\(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [221]

Optimal result

Integrand size = 37, antiderivative size = 544 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {45 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^3}-\frac {15 \left (c d^2-a e^2\right )^6 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8192 c^5 d^5 e^2 (d+e x)}+\frac {3 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2048 c^5 d^5 e (d+e x)^2}+\frac {9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac {9 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{1024 c^5 d^5 (d+e x)^3}+\frac {3 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{128 c^4 d^4 (d+e x)^2}+\frac {3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{64 c^3 d^3 (d+e x)}+\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac {45 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{16384 c^{11/2} d^{11/2} e^{7/2}} \] Output:

45/16384*(-a*e^2+c*d^2)^7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^5/d^5/ 
e^3-15/8192*(-a*e^2+c*d^2)^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^5/d 
^5/e^2/(e*x+d)+3/2048*(-a*e^2+c*d^2)^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 
5/2)/c^5/d^5/e/(e*x+d)^2+9/112*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(7/2)/c^2/d^2+9/1024*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x 
^2)^(7/2)/c^5/d^5/(e*x+d)^3+3/128*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(7/2)/c^4/d^4/(e*x+d)^2+3/64*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d 
^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/(e*x+d)+1/8*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(7/2)/c/d-45/16384*(-a*e^2+c*d^2)^8*arctanh(e^(1/2)*(a*d*e+(a*e 
^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/(e*x+d))/c^(11/2)/d^(11/2)/e^ 
(7/2)
 

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.97 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (315 a^7 e^{14}-105 a^6 c d e^{12} (23 d+2 e x)+21 a^5 c^2 d^2 e^{10} \left (383 d^2+76 d e x+8 e^2 x^2\right )-3 a^4 c^3 d^3 e^8 \left (5053 d^3+1754 d^2 e x+424 d e^2 x^2+48 e^3 x^3\right )+a^3 c^4 d^4 e^6 \left (17609 d^4+9800 d^3 e x+4176 d^2 e^2 x^2+1088 d e^3 x^3+128 e^4 x^4\right )+3 a^2 c^5 d^5 e^4 \left (2681 d^5+31014 d^4 e x+66928 d^3 e^2 x^2+68320 d^2 e^3 x^3+34432 d e^4 x^4+6912 e^5 x^5\right )+3 a c^6 d^6 e^2 \left (-805 d^6+532 d^5 e x+32344 d^4 e^2 x^2+87744 d^3 e^3 x^3+99968 d^2 e^4 x^4+53760 d e^5 x^5+11264 e^6 x^6\right )+c^7 d^7 \left (315 d^7-210 d^6 e x+168 d^5 e^2 x^2+32624 d^4 e^3 x^3+98432 d^3 e^4 x^4+119040 d^2 e^5 x^5+66560 d e^6 x^6+14336 e^7 x^7\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {315 \left (c d^2-a e^2\right )^8 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{114688 c^{11/2} d^{11/2} e^{7/2}} \] Input:

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

Output:

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(315*a^7*e^14 - 
 105*a^6*c*d*e^12*(23*d + 2*e*x) + 21*a^5*c^2*d^2*e^10*(383*d^2 + 76*d*e*x 
 + 8*e^2*x^2) - 3*a^4*c^3*d^3*e^8*(5053*d^3 + 1754*d^2*e*x + 424*d*e^2*x^2 
 + 48*e^3*x^3) + a^3*c^4*d^4*e^6*(17609*d^4 + 9800*d^3*e*x + 4176*d^2*e^2* 
x^2 + 1088*d*e^3*x^3 + 128*e^4*x^4) + 3*a^2*c^5*d^5*e^4*(2681*d^5 + 31014* 
d^4*e*x + 66928*d^3*e^2*x^2 + 68320*d^2*e^3*x^3 + 34432*d*e^4*x^4 + 6912*e 
^5*x^5) + 3*a*c^6*d^6*e^2*(-805*d^6 + 532*d^5*e*x + 32344*d^4*e^2*x^2 + 87 
744*d^3*e^3*x^3 + 99968*d^2*e^4*x^4 + 53760*d*e^5*x^5 + 11264*e^6*x^6) + c 
^7*d^7*(315*d^7 - 210*d^6*e*x + 168*d^5*e^2*x^2 + 32624*d^4*e^3*x^3 + 9843 
2*d^3*e^4*x^4 + 119040*d^2*e^5*x^5 + 66560*d*e^6*x^6 + 14336*e^7*x^7)))/(( 
a*e + c*d*x)^2*(d + e*x)^2) - (315*(c*d^2 - a*e^2)^8*ArcTanh[(Sqrt[c]*Sqrt 
[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + 
 e*x)^(5/2))))/(114688*c^(11/2)*d^(11/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1134, 1160, 1087, 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.

