^5*d^11*e - 55*a^2*c^4*d^9*e^3 - 54*a^3*c^3*d^7*e^5 - 78*a^4*c^2*d^5 
*e^7 + 525*a^5*c*d^3*e^9 - 315*a^6*d*e^11)*x^5 + 2*(35*a^2*c^4*d^10*e^2 - 
16*a^3*c^3*d^8*e^4 - 18*a^4*c^2*d^6*e^6 + 168*a^5*c*d^4*e^8 - 105*a^6*d^2* 
e^10)*x^4 - 8*(7*a^3*c^3*d^9*e^3 - 3*a^4*c^2*d^7*e^5 + 33*a^5*c*d^5*e^7 - 
21*a^6*d^3*e^9)*x^3 + 16*(3*a^4*c^2*d^8*e^4 + 14*a^5*c*d^6*e^6 - 9*a^6*d^4 
*e^8)*x^2 + 128*(13*a^5*c*d^7*e^5 + a^6*d^5*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e 
 + (c*d^2 + a*e^2)*x))/(a^5*d^6*e^5*x^6), 1/15360*(15*(7*c^6*d^12 - 6*a*c^ 
5*d^10*e^2 - 3*a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 
42*a^5*c*d^2*e^10 - 21*a^6*e^12)*sqrt(-a*d*e)*x^6*arctan(1/2*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/ 
(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) - 2*(1280*a^6 
*d^6*e^6 - (105*a*c^5*d^11*e - 55*a^2*c^4*d^9*e^3 - 54*a^3*c^3*d^7*e^5 - 7 
8*a^4*c^2*d^5*e^7 + 525*a^5*c*d^3*e^9 - 315*a^6*d*e^11)*x^5 + 2*(35*a^2*c^ 
4*d^10*e^2 - 16*a^3*c^3*d^8*e^4 - 18*a^4*c^2*d^6*e^6 + 168*a^5*c*d^4*e^8 - 
 105*a^6*d^2*e^10)*x^4 - 8*(7*a^3*c^3*d^9*e^3 - 3*a^4*c^2*d^7*e^5 + 33*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**7/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{7}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm 
="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^7), x 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3251 vs. \(2 (445) = 890\).

Time = 0.29 (sec) , antiderivative size = 3251, normalized size of antiderivative = 6.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm 
="giac")
 

Output:

1/512*(7*c^6*d^12 - 6*a*c^5*d^10*e^2 - 3*a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e 
^6 - 15*a^4*c^2*d^4*e^8 + 42*a^5*c*d^2*e^10 - 21*a^6*e^12)*arctan(-(sqrt(c 
*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sqrt 
(-a*d*e)*a^4*d^5*e^4) - 1/7680*(105*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^ 
2*x + a*e^2*x + a*d*e))*a^5*c^6*d^17*e^5 - 90*(sqrt(c*d*e)*x - sqrt(c*d*e* 
x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^6*c^5*d^15*e^7 - 15405*(sqrt(c*d*e)*x 
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^7*c^4*d^13*e^9 - 46140*(s 
qrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^8*c^3*d^11*e 
^11 - 46305*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))* 
a^9*c^2*d^9*e^13 - 14730*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2 
*x + a*d*e))*a^10*c*d^7*e^15 - 315*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 
*x + a*e^2*x + a*d*e))*a^11*d^5*e^17 - 3072*sqrt(c*d*e)*a^8*c^3*d^12*e^10 
- 6144*sqrt(c*d*e)*a^9*c^2*d^10*e^12 - 5120*sqrt(c*d*e)*a^10*c*d^8*e^14 - 
595*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^4*c^ 
6*d^16*e^4 - 30210*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a 
*d*e))^3*a^5*c^5*d^14*e^6 - 199425*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 
*x + a*e^2*x + a*d*e))^3*a^6*c^4*d^12*e^8 - 419500*(sqrt(c*d*e)*x - sqrt(c 
*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c^3*d^10*e^10 - 305925*(sqrt( 
c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*c^2*d^8*e^12 
 - 65010*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^7\,\left (d+e\,x\right )} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x)
                                                                                    
                                                                                    
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{x^{7} \left (e x +d \right )}d x \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x)
 

Output:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x)