\(\int \frac {(d+e x)^3}{x^5 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [111]

Optimal result

Integrand size = 40, antiderivative size = 439 \[ \int \frac {(d+e x)^3}{x^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c^3 d^3 \left (c d^2-a e^2\right ) (d+e x)}{a^5 e^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^2 e^2 x^4}+\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 a^3 e^3 x^3}-\frac {\left (41 c^2 d^4-34 a c d^2 e^2+a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 a^4 d e^4 x^2}+\frac {\left (187 c^3 d^6-187 a c^2 d^4 e^2+13 a^2 c d^2 e^4+3 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^5 d^2 e^5 x}-\frac {3 \left (c d^2-a e^2\right ) \left (105 c^3 d^6-35 a c^2 d^4 e^2-5 a^2 c d^2 e^4-a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 a^{11/2} d^{5/2} e^{11/2}} \] Output:

2*c^3*d^3*(-a*e^2+c*d^2)*(e*x+d)/a^5/e^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) 
^(1/2)-1/4*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/e^2/x^4+1/8*(-3*a 
*e^2+5*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/e^3/x^3-1/32*(a^ 
2*e^4-34*a*c*d^2*e^2+41*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a 
^4/d/e^4/x^2+1/64*(3*a^3*e^6+13*a^2*c*d^2*e^4-187*a*c^2*d^4*e^2+187*c^3*d^ 
6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^5/d^2/e^5/x-3/64*(-a*e^2+c*d^ 
2)*(-a^3*e^6-5*a^2*c*d^2*e^4-35*a*c^2*d^4*e^2+105*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^(11/2) 
/d^(5/2)/e^(11/2)
 

Mathematica [A] (verified)

Time = 10.28 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^3}{x^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (315 c^4 d^7 x^4+105 a c^3 d^5 e x^3 (d-3 e x)+a^2 c^2 d^3 e^2 x^2 \left (-42 d^2-119 d e x+13 e^2 x^2\right )+a^4 e^4 \left (-16 d^3-24 d^2 e x-2 d e^2 x^2+3 e^3 x^3\right )+a^3 c d e^3 x \left (24 d^3+44 d^2 e x+11 d e^2 x^2+3 e^3 x^3\right )\right )-3 \left (105 c^4 d^8-140 a c^3 d^6 e^2+30 a^2 c^2 d^4 e^4+4 a^3 c d^2 e^6+a^4 e^8\right ) x^4 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{64 a^{11/2} d^{5/2} e^{11/2} x^4 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a]*Sqrt[d]*Sqrt[e]*(d + e*x)*(315*c^4*d^7*x^4 + 105*a*c^3*d^5*e*x^3* 
(d - 3*e*x) + a^2*c^2*d^3*e^2*x^2*(-42*d^2 - 119*d*e*x + 13*e^2*x^2) + a^4 
*e^4*(-16*d^3 - 24*d^2*e*x - 2*d*e^2*x^2 + 3*e^3*x^3) + a^3*c*d*e^3*x*(24* 
d^3 + 44*d^2*e*x + 11*d*e^2*x^2 + 3*e^3*x^3)) - 3*(105*c^4*d^8 - 140*a*c^3 
*d^6*e^2 + 30*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*x^4*Sqrt[a*e + 
c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]* 
Sqrt[d + e*x])])/(64*a^(11/2)*d^(5/2)*e^(11/2)*x^4*Sqrt[(a*e + c*d*x)*(d + 
 e*x)])
 

Rubi [A] (verified)

Time = 3.74 (sec) , antiderivative size = 525, 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 = {1212, 25, 2181, 27, 2181, 27, 2181, 27, 1228, 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.