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

\(\Big \downarrow \) 1134

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {9 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\right )}{16 d}+\frac {(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d}\)

Input:

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

Output:

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

Defintions of rubi rules used

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 1134

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1559\) vs. \(2(496)=992\).

Time = 1.70 (sec) , antiderivative size = 1560, normalized size of antiderivative = 2.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1560\)

Input:

int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURNVERB 
OSE)
 

Output:

d^2*(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))))+e^2*(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)/(d...
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1520, normalized size of antiderivative = 2.79 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((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:

[1/458752*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3 
*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4* 
e^12 - 8*a^7*c*d^2*e^14 + 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*(14336*c^8*d^8*e^8*x^7 + 315*c^8*d^15*e - 2415*a*c^7*d^13*e^3 + 804 
3*a^2*c^6*d^11*e^5 + 17609*a^3*c^5*d^9*e^7 - 15159*a^4*c^4*d^7*e^9 + 8043* 
a^5*c^3*d^5*e^11 - 2415*a^6*c^2*d^3*e^13 + 315*a^7*c*d*e^15 + 1024*(65*c^8 
*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 768*(155*c^8*d^10*e^6 + 210*a*c^7*d^8*e 
^8 + 27*a^2*c^6*d^6*e^10)*x^5 + 128*(769*c^8*d^11*e^5 + 2343*a*c^7*d^9*e^7 
 + 807*a^2*c^6*d^7*e^9 + a^3*c^5*d^5*e^11)*x^4 + 16*(2039*c^8*d^12*e^4 + 1 
6452*a*c^7*d^10*e^6 + 12810*a^2*c^6*d^8*e^8 + 68*a^3*c^5*d^6*e^10 - 9*a^4* 
c^4*d^4*e^12)*x^3 + 24*(7*c^8*d^13*e^3 + 4043*a*c^7*d^11*e^5 + 8366*a^2*c^ 
6*d^9*e^7 + 174*a^3*c^5*d^7*e^9 - 53*a^4*c^4*d^5*e^11 + 7*a^5*c^3*d^3*e^13 
)*x^2 - 2*(105*c^8*d^14*e^2 - 798*a*c^7*d^12*e^4 - 46521*a^2*c^6*d^10*e^6 
- 4900*a^3*c^5*d^8*e^8 + 2631*a^4*c^4*d^6*e^10 - 798*a^5*c^3*d^4*e^12 + 10 
5*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^4), 1/229376*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 
- 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6* 
c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*sqrt(-c*d*e)*arctan(1/2*sqr...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8918 vs. \(2 (518) = 1036\).

Time = 6.60 (sec) , antiderivative size = 8918, normalized size of antiderivative = 16.39 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**4* 
x**7/8 + x**6*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15* 
a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e) + x**5*(3*a**2*c*d*e**7 + 113*a*c**2* 
d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d* 
*2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8) 
/(7*c*d*e))/(6*c*d*e) + x**4*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2* 
d**4*e**4 - 6*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(1 
5*a*e**2/2 + 15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11 
*c*d**2/2)*(3*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - 
 (13*a*e**2/2 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2 
*d**2*e**4*(15*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c*d*e))/(5*c*d*e) 
+ x**3*(5*a**3*d*e**7 + 30*a**2*c*d**3*e**5 + 30*a*c**2*d**5*e**3 - 5*a*(3 
*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 
 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(1 
5*a*e**2/2 + 15*c*d**2/2)/8)/(7*c*d*e))/(6*c) + 5*c**3*d**7*e - (9*a*e**2/ 
2 + 9*c*d**2/2)*(a**3*e**8 + 15*a**2*c*d**2*e**6 + 30*a*c**2*d**4*e**4 - 6 
*a*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4*(15*a*e**2/2 + 
15*c*d**2/2)/8)/(7*c) + 10*c**3*d**6*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3 
*a**2*c*d*e**7 + 113*a*c**2*d**3*e**5/8 + 10*c**3*d**5*e**3 - (13*a*e**2/2 
 + 13*c*d**2/2)*(3*a*c**2*d**2*e**6 + 5*c**3*d**4*e**4 - c**2*d**2*e**4...
 

Maxima [F(-2)]

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

integrate((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>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.44 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((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:

1/114688*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(4*(14 
*c^2*d^2*e^4*x + (65*c^9*d^10*e^10 + 33*a*c^8*d^8*e^12)/(c^7*d^7*e^7))*x + 
 3*(155*c^9*d^11*e^9 + 210*a*c^8*d^9*e^11 + 27*a^2*c^7*d^7*e^13)/(c^7*d^7* 
e^7))*x + (769*c^9*d^12*e^8 + 2343*a*c^8*d^10*e^10 + 807*a^2*c^7*d^8*e^12 
+ a^3*c^6*d^6*e^14)/(c^7*d^7*e^7))*x + (2039*c^9*d^13*e^7 + 16452*a*c^8*d^ 
11*e^9 + 12810*a^2*c^7*d^9*e^11 + 68*a^3*c^6*d^7*e^13 - 9*a^4*c^5*d^5*e^15 
)/(c^7*d^7*e^7))*x + 3*(7*c^9*d^14*e^6 + 4043*a*c^8*d^12*e^8 + 8366*a^2*c^ 
7*d^10*e^10 + 174*a^3*c^6*d^8*e^12 - 53*a^4*c^5*d^6*e^14 + 7*a^5*c^4*d^4*e 
^16)/(c^7*d^7*e^7))*x - (105*c^9*d^15*e^5 - 798*a*c^8*d^13*e^7 - 46521*a^2 
*c^7*d^11*e^9 - 4900*a^3*c^6*d^9*e^11 + 2631*a^4*c^5*d^7*e^13 - 798*a^5*c^ 
4*d^5*e^15 + 105*a^6*c^3*d^3*e^17)/(c^7*d^7*e^7))*x + (315*c^9*d^16*e^4 - 
2415*a*c^8*d^14*e^6 + 8043*a^2*c^7*d^12*e^8 + 17609*a^3*c^6*d^10*e^10 - 15 
159*a^4*c^5*d^8*e^12 + 8043*a^5*c^4*d^6*e^14 - 2415*a^6*c^3*d^4*e^16 + 315 
*a^7*c^2*d^2*e^18)/(c^7*d^7*e^7)) + 45/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 
+ 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5* 
c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + 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^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 1615, normalized size of antiderivative = 2.97 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(315*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**7*c*d*e**15 - 2415*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**6*c**2*d**3*e**13 - 210*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**6*c**2*d**2*e**14*x + 8043*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**3* 
d**5*e**11 + 1596*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**3*d**4*e**12*x + 
 168*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c**3*d**3*e**13*x**2 - 15159*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**4*d**7*e**9 - 5262*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a**4*c**4*d**6*e**10*x - 1272*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**4*c**4*d**5*e**11*x**2 - 144*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c** 
4*d**4*e**12*x**3 + 17609*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**5*d**9*e 
**7 + 9800*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**5*d**8*e**8*x + 4176*sq 
rt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**5*d**7*e**9*x**2 + 1088*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a**3*c**5*d**6*e**10*x**3 + 128*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*a**3*c**5*d**5*e**11*x**4 + 8043*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**2*c**6*d**11*e**5 + 93042*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d* 
*10*e**6*x + 200784*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**9*e**7*x* 
*2 + 204960*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**8*e**8*x**3 + 103 
296*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**7*e**9*x**4 + 20736*sqrt( 
d + e*x)*sqrt(a*e + c*d*x)*a**2*c**6*d**6*e**10*x**5 - 2415*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a*c**7*d**13*e**3 + 1596*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a*c**7*d**12*e**4*x + 97032*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**7*d**...