\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [299]

Optimal result

Integrand size = 39, antiderivative size = 175 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^3 d^3 (d+e x)^{7/2}}+\frac {4 e \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^3 d^3 (d+e x)^{9/2}}+\frac {2 e^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{11/2}}{11 c^3 d^3 (d+e x)^{11/2}} \] Output:

2/7*(-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)^(7/2)+4/9*e*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/c^3/ 
d^3/(e*x+d)^(9/2)+2/11*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(11/2)/c^3/d^ 
3/(e*x+d)^(11/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (8 a^2 e^4-4 a c d e^2 (11 d+7 e x)+c^2 d^2 \left (99 d^2+154 d e x+63 e^2 x^2\right )\right )}{693 c^3 d^3 \sqrt {d+e x}} \] Input:

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

Output:

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^4 - 4*a*c*d*e^2* 
(11*d + 7*e*x) + c^2*d^2*(99*d^2 + 154*d*e*x + 63*e^2*x^2)))/(693*c^3*d^3* 
Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1128, 1128, 1122}

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.

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c d^2 (d+e x)^{7/2}}\right )}{11 d}\)

Input:

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

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(3/2)) 
 + (4*(d^2 - (a*e^2)/c)*((4*(d^2 - (a*e^2)/c)*(a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2)^(7/2))/(63*c*d^2*(d + e*x)^(7/2)) + (2*(a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2)^(7/2))/(9*c*d*(d + e*x)^(5/2))))/(11*d)
 

Defintions of rubi rules used

rule 1122

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*(p + 1))), 
 x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
EqQ[m + p, 0]
 

rule 1128

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[Simplify[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, m, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
 

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.58

method	result	size
default	\(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{3} \left (63 x^{2} c^{2} d^{2} e^{2}-28 x a c d \,e^{3}+154 x \,c^{2} d^{3} e +8 a^{2} e^{4}-44 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right )}{693 \sqrt {e x +d}\, d^{3} c^{3}}\)	\(102\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (63 x^{2} c^{2} d^{2} e^{2}-28 x a c d \,e^{3}+154 x \,c^{2} d^{3} e +8 a^{2} e^{4}-44 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 d^{3} c^{3} \left (e x +d \right )^{\frac {5}{2}}}\)	\(110\)
orering	\(\frac {2 \left (63 x^{2} c^{2} d^{2} e^{2}-28 x a c d \,e^{3}+154 x \,c^{2} d^{3} e +8 a^{2} e^{4}-44 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{693 d^{3} c^{3} \left (e x +d \right )^{\frac {5}{2}}}\)	\(111\)

Input:

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

Output:

2/693*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(63*c^2*d^2* 
e^2*x^2-28*a*c*d*e^3*x+154*c^2*d^3*e*x+8*a^2*e^4-44*a*c*d^2*e^2+99*c^2*d^4 
)/d^3/c^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (63 \, c^{5} d^{5} e^{2} x^{5} + 99 \, a^{3} c^{2} d^{4} e^{3} - 44 \, a^{4} c d^{2} e^{5} + 8 \, a^{5} e^{7} + 7 \, {\left (22 \, c^{5} d^{6} e + 23 \, a c^{4} d^{4} e^{3}\right )} x^{4} + {\left (99 \, c^{5} d^{7} + 418 \, a c^{4} d^{5} e^{2} + 113 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (99 \, a c^{4} d^{6} e + 110 \, a^{2} c^{3} d^{4} e^{3} + a^{3} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{5} e^{2} + 22 \, a^{3} c^{2} d^{3} e^{4} - 4 \, a^{4} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{693 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \] Input:

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

Output:

