3.20.29 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^7} \, dx\) [1929]

Optimal. Leaf size=171 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 \left (c d^2-a e^2\right )^3 (d+e x)^5} \]

[Out]

2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)/(e*x+d)^7+8/63*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(5/2)/(-a*e^2+c*d^2)^2/(e*x+d)^6+16/315*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e^2+c*d^2)^3/(e*
x+d)^5

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Rubi [A]
time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \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 )^{5/2}}{9 (d+e x)^7 \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)^(3/2)/(d + e*x)^7,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^7) + (8*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^6) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(5/2))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^5)

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 )^{3/2}}{(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 )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {(4 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, 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 )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^7}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^6}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 \left (c d^2-a e^2\right )^3 (d+e x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 94, normalized size = 0.55 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (35 a^2 e^4-10 a c d e^2 (9 d+2 e x)+c^2 d^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )}{315 \left (c d^2-a e^2\right )^3 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(35*a^2*e^4 - 10*a*c*d*e^2*(9*d + 2*e*x) + c^2*d^2*(63*d^2 + 36*d*e*x + 8*e
^2*x^2)))/(315*(c*d^2 - a*e^2)^3*(d + e*x)^7)

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Maple [A]
time = 0.71, size = 212, normalized size = 1.24

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 e^{2} x^{2} c^{2} d^{2}-20 a c d \,e^{3} x +36 c^{2} d^{3} e x +35 a^{2} e^{4}-90 a c \,d^{2} e^{2}+63 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{6} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}\)	\(146\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}}{e^{7}}\)	\(212\)
trager	\(-\frac {2 \left (8 c^{4} d^{4} e^{2} x^{4}-4 a \,c^{3} d^{3} e^{3} x^{3}+36 c^{4} d^{5} e \,x^{3}+3 a^{2} c^{2} d^{2} e^{4} x^{2}-18 a \,c^{3} d^{4} e^{2} x^{2}+63 c^{4} d^{6} x^{2}+50 d \,e^{5} c \,a^{3} x -144 a^{2} c^{2} d^{3} e^{3} x +126 a \,c^{3} d^{5} e x +35 a^{4} e^{6}-90 a^{3} c \,d^{2} e^{4}+63 a^{2} c^{2} d^{4} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \left (e x +d \right )^{5}}\)	\(234\)

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)^(3/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

1/e^7*(-2/9/(a*e^2-c*d^2)/(x+d/e)^7*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-4/9*c*d*e/(a*e^2-c*d^2)*(-2/
7/(a*e^2-c*d^2)/(x+d/e)^6*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+4/35*c*d*e/(a*e^2-c*d^2)^2/(x+d/e)^5*(
c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)))

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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)^(3/2)/(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(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (162) = 324\).
time = 28.64, size = 470, normalized size = 2.75 \begin {gather*} \frac {2 \, {\left (63 \, c^{4} d^{6} x^{2} + 50 \, a^{3} c d x e^{5} + 35 \, a^{4} e^{6} + 3 \, {\left (a^{2} c^{2} d^{2} x^{2} - 30 \, a^{3} c d^{2}\right )} e^{4} - 4 \, {\left (a c^{3} d^{3} x^{3} + 36 \, a^{2} c^{2} d^{3} x\right )} e^{3} + {\left (8 \, c^{4} d^{4} x^{4} - 18 \, a c^{3} d^{4} x^{2} + 63 \, a^{2} c^{2} d^{4}\right )} e^{2} + 18 \, {\left (2 \, c^{4} d^{5} x^{3} + 7 \, a c^{3} d^{5} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{315 \, {\left (5 \, c^{3} d^{10} x e + c^{3} d^{11} - a^{3} x^{5} e^{11} - 5 \, a^{3} d x^{4} e^{10} + {\left (3 \, a^{2} c d^{2} x^{5} - 10 \, a^{3} d^{2} x^{3}\right )} e^{9} + 5 \, {\left (3 \, a^{2} c d^{3} x^{4} - 2 \, a^{3} d^{3} x^{2}\right )} e^{8} - {\left (3 \, a c^{2} d^{4} x^{5} - 30 \, a^{2} c d^{4} x^{3} + 5 \, a^{3} d^{4} x\right )} e^{7} - {\left (15 \, a c^{2} d^{5} x^{4} - 30 \, a^{2} c d^{5} x^{2} + a^{3} d^{5}\right )} e^{6} + {\left (c^{3} d^{6} x^{5} - 30 \, a c^{2} d^{6} x^{3} + 15 \, a^{2} c d^{6} x\right )} e^{5} + {\left (5 \, c^{3} d^{7} x^{4} - 30 \, a c^{2} d^{7} x^{2} + 3 \, a^{2} c d^{7}\right )} e^{4} + 5 \, {\left (2 \, c^{3} d^{8} x^{3} - 3 \, a c^{2} d^{8} x\right )} e^{3} + {\left (10 \, c^{3} d^{9} x^{2} - 3 \, a c^{2} d^{9}\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)^(3/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