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

\(\Big \downarrow \) 1212

\(\displaystyle \frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-c^3 d^3 e^5 \int -\frac {\frac {\left (c d^2-a e^2\right )^2 x^4}{a^5 c d e^{10}}-\frac {\left (c d^2-a e^2\right )^2 x^3}{a^4 c^2 d^2 e^9}+\frac {\left (c d^2-a e^2\right )^2 x^2}{a^3 c^3 d^3 e^8}-\frac {\left (c d^2-2 a e^2\right ) x}{a^2 c^3 d^2 e^7}+\frac {1}{a c^3 d e^6}}{x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle c^3 d^3 e^5 \int \frac {\frac {\left (c d^2-a e^2\right )^2 x^4}{a^5 c d e^{10}}-\frac {\left (c d^2-a e^2\right )^2 x^3}{a^4 c^2 d^2 e^9}+\frac {\left (c d^2-a e^2\right )^2 x^2}{a^3 c^3 d^3 e^8}-\frac {\left (c d^2-2 a e^2\right ) x}{a^2 c^3 d^2 e^7}+\frac {1}{a c^3 d e^6}}{x^5 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle c^3 d^3 e^5 \left (-\frac {\int \frac {-\frac {8 \left (c d^2-a e^2\right )^2 x^3}{a^4 c e^9}+\frac {8 \left (c d^2-a e^2\right )^2 x^2}{a^3 c^2 d e^8}-\frac {2 \left (4 c^2 d^4-11 a c e^2 d^2+4 a^2 e^4\right ) x}{a^2 c^3 d^2 e^7}+\frac {3 \left (5 c d^2-3 a e^2\right )}{a c^3 d e^6}}{2 x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle c^3 d^3 e^5 \left (-\frac {\int \frac {-\frac {8 \left (c d^2-a e^2\right )^2 x^3}{a^4 c e^9}+\frac {8 \left (c d^2-a e^2\right )^2 x^2}{a^3 c^2 d e^8}-\frac {2 \left (4 c^2 d^4-11 a c e^2 d^2+4 a^2 e^4\right ) x}{a^2 c^3 d^2 e^7}+\frac {3 \left (5 c d^2-3 a e^2\right )}{a c^3 d e^6}}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle c^3 d^3 e^5 \left (-\frac {-\frac {\int \frac {3 \left (\frac {16 d \left (c d^2-a e^2\right )^2 x^2}{a^3 c e^8}-\frac {4 \left (4 c^2 d^4-13 a c e^2 d^2+7 a^2 e^4\right ) x}{a^2 c^2 e^7}+\frac {41 c^2 d^4-34 a c e^2 d^2+a^2 e^4}{a c^3 d e^6}\right )}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle c^3 d^3 e^5 \left (-\frac {-\frac {\int \frac {\frac {16 d \left (c d^2-a e^2\right )^2 x^2}{a^3 c e^8}-\frac {4 \left (4 c^2 d^4-13 a c e^2 d^2+7 a^2 e^4\right ) x}{a^2 c^2 e^7}+\frac {41 c^2 d^4-34 a c e^2 d^2+a^2 e^4}{a c^3 d e^6}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle c^3 d^3 e^5 \left (-\frac {-\frac {-\frac {\int \frac {\left (\frac {187 d^6}{a e^6}-\frac {187 d^4}{c e^4}+\frac {13 a d^2}{c^2 e^2}+\frac {3 a^2}{c^3}\right ) e+2 d \left (-\frac {32 c d^6}{a^2 e^6}+\frac {105 d^4}{a e^4}-\frac {66 d^2}{c e^2}+\frac {a}{c^2}\right ) x}{2 d e x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^3 d^2 e^7 x^2}}{2 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle c^3 d^3 e^5 \left (-\frac {-\frac {-\frac {\int \frac {\frac {187 d^6}{a e^5}-\frac {187 d^4}{c e^3}+\frac {13 a d^2}{c^2 e}+2 \left (-\frac {32 c d^6}{a^2 e^6}+\frac {105 d^4}{a e^4}-\frac {66 d^2}{c e^2}+\frac {a}{c^2}\right ) x d+\frac {3 a^2 e}{c^3}}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d^2 e^2}-\frac {\left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^3 d^2 e^7 x^2}}{2 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle c^3 d^3 e^5 \left (-\frac {-\frac {-\frac {-\frac {3 \left (c d^2-a e^2\right ) \left (-a^3 e^6-5 a^2 c d^2 e^4-35 a c^2 d^4 e^2+105 c^3 d^6\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a^2 c^3 d e^6}-\frac {\left (3 a^3 e^6+13 a^2 c d^2 e^4-187 a c^2 d^4 e^2+187 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d e^6 x}}{4 a d^2 e^2}-\frac {\left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^3 d^2 e^7 x^2}}{2 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle c^3 d^3 e^5 \left (-\frac {-\frac {-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (-a^3 e^6-5 a^2 c d^2 e^4-35 a c^2 d^4 e^2+105 c^3 d^6\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a^2 c^3 d e^6}-\frac {\left (3 a^3 e^6+13 a^2 c d^2 e^4-187 a c^2 d^4 e^2+187 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d e^6 x}}{4 a d^2 e^2}-\frac {\left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^3 d^2 e^7 x^2}}{2 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}}{8 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}\right )+\frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )}{a^5 e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+c^3 d^3 e^5 \left (-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 c^3 d^2 e^7 x^4}-\frac {-\frac {\left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d^2 e^7 x^3}-\frac {-\frac {\left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^3 d^2 e^7 x^2}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (-a^3 e^6-5 a^2 c d^2 e^4-35 a c^2 d^4 e^2+105 c^3 d^6\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} c^3 d^{3/2} e^{13/2}}-\frac {\left (3 a^3 e^6+13 a^2 c d^2 e^4-187 a c^2 d^4 e^2+187 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^3 d e^6 x}}{4 a d^2 e^2}}{2 a d e}}{8 a d e}\right )\)