2/693*(63*c^5*d^5*e^2*x^5 + 99*a^3*c^2*d^4*e^3 - 44*a^4*c*d^2*e^5 + 8*a^5* 
e^7 + 7*(22*c^5*d^6*e + 23*a*c^4*d^4*e^3)*x^4 + (99*c^5*d^7 + 418*a*c^4*d^ 
5*e^2 + 113*a^2*c^3*d^3*e^4)*x^3 + 3*(99*a*c^4*d^6*e + 110*a^2*c^3*d^4*e^3 
 + a^3*c^2*d^2*e^5)*x^2 + (297*a^2*c^3*d^5*e^2 + 22*a^3*c^2*d^3*e^4 - 4*a^ 
4*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c 
^3*d^3*e*x + c^3*d^4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (63 \, c^{5} d^{5} e^{2} x^{5} + 99 \, a^{3} c^{2} d^{4} e^{3} - 44 \, a^{4} c d^{2} e^{5} + 8 \, a^{5} e^{7} + 7 \, {\left (22 \, c^{5} d^{6} e + 23 \, a c^{4} d^{4} e^{3}\right )} x^{4} + {\left (99 \, c^{5} d^{7} + 418 \, a c^{4} d^{5} e^{2} + 113 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (99 \, a c^{4} d^{6} e + 110 \, a^{2} c^{3} d^{4} e^{3} + a^{3} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{5} e^{2} + 22 \, a^{3} c^{2} d^{3} e^{4} - 4 \, a^{4} c d e^{6}\right )} x\right )} \sqrt {c d x + a e}}{693 \, c^{3} d^{3}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(1/2),x, algorit 
hm="maxima")
 

Output:

2/693*(63*c^5*d^5*e^2*x^5 + 99*a^3*c^2*d^4*e^3 - 44*a^4*c*d^2*e^5 + 8*a^5* 
e^7 + 7*(22*c^5*d^6*e + 23*a*c^4*d^4*e^3)*x^4 + (99*c^5*d^7 + 418*a*c^4*d^ 
5*e^2 + 113*a^2*c^3*d^3*e^4)*x^3 + 3*(99*a*c^4*d^6*e + 110*a^2*c^3*d^4*e^3 
 + a^3*c^2*d^2*e^5)*x^2 + (297*a^2*c^3*d^5*e^2 + 22*a^3*c^2*d^3*e^4 - 4*a^ 
4*c*d*e^6)*x)*sqrt(c*d*x + a*e)/(c^3*d^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 780 vs. \(2 (157) = 314\).

Time = 0.18 (sec) , antiderivative size = 780, normalized size of antiderivative = 4.46 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/3465*(3465*sqrt(c*d*x + a*e)*a^3*d^2*e^3 - 3465*(3*sqrt(c*d*x + a*e)*a*e 
 - (c*d*x + a*e)^(3/2))*a^2*d^2*e^2 - 2310*(3*sqrt(c*d*x + a*e)*a*e - (c*d 
*x + a*e)^(3/2))*a^3*e^4/c + 693*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x 
 + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a*d^2*e + 1386*(15*sqrt(c*d*x + 
 a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a^2*e^ 
3/c + 231*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*( 
c*d*x + a*e)^(5/2))*a^3*e^5/(c^2*d^2) - 99*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 
 35*(c*d*x + a*e)^(3/2)*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + 
a*e)^(7/2))*d^2 - 594*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/ 
2)*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a*e^2/c - 
 297*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2*e^2 + 21*( 
c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a^2*e^4/(c^2*d^2) + 22*(31 
5*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3*e^3 + 378*(c*d*x 
 + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a*e + 35*(c*d*x + a*e)^(9/ 
2))*e/c + 33*(315*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3* 
e^3 + 378*(c*d*x + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a*e + 35*( 
c*d*x + a*e)^(9/2))*a*e^3/(c^2*d^2) - 5*(693*sqrt(c*d*x + a*e)*a^5*e^5 - 1 
155*(c*d*x + a*e)^(3/2)*a^4*e^4 + 1386*(c*d*x + a*e)^(5/2)*a^3*e^3 - 990*( 
c*d*x + a*e)^(7/2)*a^2*e^2 + 385*(c*d*x + a*e)^(9/2)*a*e - 63*(c*d*x + a*e 
)^(11/2))*e^2/(c^2*d^2))/(c*d)
 

