\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^7} \, dx\) [1942]

Optimal result

Integrand size = 37, antiderivative size = 54 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^7} \]

[Out]

2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e^2+c*d^2)/(e*x+d)^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {664} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^7 \left (c d^2-a e^2\right )} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7)

Rule 664

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {2 ((a e+c d x) (d+e x))^{7/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^7} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7)

Maple [A] (verified)

Time = 5.56 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{6} \left (e^{2} a -c \,d^{2}\right )}\)	\(58\)
default	\(-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7 e^{7} \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}\)	\(65\)
trager	\(-\frac {2 \left (c^{3} d^{3} x^{3}+3 c^{2} d^{2} a e \,x^{2}+3 d \,e^{2} a^{2} c x +a^{3} e^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{7 \left (e x +d \right )^{4} \left (e^{2} a -c \,d^{2}\right )}\)	\(92\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

-2/7*(c*d*x+a*e)/(e*x+d)^6/(a*e^2-c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (50) = 100\).

Time = 2.41 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.15 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7 \, {\left (c d^{6} - a d^{4} e^{2} + {\left (c d^{2} e^{4} - a e^{6}\right )} x^{4} + 4 \, {\left (c d^{3} e^{3} - a d e^{5}\right )} x^{3} + 6 \, {\left (c d^{4} e^{2} - a d^{2} e^{4}\right )} x^{2} + 4 \, {\left (c d^{5} e - a d^{3} e^{3}\right )} x\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/
(c*d^6 - a*d^4*e^2 + (c*d^2*e^4 - a*e^6)*x^4 + 4*(c*d^3*e^3 - a*d*e^5)*x^3 + 6*(c*d^4*e^2 - a*d^2*e^4)*x^2 + 4
*(c*d^5*e - a*d^3*e^3)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**7,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assu
me?` for mor

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[0,0,4]%%%},[8]%%%}+%%%{%%{[%%%{-8,[0,1,3]%%%},0]:
[1,0,%%%{-1

Mupad [B] (verification not implemented)

Time = 13.14 (sec) , antiderivative size = 2156, normalized size of antiderivative = 39.93 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^7,x)

[Out]

(((d*((d*((16*c^6*d^7)/(105*e*(a*e^2 - c*d^2)^4) - (16*c^5*d^5*(4*a*e^2 - 3*c*d^2))/(35*e*(a*e^2 - c*d^2)^4)))
/e + (16*c^4*d^4*(56*a^2*e^4 + 35*c^2*d^4 - 88*a*c*d^2*e^2))/(105*e^2*(a*e^2 - c*d^2)^4)))/e - (16*a*c^3*d^3*(
45*a^2*e^4 + 35*c^2*d^4 - 79*a*c*d^2*e^2))/(105*e*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^
(1/2))/(d + e*x) - (((2*a^3*e^4)/(7*a*e^3 - 7*c*d^2*e) - (d*((d*((2*c^3*d^4)/(7*a*e^3 - 7*c*d^2*e) - (6*a*c^2*
d^2*e^2)/(7*a*e^3 - 7*c*d^2*e)))/e + (6*a^2*c*d*e^3)/(7*a*e^3 - 7*c*d^2*e)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c
*d*e*x^2)^(1/2))/(d + e*x)^4 + (((d*((d*((4*c^4*d^5)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*c^3*d^3*(7
*a*e^2 - 4*c*d^2))/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e))))/e + (16*c^4*d^6 - 64*a*c^3*d^4*e^2 + 60*a^2*c^2
*d^2*e^4)/(7*e*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e))))/e - (4*a*c*d*(3*a*e^2 - 2*c*d^2)^2)/(7*(a*e^2 - c*d^2)
*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((d*((4*c^4*d^5)/(7*e*(
a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (2*c^3*d^3*(7*a*e^2 - 3*c*d^2))/(7*e*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^
2*e))))/e + (16*c^4*d^6 - 62*a*c^3*d^4*e^2 + 66*a^2*c^2*d^2*e^4)/(35*e^2*(a*e^2 - c*d^2)*(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 + (((216*c^6*d^9 - 280*a*c^5*d^7*e^2 - 208*a^2*c
^4*d^5*e^4 + 288*a^3*c^3*d^3*e^6)/(105*e^3*(a*e^2 - c*d^2)^4) - (d*((d*((16*c^6*d^7)/(105*e*(a*e^2 - c*d^2)^4)
- (8*c^5*d^5*(7*a*e^2 - 5*c*d^2))/(35*e*(a*e^2 - c*d^2)^4)))/e + (16*c^4*d^4*(41*a^2*e^4 + 23*c^2*d^4 - 61*a*
c*d^2*e^2))/(105*e^2*(a*e^2 - c*d^2)^4)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((d*(
(d*((8*c^5*d^6)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*c^4*d^4*(10*a*e^2 - 7*c*d^2))/(35*(a*e^2 - c
*d^2)^2*(3*a*e^3 - 3*c*d^2*e))))/e + (8*c^3*d^3*(36*a^2*e^4 + 19*c^2*d^4 - 52*a*c*d^2*e^2))/(35*e*(a*e^2 - c*d
^2)^2*(3*a*e^3 - 3*c*d^2*e))))/e - (8*a*c^2*d^2*(27*a^2*e^4 + 19*c^2*d^4 - 45*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2
)^2*(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 + (((d*((8*c^5*d^6)/(35
*e^2*(a*e^2 - c*d^2)^3) - (4*c^4*d^4*(15*a*e^2 - 11*c*d^2))/(35*e^2*(a*e^2 - c*d^2)^3)))/e + (4*c^3*d^3*(27*a^
2*e^4 + 14*c^2*d^4 - 39*a*c*d^2*e^2))/(35*e^3*(a*e^2 - c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2
))/(d + e*x) - (((d*((d*((4*c^4*d^5)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (2*c^3*d^3*(7*a*e^2 - c*d^2))
/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e))))/e + (2*c^2*d^2*(9*a^2*e^4 + c^2*d^4 - 4*a*c*d^2*e^2))/(7*e*(a*e^2
- c*d^2)*(5*a*e^3 - 5*c*d^2*e))))/e - (2*c^4*d^7 - 4*a*c^3*d^5*e^2 + 6*a^3*c*d*e^6)/(7*e^2*(a*e^2 - c*d^2)*(5
*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((d*((8*c^5*d^6)/(35*(
a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (12*c^4*d^4*(5*a*e^2 - 3*c*d^2))/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*
c*d^2*e))))/e + (4*c^3*d^3*(37*a^2*e^4 + 13*c^2*d^4 - 44*a*c*d^2*e^2))/(35*e*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*
d^2*e))))/e - (36*c^5*d^8 - 56*a*c^4*d^6*e^2 - 32*a^2*c^3*d^4*e^4 + 60*a^3*c^2*d^2*e^6)/(35*e^2*(a*e^2 - c*d^2
)^2*(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 - (((6*c^4*d^5 - 26*a*c
^3*d^3*e^2)/(35*e^3*(a*e^2 - c*d^2)^2) + (4*c^4*d^5)/(7*e^3*(a*e^2 - c*d^2)^2))*(x*(a*e^2 + c*d^2) + a*d*e + c
*d*e*x^2)^(1/2))/(d + e*x)