2/315*(63*c^4*d^6*x^2 + 50*a^3*c*d*x*e^5 + 35*a^4*e^6 + 3*(a^2*c^2*d^2*x^2 - 30*a^3*c*d^2)*e^4 - 4*(a*c^3*d^3*
x^3 + 36*a^2*c^2*d^3*x)*e^3 + (8*c^4*d^4*x^4 - 18*a*c^3*d^4*x^2 + 63*a^2*c^2*d^4)*e^2 + 18*(2*c^4*d^5*x^3 + 7*
a*c^3*d^5*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(5*c^3*d^10*x*e + c^3*d^11 - a^3*x^5*e^11 - 5*a^3*
d*x^4*e^10 + (3*a^2*c*d^2*x^5 - 10*a^3*d^2*x^3)*e^9 + 5*(3*a^2*c*d^3*x^4 - 2*a^3*d^3*x^2)*e^8 - (3*a*c^2*d^4*x
^5 - 30*a^2*c*d^4*x^3 + 5*a^3*d^4*x)*e^7 - (15*a*c^2*d^5*x^4 - 30*a^2*c*d^5*x^2 + a^3*d^5)*e^6 + (c^3*d^6*x^5
- 30*a*c^2*d^6*x^3 + 15*a^2*c*d^6*x)*e^5 + (5*c^3*d^7*x^4 - 30*a*c^2*d^7*x^2 + 3*a^2*c*d^7)*e^4 + 5*(2*c^3*d^8
*x^3 - 3*a*c^2*d^8*x)*e^3 + (10*c^3*d^9*x^2 - 3*a*c^2*d^9)*e^2)

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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)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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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)^(3/2)/(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,[0,0,5]%%%},[10]%%%}+%%%{%%{[%%%{-10,[0,1,4]%%%},0
]:[1,0,%%%{

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Mupad [B]
time = 3.85, size = 2067, normalized size = 12.09 \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)^(3/2)/(d + e*x)^7,x)

[Out]

(((d*((4*c^3*d^4)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (2*c^2*d^2*(5*a*e^2 - c*d^2))/(9*(a*e^2 - c*d^2)
*(7*a*e^3 - 7*c*d^2*e))))/e + (2*a*c^2*d^3*e^2 - 2*c^3*d^5 + 4*a^2*c*d*e^4)/(9*e*(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 - (((148*c^4*d^5 - 188*a*c^3*d^3*e^2)/(3
15*e*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) + (8*c^4*d^5)/(63*e*(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*((16*c^5*d^6)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3
- 3*c*d^2*e)) - (32*c^4*d^4*(7*a*e^2 - 6*c*d^2))/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (16*a*c^3
*d^3*e*(13*a*e^2 - 12*c*d^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 - (((18*c^3*d^4 - 46*a*c^2*d^2*e^2)/(63*e*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) + (
4*c^3*d^4)/(9*e*(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 - (((16*c^5*d^6)/(315*e^2*(a*e^2 - c*d^2)^4) - (8*c^4*d^4*(47*a*e^2 - 41*c*d^2))/(945*e^2*(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*((72*c^4*d^5 - 88*a*c^3*d^3*e^2)/(63*(a*e
^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) + (8*c^4*d^5)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e))))/e + (8*a*c^2
*d^2*e*(10*a*e^2 - 9*c*d^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*((32*c^6*d^7)/(945*e*(a*e^2 - c*d^2)^5) - (16*c^5*d^5*(29*a*e^2 - 25*c*d^2))/(9
45*e*(a*e^2 - c*d^2)^5)))/e + (16*c^4*d^4*(14*a^2*e^4 - 13*c^2*d^4 + a*c*d^2*e^2))/(945*e^2*(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*((16*c^5*d^6)/(315*(a*e^2 - c*d^2)^3*(3*a*e^
3 - 3*c*d^2*e)) - (8*c^4*d^4*(23*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 + (8*c^3
*d^3*(11*a^2*e^4 - 10*c^2*d^4 + a*c*d^2*e^2))/(315*e*(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 - (((2*a^2*e^3)/(9*a*e^3 - 9*c*d^2*e) + (d*((2*c^2*d^3)/(9*a*e^3 -
9*c*d^2*e) - (4*a*c*d*e^2)/(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 - (((d*((32*c^6*d^7)/(945*e*(a*e^2 - c*d^2)^5) - (64*c^5*d^5*(8*a*e^2 - 7*c*d^2))/(945*e*(a*e^2 - c*d^2)^5
)))/e + (32*a*c^4*d^4*(15*a*e^2 - 14*c*d^2))/(945*(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*((20*c^3*d^4 - 28*a*c^2*d^2*e^2)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) + (4*c^3*d^
4)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e))))/e + (4*a*c*d*e*(6*a*e^2 - 5*c*d^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*((8*c^4*d^5)/(63*(a*e^2 - c*
d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (4*c^3*d^3*(15*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 + (4*a*c^3*d^4*e^2 - 24*c^4*d^6 + 28*a^2*c^2*d^2*e^4)/(63*e*(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 - (8*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(
1/2))/(45*e^2*(a*e^2 - c*d^2)^3*(d + e*x)) - (44*c^3*d^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(315*e
*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2)

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