3.20.43 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^8} \, dx\) [1943]

Optimal. Leaf size=111 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^7} \]

[Out]

2/9*(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)^8+4/63*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(7/2)/(-a*e^2+c*d^2)^2/(e*x+d)^7

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(7/2))/(63*(c*d^2 - a*e^2)^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]

Rule 672

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)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[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] && NeQ[b^2 - 4*a
*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rubi steps

\begin {align*} \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)^8} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {(2 c d) \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}{9 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 71, normalized size = 0.64 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-7 a e^2+c d (9 d+2 e x)\right )}{63 \left (c d^2-a e^2\right )^2 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-7*a*e^2 + c*d*(9*d + 2*e*x)))/(63*(c*d^2 - a*e^2)^2*(d + e*
x)^5)

________________________________________________________________________________________

Maple [A]
time = 0.71, size = 131, normalized size = 1.18

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d e x +7 e^{2} a -9 c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\)	\(90\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}}{e^{8}}\)	\(131\)
trager	\(-\frac {2 \left (-2 c^{4} d^{4} e \,x^{4}+a \,c^{3} d^{3} e^{2} x^{3}-9 c^{4} d^{5} x^{3}+15 a^{2} c^{2} d^{2} e^{3} x^{2}-27 a \,c^{3} d^{4} e \,x^{2}+19 a^{3} c d \,e^{4} x -27 a^{2} c^{2} d^{3} e^{2} x +7 a^{4} e^{5}-9 a^{3} c \,d^{2} e^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{63 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{5}}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

1/e^8*(-2/9/(a*e^2-c*d^2)/(x+d/e)^8*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(7/2)+4/63*c*d*e/(a*e^2-c*d^2)^2/(
x+d/e)^7*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(7/2))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,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(c*d^2-%e^2*a>0)', see `assume?
` for more d

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (105) = 210\).
time = 32.91, size = 343, normalized size = 3.09 \begin {gather*} \frac {2 \, {\left (9 \, c^{4} d^{5} x^{3} - 19 \, a^{3} c d x e^{4} - 7 \, a^{4} e^{5} - 3 \, {\left (5 \, a^{2} c^{2} d^{2} x^{2} - 3 \, a^{3} c d^{2}\right )} e^{3} - {\left (a c^{3} d^{3} x^{3} - 27 \, a^{2} c^{2} d^{3} x\right )} e^{2} + {\left (2 \, c^{4} d^{4} x^{4} + 27 \, a c^{3} d^{4} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{63 \, {\left (5 \, c^{2} d^{8} x e + c^{2} d^{9} + a^{2} x^{5} e^{9} + 5 \, a^{2} d x^{4} e^{8} - 2 \, {\left (a c d^{2} x^{5} - 5 \, a^{2} d^{2} x^{3}\right )} e^{7} - 10 \, {\left (a c d^{3} x^{4} - a^{2} d^{3} x^{2}\right )} e^{6} + {\left (c^{2} d^{4} x^{5} - 20 \, a c d^{4} x^{3} + 5 \, a^{2} d^{4} x\right )} e^{5} + {\left (5 \, c^{2} d^{5} x^{4} - 20 \, a c d^{5} x^{2} + a^{2} d^{5}\right )} e^{4} + 10 \, {\left (c^{2} d^{6} x^{3} - a c d^{6} x\right )} e^{3} + 2 \, {\left (5 \, c^{2} d^{7} x^{2} - a c d^{7}\right )} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

2/63*(9*c^4*d^5*x^3 - 19*a^3*c*d*x*e^4 - 7*a^4*e^5 - 3*(5*a^2*c^2*d^2*x^2 - 3*a^3*c*d^2)*e^3 - (a*c^3*d^3*x^3
- 27*a^2*c^2*d^3*x)*e^2 + (2*c^4*d^4*x^4 + 27*a*c^3*d^4*x^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(5
*c^2*d^8*x*e + c^2*d^9 + a^2*x^5*e^9 + 5*a^2*d*x^4*e^8 - 2*(a*c*d^2*x^5 - 5*a^2*d^2*x^3)*e^7 - 10*(a*c*d^3*x^4
- a^2*d^3*x^2)*e^6 + (c^2*d^4*x^5 - 20*a*c*d^4*x^3 + 5*a^2*d^4*x)*e^5 + (5*c^2*d^5*x^4 - 20*a*c*d^5*x^2 + a^2
*d^5)*e^4 + 10*(c^2*d^6*x^3 - a*c*d^6*x)*e^3 + 2*(5*c^2*d^7*x^2 - a*c*d^7)*e^2)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^8,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,[0,0,5]%%%},[10]%%%}+%%%{%%{[%%%{-10,[0,1,4]%%%},0
]:[1,0,%%%{

________________________________________________________________________________________

Mupad [B]
time = 5.29, size = 3125, normalized size = 28.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^8,x)

[Out]

