3.1061 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=41 \[ -\frac {c^3}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

-1/2*c^3/e/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {642, 607} \[ -\frac {c^3}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c^3/(2*e*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 642

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

Rubi steps

\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=c^4 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {c^3}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.66 \[ -\frac {\left (c (d+e x)^2\right )^{5/2}}{2 e (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c*(d + e*x)^2)^(5/2)/(e*(d + e*x)^7)

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fricas [A]  time = 0.92, size = 60, normalized size = 1.46 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c^{2}}{2 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-105*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+
c*x^2*exp(2))-sqrt(c*exp(2))*x)^13*exp(1)^12+1365*c^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-
sqrt(c*exp(2))*x)^12*d*exp(1)^11-336*c^3*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt
(c*exp(2))*x)^12*d*exp(1)^5-700*c^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^11*d^2*exp(1)^1
2-7490*c^4*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^11*d^2*exp(1)^10+2912*c^4*exp(2)^
3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^11*d^2*exp(1)^6+1120*c^4*exp(2)^4*(sqrt(c*d^2+2*c
*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^11*d^2*exp(1)^4+7700*c^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)
+c*x^2*exp(2))-sqrt(c*exp(2))*x)^10*d^3*exp(1)^11+22330*c^4*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c
*x^2*exp(2))-sqrt(c*exp(2))*x)^10*d^3*exp(1)^9-12096*c^4*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*
x^2*exp(2))-sqrt(c*exp(2))*x)^10*d^3*exp(1)^7-7840*c^4*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^
2*exp(2))-sqrt(c*exp(2))*x)^10*d^3*exp(1)^5-2240*c^4*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*
exp(2))-sqrt(c*exp(2))*x)^10*d^3*exp(1)^3-1981*c^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^
9*d^4*exp(1)^12-34538*c^5*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^4*exp(1)^10-89
88*c^5*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^4*exp(1)^8+29792*c^5*exp(2)^3*(
sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^4*exp(1)^6+11872*c^5*exp(2)^4*(sqrt(c*d^2+2*c*d*
x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^4*exp(1)^4+2688*c^5*exp(2)^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp
(2))-sqrt(c*exp(2))*x)^9*d^4*exp(1)^2+17829*c^5*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c
*exp(2))*x)^8*d^5*exp(1)^11+35042*c^5*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*ex
p(2))*x)^8*d^5*exp(1)^9-36008*c^5*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(
2))*x)^8*d^5*exp(1)^7-34608*c^5*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2)
)*x)^8*d^5*exp(1)^5-11648*c^5*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*
x)^8*d^5*exp(1)^3-1792*c^5*exp(2)^5*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^
8*d^5*exp(1)-3072*c^6*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^6*exp(1)^12-20628*c^6*exp
(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^6*exp(1)^10+22352*c^6*exp(2)^2*(sqrt(c*d^2+
2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^6*exp(1)^8+54832*c^6*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*
x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^6*exp(1)^6+25152*c^6*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(
c*exp(2))*x)^7*d^6*exp(1)^4+6784*c^6*exp(2)^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^6
*exp(1)^2+512*c^6*exp(2)^6*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^6-23548*c^6*exp(2)*s
qrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^7*exp(1)^9-58688*c^6*exp(2)^2*sqr
t(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^7*exp(1)^7-34608*c^6*exp(2)^3*sqrt(
c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^7*exp(1)^5-11648*c^6*exp(2)^4*sqrt(c*
exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^7*exp(1)^3-1792*c^6*exp(2)^5*sqrt(c*exp
(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^7*exp(1)+1981*c^7*(sqrt(c*d^2+2*c*d*x*exp(
1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^8*exp(1)^12+24332*c^7*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-s
qrt(c*exp(2))*x)^5*d^8*exp(1)^10+55272*c^7*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)
^5*d^8*exp(1)^8+34832*c^7*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^8*exp(1)^6+1
1872*c^7*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^8*exp(1)^4+2688*c^7*exp(2)^5*
(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^8*exp(1)^2-9905*c^7*sqrt(c*exp(2))*(sqrt(c*d^2+
2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^9*exp(1)^11-41020*c^7*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c
*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^9*exp(1)^9-30240*c^7*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d
*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^9*exp(1)^7-7840*c^7*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*
exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^9*exp(1)^5-2240*c^7*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp
(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^9*exp(1)^3+700*c^8*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp
(2))*x)^3*d^10*exp(1)^12+16310*c^8*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^10*ex
p(1)^10+21840*c^8*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^10*exp(1)^8+3920*c^8
*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^10*exp(1)^6+1120*c^8*exp(2)^4*(sqrt(c
*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^10*exp(1)^4-2100*c^8*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*
x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^11*exp(1)^11-9310*c^8*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*e
xp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^11*exp(1)^9-2240*c^8*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp
(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^11*exp(1)^7-336*c^8*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)
+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^11*exp(1)^5+105*c^9*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2
))*x)*d^12*exp(1)^12+1470*c^9*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^12*exp(1)^10
+980*c^9*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^12*exp(1)^8+112*c^9*exp(2)^3*(s
qrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^12*exp(1)^6-105*c^9*sqrt(c*exp(2))*d^13*exp(1)^11-7
0*c^9*exp(2)*sqrt(c*exp(2))*d^13*exp(1)^9-56*c^9*exp(2)^2*sqrt(c*exp(2))*d^13*exp(1)^7)/336/d/exp(1)^6/(-(sqrt
(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*exp(1)+2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x
^2*exp(2))-sqrt(c*exp(2))*x)*d-c*d^2*exp(1))^7+5*c^3/8/d/2/d/sqrt(c*exp(1)^2-c*exp(2))*atan((-d*sqrt(c*exp(2))
+(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1))/d/sqrt(c*exp(1)^2-c*exp(2))))

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maple [A]  time = 0.05, size = 35, normalized size = 0.85 \[ -\frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{2 \left (e x +d \right )^{7} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(e*x+d)^7/e*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.45, size = 37, normalized size = 0.90 \[ -\frac {c^2\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,e\,{\left (d+e\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^2*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2))/(2*e*(d + e*x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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