Input:

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

Output:

(2*c^3*d^3*(c*d^2 - a*e^2)*(d + e*x))/(a^5*e^5*Sqrt[a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2]) + c^3*d^3*e^5*(-1/4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d 
*e*x^2]/(a^2*c^3*d^2*e^7*x^4) - (-(((5*c*d^2 - 3*a*e^2)*Sqrt[a*d*e + (c*d^ 
2 + a*e^2)*x + c*d*e*x^2])/(a^2*c^3*d^2*e^7*x^3)) - (-1/2*((41*c^2*d^4 - 3 
4*a*c*d^2*e^2 + a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a^2 
*c^3*d^2*e^7*x^2) - (-(((187*c^3*d^6 - 187*a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^ 
4 + 3*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a^2*c^3*d*e^6 
*x)) + (3*(c*d^2 - a*e^2)*(105*c^3*d^6 - 35*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^ 
4 - a^3*e^6)*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])])/(2*a^(5/2)*c^3*d^(3/2)* 
e^(13/2)))/(4*a*d^2*e^2))/(2*a*d*e))/(8*a*d*e))
 

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 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 1212

Int[((x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.))/((a_) + (b_.)*(x_) + (c_.)*(x_) 
^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*d - b*e)^n*((d + e 
*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c*x^2])), x] - Simp[e^2/c^(m + 
n - 1)   Int[ExpandToSum[(c^(m + n - 1)*(d + e*x)^(m - 1) - ((c*d - b*e)^n* 
(2*c*d - b*e)^(m - 1))/(e^n*x^n))/(c*d - b*e - c*e*x), x]/(Sqrt[a + b*x + c 
*x^2]/x^n), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^ 
2, 0] && IGtQ[m, 0] && ILtQ[n, 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 2181

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3793\) vs. \(2(405)=810\).

Time = 2.80 (sec) , antiderivative size = 3794, normalized size of antiderivative = 8.64

method	result	size
default	\(\text {Expression too large to display}\)	\(3794\)

Input:

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

Output:

d^3*(-1/4/a/d/e/x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-9/8*(a*e^2+c*d 
^2)/a/d/e*(-1/3/a/d/e/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-7/6*(a*e 
^2+c*d^2)/a/d/e*(-1/2/a/d/e/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-5/ 
4*(a*e^2+c*d^2)/a/d/e*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)- 
3/2*(a*e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-( 
a*e^2+c*d^2)/a/d/e*(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/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e 
+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/ 
x))-4*c/a*(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))-3/2*c/a*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c 
*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/a/d/e*(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/a/d/e/(a*d*e)^ 
(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c 
*d*x^2*e)^(1/2))/x)))-4/3*c/a*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e 
)^(1/2)-3/2*(a*e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e) 
^(1/2)-(a*e^2+c*d^2)/a/d/e*(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/a/d/e/(a*d*e)^(1/2)*ln( 
(2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e) 
^(1/2))/x))-4*c/a*(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)))-5/4*c/a*(-1/2/a/d/e/x^2/(a*d*...
 