(((d*((8*c^5*d^6)/(63*e*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (4*c^4*d^4*(17*a*e^2 - 13*c*d^2))/(63*e*(a*
e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e))))/e + (4*c^3*d^3*(148*a^2*e^4 + 73*c^2*d^4 - 211*a*c*d^2*e^2))/(315*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 - (((2*a^
3*e^4)/(9*a*e^3 - 9*c*d^2*e) - (d*((d*((2*c^3*d^4)/(9*a*e^3 - 9*c*d^2*e) - (6*a*c^2*d^2*e^2)/(9*a*e^3 - 9*c*d^
2*e)))/e + (6*a^2*c*d*e^3)/(9*a*e^3 - 9*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)
^5 + (((2*c^4*d^5 + 26*a*c^3*d^3*e^2)/(63*e^2*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (4*c^4*d^5)/(9*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 + (((d*((d*((32
*c^7*d^8)/(945*e*(a*e^2 - c*d^2)^5) - (32*c^6*d^6*(17*a*e^2 - 14*c*d^2))/(945*e*(a*e^2 - c*d^2)^5)))/e + (32*c
^5*d^5*(116*a^2*e^4 + 85*c^2*d^4 - 198*a*c*d^2*e^2))/(945*e^2*(a*e^2 - c*d^2)^5)))/e - (32*a*c^4*d^4*(100*a^2*
e^4 + 85*c^2*d^4 - 184*a*c*d^2*e^2))/(945*e*(a*e^2 - c*d^2)^5))*(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)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^3*d^3*(8*a*e^2 - 5*c*d^2))
/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e))))/e + (20*c^4*d^6 - 80*a*c^3*d^4*e^2 + 72*a^2*c^2*d^2*e^4)/(9*e*(a*
e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e))))/e - (4*a*c*d*(11*a^2*e^4 + 5*c^2*d^4 - 15*a*c*d^2*e^2))/(9*(a*e^2 - c*d^
2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 + (((d*((4*c^4*d^5)/(9*e
*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (2*c^3*d^3*(7*a*e^2 - 3*c*d^2))/(9*e*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*
d^2*e))))/e + (20*c^4*d^6 - 82*a*c^3*d^4*e^2 + 90*a^2*c^2*d^2*e^4)/(63*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 + (((1152*c^7*d^10 - 1360*a*c^6*d^8*e^2 - 1136
*a^2*c^5*d^6*e^4 + 1376*a^3*c^4*d^4*e^6)/(945*e^3*(a*e^2 - c*d^2)^5) - (d*((d*((32*c^7*d^8)/(945*e*(a*e^2 - c*
d^2)^5) - (16*c^6*d^6*(31*a*e^2 - 25*c*d^2))/(945*e*(a*e^2 - c*d^2)^5)))/e + (16*c^5*d^5*(187*a^2*e^4 + 131*c^
2*d^4 - 312*a*c*d^2*e^2))/(945*e^2*(a*e^2 - c*d^2)^5)))/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)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (8*c^4*d^4*(4*a*e^2 - 3*c*d^2))/(
21*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))))/e + (8*c^3*d^3*(50*a^2*e^4 + 29*c^2*d^4 - 76*a*c*d^2*e^2))/(63*e
*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))))/e - (8*a*c^2*d^2*(39*a^2*e^4 + 29*c^2*d^4 - 67*a*c*d^2*e^2))/(63*(
a*e^2 - c*d^2)^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
*((16*c^6*d^7)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (16*c^5*d^5*(5*a*e^2 - 4*c*d^2))/(105*(a*e^2 -
c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (16*c^4*d^4*(86*a^2*e^4 + 59*c^2*d^4 - 142*a*c*d^2*e^2))/(315*e*(a*e^2 -
c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e - (16*a*c^3*d^3*(72*a^2*e^4 + 59*c^2*d^4 - 130*a*c*d^2*e^2))/(315*(a*e^2
- c*d^2)^3*(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*((16*c^6*
d^7)/(315*e^2*(a*e^2 - c*d^2)^4) - (8*c^5*d^5*(25*a*e^2 - 21*c*d^2))/(315*e^2*(a*e^2 - c*d^2)^4)))/e + (8*c^4*
d^4*(236*a^2*e^4 + 167*c^2*d^4 - 397*a*c*d^2*e^2))/(945*e^3*(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) - (((d*((d*((4*c^4*d^5)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (2*c^3*d^3*(7*a
*e^2 - c*d^2))/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*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
))/(9*e*(a*e^2 - c*d^2)*(7*a*e^3 - 7*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)/(9*e^2*(a*e
^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((d*((d*((8*
c^5*d^6)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (4*c^4*d^4*(17*a*e^2 - 11*c*d^2))/(63*(a*e^2 - c*d^2)^
2*(5*a*e^3 - 5*c*d^2*e))))/e + (8*c^3*d^3*(22*a^2*e^4 + 8*c^2*d^4 - 27*a*c*d^2*e^2))/(63*e*(a*e^2 - c*d^2)^2*(
5*a*e^3 - 5*c*d^2*e))))/e - (44*c^5*d^8 - 68*a*c^4*d^6*e^2 - 40*a^2*c^3*d^4*e^4 + 72*a^3*c^2*d^2*e^6)/(63*e^2*
(a*e^2 - c*d^2)^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*((16*c^6*d^7)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^5*d^5*(25*a*e^2 - 19*c*d^2))/(315*(a*e^2
- c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (16*c^4*d^4*(56*a^2*e^4 + 34*c^2*d^4 - 87*a*c*d^2*e^2))/(315*e*(a*e^2
- c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e - (312*c^6*d^9 - 392*a*c^5*d^7*e^2 - 304*a^2*c^4*d^5*e^4 + 400*a^3*c^3*d
^3*e^6)/(315*e^2*(a*e^2 - c*d^2)^3*(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 + (((284*c^5*d^6 - 164*a*c^4*d^4*e^2)/(945*e^3*(a*e^2 - c*d^2)^3) - (8*c^5*d^6)/(63*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) + (32*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*
e*x^2)^(1/2))/(189*e^3*(a*e^2 - c*d^2)^2*(d + e*x))

________________________________________________________________________________________