Mupad [B] (verification not implemented)

Time = 5.52 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,a^5\,e^7-88\,a^4\,c\,d^2\,e^5+198\,a^3\,c^2\,d^4\,e^3}{693\,c^3\,d^3}+\frac {x^3\,\left (226\,a^2\,c^3\,d^3\,e^4+836\,a\,c^4\,d^5\,e^2+198\,c^5\,d^7\right )}{693\,c^3\,d^3}+\frac {2\,c^2\,d^2\,e^2\,x^5}{11}+\frac {2\,c\,d\,e\,x^4\,\left (22\,c\,d^2+23\,a\,e^2\right )}{99}+\frac {2\,a\,e\,x^2\,\left (a^2\,e^4+110\,a\,c\,d^2\,e^2+99\,c^2\,d^4\right )}{231\,c\,d}+\frac {2\,a^2\,e^2\,x\,\left (-4\,a^2\,e^4+22\,a\,c\,d^2\,e^2+297\,c^2\,d^4\right )}{693\,c^2\,d^2}\right )}{\sqrt {d+e\,x}} \] Input:

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

Output:

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*a^5*e^7 - 88*a^4*c*d^2 
*e^5 + 198*a^3*c^2*d^4*e^3)/(693*c^3*d^3) + (x^3*(198*c^5*d^7 + 836*a*c^4* 
d^5*e^2 + 226*a^2*c^3*d^3*e^4))/(693*c^3*d^3) + (2*c^2*d^2*e^2*x^5)/11 + ( 
2*c*d*e*x^4*(23*a*e^2 + 22*c*d^2))/99 + (2*a*e*x^2*(a^2*e^4 + 99*c^2*d^4 + 
 110*a*c*d^2*e^2))/(231*c*d) + (2*a^2*e^2*x*(297*c^2*d^4 - 4*a^2*e^4 + 22* 
a*c*d^2*e^2))/(693*c^2*d^2)))/(d + e*x)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (63 c^{5} d^{5} e^{2} x^{5}+161 a \,c^{4} d^{4} e^{3} x^{4}+154 c^{5} d^{6} e \,x^{4}+113 a^{2} c^{3} d^{3} e^{4} x^{3}+418 a \,c^{4} d^{5} e^{2} x^{3}+99 c^{5} d^{7} x^{3}+3 a^{3} c^{2} d^{2} e^{5} x^{2}+330 a^{2} c^{3} d^{4} e^{3} x^{2}+297 a \,c^{4} d^{6} e \,x^{2}-4 a^{4} c d \,e^{6} x +22 a^{3} c^{2} d^{3} e^{4} x +297 a^{2} c^{3} d^{5} e^{2} x +8 a^{5} e^{7}-44 a^{4} c \,d^{2} e^{5}+99 a^{3} c^{2} d^{4} e^{3}\right )}{693 c^{3} d^{3}} \] Input:

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

Output:

(2*sqrt(a*e + c*d*x)*(8*a**5*e**7 - 44*a**4*c*d**2*e**5 - 4*a**4*c*d*e**6* 
x + 99*a**3*c**2*d**4*e**3 + 22*a**3*c**2*d**3*e**4*x + 3*a**3*c**2*d**2*e 
**5*x**2 + 297*a**2*c**3*d**5*e**2*x + 330*a**2*c**3*d**4*e**3*x**2 + 113* 
a**2*c**3*d**3*e**4*x**3 + 297*a*c**4*d**6*e*x**2 + 418*a*c**4*d**5*e**2*x 
**3 + 161*a*c**4*d**4*e**3*x**4 + 99*c**5*d**7*x**3 + 154*c**5*d**6*e*x**4 
 + 63*c**5*d**5*e**2*x**5))/(693*c**3*d**3)