Fricas [A] (verification not implemented)

Time = 18.06 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.26 \[ \int \frac {(d+e x)^3}{x^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/256*(3*((105*c^5*d^9 - 140*a*c^4*d^7*e^2 + 30*a^2*c^3*d^5*e^4 + 4*a^3*c 
^2*d^3*e^6 + a^4*c*d*e^8)*x^5 + (105*a*c^4*d^8*e - 140*a^2*c^3*d^6*e^3 + 3 
0*a^3*c^2*d^4*e^5 + 4*a^4*c*d^2*e^7 + a^5*e^9)*x^4)*sqrt(a*d*e)*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*(16*a^5*d^4*e^5 - (315*a*c^4*d^8*e - 315* 
a^2*c^3*d^6*e^3 + 13*a^3*c^2*d^4*e^5 + 3*a^4*c*d^2*e^7)*x^4 - (105*a^2*c^3 
*d^7*e^2 - 119*a^3*c^2*d^5*e^4 + 11*a^4*c*d^3*e^6 + 3*a^5*d*e^8)*x^3 + 2*( 
21*a^3*c^2*d^6*e^3 - 22*a^4*c*d^4*e^5 + a^5*d^2*e^7)*x^2 - 24*(a^4*c*d^5*e 
^4 - a^5*d^3*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^6*c*d 
^4*e^6*x^5 + a^7*d^3*e^7*x^4), 1/128*(3*((105*c^5*d^9 - 140*a*c^4*d^7*e^2 
+ 30*a^2*c^3*d^5*e^4 + 4*a^3*c^2*d^3*e^6 + a^4*c*d*e^8)*x^5 + (105*a*c^4*d 
^8*e - 140*a^2*c^3*d^6*e^3 + 30*a^3*c^2*d^4*e^5 + 4*a^4*c*d^2*e^7 + a^5*e^ 
9)*x^4)*sqrt(-a*d*e)*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*(16*a^5*d^4*e^5 - (315*a*c^4*d^8*e - 3 
15*a^2*c^3*d^6*e^3 + 13*a^3*c^2*d^4*e^5 + 3*a^4*c*d^2*e^7)*x^4 - (105*a^2* 
c^3*d^7*e^2 - 119*a^3*c^2*d^5*e^4 + 11*a^4*c*d^3*e^6 + 3*a^5*d*e^8)*x^3 + 
2*(21*a^3*c^2*d^6*e^3 - 22*a^4*c*d^4*e^5 + a^5*d^2*e^7)*x^2 - 24*(a^4*c*d^ 
5*e^4 - a^5*d^3*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^3/x^5/(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(-2)]

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

integrate((e*x+d)^3/x^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="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%%%{%%%{1,[1,1,17]%%%},[2,11]%%%}+%%%{%%%{-6,[2,3,15]%%%},[2,1 
0]%%%}+%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 8.45 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.33 \[ \int \frac {(d+e x)^3}{x^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(21*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x 
) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d 
 + e*x))*a**5*e**10*x**4 + 111*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*l 
og(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d** 
2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**4*c*d**2*e**8*x**4 + 738*sqrt(e)*sq 
rt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqr 
t(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**3* 
c**2*d**4*e**6*x**4 - 2130*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(s 
qrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + 
 sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2*c**3*d**6*e**4*x**4 - 1575*sqrt(e)*sq 
rt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqr 
t(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c** 
4*d**8*e**2*x**4 + 2835*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt 
(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sq 
rt(d)*sqrt(c)*sqrt(d + e*x))*c**5*d**10*x**4 + 21*sqrt(e)*sqrt(d)*sqrt(a)* 
sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(c)*sqrt(a)*d 
*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**5*e**10*x**4 + 1 
11*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) 
 + sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d 
+ e*x))*a**4*c*d**2*e**8*x**4 + 738*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + ...