\(\int \frac {1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [273]

Optimal result

Integrand size = 37, antiderivative size = 352 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {20 e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {160 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{21 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {64 c d e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right )^4 (d+e x)^3}+\frac {256 c^2 d^2 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{21 \left (c d^2-a e^2\right )^5 (d+e x)^2}+\frac {512 c^3 d^3 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{21 \left (c d^2-a e^2\right )^6 (d+e x)} \] Output:

-2/3/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+20/3 
*e/(-a*e^2+c*d^2)^2/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+160/ 
21*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^3/(e*x+d)^4+ 
64/7*c*d*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^4/(e*x 
+d)^3+256/21*c^2*d^2*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c 
*d^2)^5/(e*x+d)^2+512/21*c^3*d^3*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 
2)/(-a*e^2+c*d^2)^6/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (-3 a^5 e^{10}+3 a^4 c d e^8 (7 d+2 e x)-2 a^3 c^2 d^2 e^6 \left (35 d^2+28 d e x+8 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )+3 a c^4 d^4 e^2 \left (35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (-7 d^5+70 d^4 e x+560 d^3 e^2 x^2+1120 d^2 e^3 x^3+896 d e^4 x^4+256 e^5 x^5\right )\right )}{21 \left (c d^2-a e^2\right )^6 (d+e x)^2 ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
 

Output:

(2*(-3*a^5*e^10 + 3*a^4*c*d*e^8*(7*d + 2*e*x) - 2*a^3*c^2*d^2*e^6*(35*d^2 
+ 28*d*e*x + 8*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(35*d^3 + 70*d^2*e*x + 56*d*e^ 
2*x^2 + 16*e^3*x^3) + 3*a*c^4*d^4*e^2*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2* 
x^2 + 448*d*e^3*x^3 + 128*e^4*x^4) + c^5*d^5*(-7*d^5 + 70*d^4*e*x + 560*d^ 
3*e^2*x^2 + 1120*d^2*e^3*x^3 + 896*d*e^4*x^4 + 256*e^5*x^5)))/(21*(c*d^2 - 
 a*e^2)^6*(d + e*x)^2*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1129, 1129, 1089, 1088}

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.

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1129

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

\(\Big \downarrow \) 1089

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

\(\Big \downarrow \) 1088

\(\displaystyle \frac {10 c d \left (\frac {8 c d \left (\frac {16 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{7 \left (c d^2-a e^2\right )}+\frac {2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

Input:

Int[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
 

Output:

2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^( 
3/2)) + (10*c*d*(2/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x 
 + c*d*e*x^2)^(3/2)) + (8*c*d*((-2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 
- a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (16*c*d*e*(c*d 
^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2) 
*x + c*d*e*x^2])))/(5*(c*d^2 - a*e^2))))/(7*(c*d^2 - a*e^2))
 

Defintions of rubi rules used

rule 1088

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

rule 1089

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

rule 1129

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {-\frac {2}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {10 d e c \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 d e c \left (-\frac {2 \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{7 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{2}}\)	\(323\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-256 x^{5} e^{5} d^{5} c^{5}-384 x^{4} a \,c^{4} d^{4} e^{6}-896 x^{4} c^{5} d^{6} e^{4}-96 x^{3} a^{2} c^{3} d^{3} e^{7}-1344 x^{3} a \,c^{4} d^{5} e^{5}-1120 x^{3} c^{5} d^{7} e^{3}+16 x^{2} a^{3} c^{2} d^{2} e^{8}-336 x^{2} a^{2} c^{3} d^{4} e^{6}-1680 x^{2} a \,c^{4} d^{6} e^{4}-560 x^{2} c^{5} d^{8} e^{2}-6 a^{4} c d \,e^{9} x +56 a^{3} c^{2} d^{3} e^{7} x -420 a^{2} c^{3} d^{5} e^{5} x -840 x a \,c^{4} d^{7} e^{3}-70 c^{5} d^{9} e x +3 a^{5} e^{10}-21 a^{4} c \,d^{2} e^{8}+70 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}-105 a \,c^{4} d^{8} e^{2}+7 c^{5} d^{10}\right )}{21 \left (e x +d \right ) \left (a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(412\)
orering	\(-\frac {2 \left (-256 x^{5} e^{5} d^{5} c^{5}-384 x^{4} a \,c^{4} d^{4} e^{6}-896 x^{4} c^{5} d^{6} e^{4}-96 x^{3} a^{2} c^{3} d^{3} e^{7}-1344 x^{3} a \,c^{4} d^{5} e^{5}-1120 x^{3} c^{5} d^{7} e^{3}+16 x^{2} a^{3} c^{2} d^{2} e^{8}-336 x^{2} a^{2} c^{3} d^{4} e^{6}-1680 x^{2} a \,c^{4} d^{6} e^{4}-560 x^{2} c^{5} d^{8} e^{2}-6 a^{4} c d \,e^{9} x +56 a^{3} c^{2} d^{3} e^{7} x -420 a^{2} c^{3} d^{5} e^{5} x -840 x a \,c^{4} d^{7} e^{3}-70 c^{5} d^{9} e x +3 a^{5} e^{10}-21 a^{4} c \,d^{2} e^{8}+70 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}-105 a \,c^{4} d^{8} e^{2}+7 c^{5} d^{10}\right ) \left (c d x +a e \right )}{21 \left (a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}\right ) \left (e x +d \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\)	\(413\)
trager	\(-\frac {2 \left (-256 x^{5} e^{5} d^{5} c^{5}-384 x^{4} a \,c^{4} d^{4} e^{6}-896 x^{4} c^{5} d^{6} e^{4}-96 x^{3} a^{2} c^{3} d^{3} e^{7}-1344 x^{3} a \,c^{4} d^{5} e^{5}-1120 x^{3} c^{5} d^{7} e^{3}+16 x^{2} a^{3} c^{2} d^{2} e^{8}-336 x^{2} a^{2} c^{3} d^{4} e^{6}-1680 x^{2} a \,c^{4} d^{6} e^{4}-560 x^{2} c^{5} d^{8} e^{2}-6 a^{4} c d \,e^{9} x +56 a^{3} c^{2} d^{3} e^{7} x -420 a^{2} c^{3} d^{5} e^{5} x -840 x a \,c^{4} d^{7} e^{3}-70 c^{5} d^{9} e x +3 a^{5} e^{10}-21 a^{4} c \,d^{2} e^{8}+70 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}-105 a \,c^{4} d^{8} e^{2}+7 c^{5} d^{10}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{21 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{4}}\)	\(415\)

Input:

int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (328) = 656\).

Time = 33.17 (sec) , antiderivative size = 1058, normalized size of antiderivative = 3.01 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm 
="fricas")
 

Output:

2/21*(256*c^5*d^5*e^5*x^5 - 7*c^5*d^10 + 105*a*c^4*d^8*e^2 + 210*a^2*c^3*d 
^6*e^4 - 70*a^3*c^2*d^4*e^6 + 21*a^4*c*d^2*e^8 - 3*a^5*e^10 + 128*(7*c^5*d 
^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + 32*(35*c^5*d^7*e^3 + 42*a*c^4*d^5*e^5 + 3* 
a^2*c^3*d^3*e^7)*x^3 + 16*(35*c^5*d^8*e^2 + 105*a*c^4*d^6*e^4 + 21*a^2*c^3 
*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 2*(35*c^5*d^9*e + 420*a*c^4*d^7*e^3 + 21 
0*a^2*c^3*d^5*e^5 - 28*a^3*c^2*d^3*e^7 + 3*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 
+ a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c^6*d^16*e^2 - 6*a^3*c^5*d^14*e^4 + 15*a 
^4*c^4*d^12*e^6 - 20*a^5*c^3*d^10*e^8 + 15*a^6*c^2*d^8*e^10 - 6*a^7*c*d^6* 
e^12 + a^8*d^4*e^14 + (c^8*d^14*e^4 - 6*a*c^7*d^12*e^6 + 15*a^2*c^6*d^10*e 
^8 - 20*a^3*c^5*d^8*e^10 + 15*a^4*c^4*d^6*e^12 - 6*a^5*c^3*d^4*e^14 + a^6* 
c^2*d^2*e^16)*x^6 + 2*(2*c^8*d^15*e^3 - 11*a*c^7*d^13*e^5 + 24*a^2*c^6*d^1 
1*e^7 - 25*a^3*c^5*d^9*e^9 + 10*a^4*c^4*d^7*e^11 + 3*a^5*c^3*d^5*e^13 - 4* 
a^6*c^2*d^3*e^15 + a^7*c*d*e^17)*x^5 + (6*c^8*d^16*e^2 - 28*a*c^7*d^14*e^4 
 + 43*a^2*c^6*d^12*e^6 - 6*a^3*c^5*d^10*e^8 - 55*a^4*c^4*d^8*e^10 + 64*a^5 
*c^3*d^6*e^12 - 27*a^6*c^2*d^4*e^14 + 2*a^7*c*d^2*e^16 + a^8*e^18)*x^4 + 4 
*(c^8*d^17*e - 3*a*c^7*d^15*e^3 - 2*a^2*c^6*d^13*e^5 + 19*a^3*c^5*d^11*e^7 
 - 30*a^4*c^4*d^9*e^9 + 19*a^5*c^3*d^7*e^11 - 2*a^6*c^2*d^5*e^13 - 3*a^7*c 
*d^3*e^15 + a^8*d*e^17)*x^3 + (c^8*d^18 + 2*a*c^7*d^16*e^2 - 27*a^2*c^6*d^ 
14*e^4 + 64*a^3*c^5*d^12*e^6 - 55*a^4*c^4*d^10*e^8 - 6*a^5*c^3*d^8*e^10 + 
43*a^6*c^2*d^6*e^12 - 28*a^7*c*d^4*e^14 + 6*a^8*d^2*e^16)*x^2 + 2*(a*c^...
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \] Input:

integrate(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
 

Output:

Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*(d + e*x)**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10592 vs. \(2 (328) = 656\).

Time = 0.43 (sec) , antiderivative size = 10592, normalized size of antiderivative = 30.09 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm 
="giac")
 

Output:

-2/21*(256*c^4*d^4*e^2*abs(e)*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d*e)*c^6*d^1 
2 - 6*sqrt(c*d*e)*a*c^5*d^10*e^2 + 15*sqrt(c*d*e)*a^2*c^4*d^8*e^4 - 20*sqr 
t(c*d*e)*a^3*c^3*d^6*e^6 + 15*sqrt(c*d*e)*a^4*c^2*d^4*e^8 - 6*sqrt(c*d*e)* 
a^5*c*d^2*e^10 + sqrt(c*d*e)*a^6*e^12) - e^6*((210*sqrt(c*d*e - c*d^2*e/(e 
*x + d) + a*e^3/(e*x + d))*c^39*d^75*e^51*sgn(1/(e*x + d))^6*sgn(e)^6 - 75 
60*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^38*d^73*e^53*sgn( 
1/(e*x + d))^6*sgn(e)^6 + 132300*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e 
*x + d))*a^2*c^37*d^71*e^55*sgn(1/(e*x + d))^6*sgn(e)^6 - 1499400*sqrt(c*d 
*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^36*d^69*e^57*sgn(1/(e*x + 
d))^6*sgn(e)^6 + 12370050*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d) 
)*a^4*c^35*d^67*e^59*sgn(1/(e*x + d))^6*sgn(e)^6 - 79168320*sqrt(c*d*e - c 
*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^5*c^34*d^65*e^61*sgn(1/(e*x + d))^6* 
sgn(e)^6 + 409036320*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^6 
*c^33*d^63*e^63*sgn(1/(e*x + d))^6*sgn(e)^6 - 1753012800*sqrt(c*d*e - c*d^ 
2*e/(e*x + d) + a*e^3/(e*x + d))*a^7*c^32*d^61*e^65*sgn(1/(e*x + d))^6*sgn 
(e)^6 + 6354671400*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^8*c 
^31*d^59*e^67*sgn(1/(e*x + d))^6*sgn(e)^6 - 19770088800*sqrt(c*d*e - c*d^2 
*e/(e*x + d) + a*e^3/(e*x + d))*a^9*c^30*d^57*e^69*sgn(1/(e*x + d))^6*sgn( 
e)^6 + 53379239760*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^10* 
c^29*d^55*e^71*sgn(1/(e*x + d))^6*sgn(e)^6 - 126169112160*sqrt(c*d*e - ...
 

Mupad [B] (verification not implemented)

Time = 6.68 (sec) , antiderivative size = 3654, normalized size of antiderivative = 10.38 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

int(1/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
 

Output:

(((d*((12*c^3*d^4*e^4)/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15* 
a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)) - (2*c^2*d^2*e^4*(19*a*e^2 - 7*c*d^2))/ 
(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2* 
c*d^2*e^5))))/e + (e^3*(14*c^3*d^5 - 42*a*c^2*d^3*e^2 + 40*a^2*c*d*e^4))/( 
7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c 
*d^2*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - ( 
((d*((24*c^4*d^5*e^4)/(35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)) - (8*c^ 
3*d^3*e^4*(11*a*e^2 - 5*c*d^2))/(35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e 
))))/e + (2*c^2*d^2*e^3*(19*a^2*e^4 - 13*c^2*d^4 + 6*a*c*d^2*e^2))/(35*(a* 
e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e* 
x^2)^(1/2))/(d + e*x)^2 + (((24*c^4*d^5*e^2)/(35*(a*e^2 - c*d^2)^7) - (4*c 
^3*d^3*e^2*(47*a*e^2 - 29*c*d^2))/(105*(a*e^2 - c*d^2)^7))*(x*(a*e^2 + c*d 
^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (2*e^3*(x*(a*e^2 + c*d^2) + a* 
d*e + c*d*e*x^2)^(1/2))/((d + e*x)^4*(7*a^3*e^7 - 7*c^3*d^6*e + 21*a*c^2*d 
^4*e^3 - 21*a^2*c*d^2*e^5)) - ((x*((a*(((a*e^2 + c*d^2)*((8*c^7*d^7*e^5*(a 
*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c* 
d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^ 
3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^6*d^6*e^4*(13*a 
^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 
2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2...
 

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 1384, normalized size of antiderivative = 3.93 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

(2*( - 256*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**3*d**7*e**2 - 10 
24*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**3*d**6*e**3*x - 1536*sqr 
t(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**3*d**5*e**4*x**2 - 1024*sqrt(e 
)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**5*x**3 - 256*sqrt(e)*sq 
rt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**3*d**3*e**6*x**4 - 256*sqrt(e)*sqrt(d 
)*sqrt(c)*sqrt(a*e + c*d*x)*c**4*d**8*e*x - 1024*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(a*e + c*d*x)*c**4*d**7*e**2*x**2 - 1536*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a 
*e + c*d*x)*c**4*d**6*e**3*x**3 - 1024*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + 
c*d*x)*c**4*d**5*e**4*x**4 - 256*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x) 
*c**4*d**4*e**5*x**5 - 3*sqrt(d + e*x)*a**5*e**10 + 21*sqrt(d + e*x)*a**4* 
c*d**2*e**8 + 6*sqrt(d + e*x)*a**4*c*d*e**9*x - 70*sqrt(d + e*x)*a**3*c**2 
*d**4*e**6 - 56*sqrt(d + e*x)*a**3*c**2*d**3*e**7*x - 16*sqrt(d + e*x)*a** 
3*c**2*d**2*e**8*x**2 + 210*sqrt(d + e*x)*a**2*c**3*d**6*e**4 + 420*sqrt(d 
 + e*x)*a**2*c**3*d**5*e**5*x + 336*sqrt(d + e*x)*a**2*c**3*d**4*e**6*x**2 
 + 96*sqrt(d + e*x)*a**2*c**3*d**3*e**7*x**3 + 105*sqrt(d + e*x)*a*c**4*d* 
*8*e**2 + 840*sqrt(d + e*x)*a*c**4*d**7*e**3*x + 1680*sqrt(d + e*x)*a*c**4 
*d**6*e**4*x**2 + 1344*sqrt(d + e*x)*a*c**4*d**5*e**5*x**3 + 384*sqrt(d + 
e*x)*a*c**4*d**4*e**6*x**4 - 7*sqrt(d + e*x)*c**5*d**10 + 70*sqrt(d + e*x) 
*c**5*d**9*e*x + 560*sqrt(d + e*x)*c**5*d**8*e**2*x**2 + 1120*sqrt(d + e*x 
)*c**5*d**7*e**3*x**3 + 896*sqrt(d + e*x)*c**5*d**6*e**4*x**4 + 256*sqr...