3.1038 \(\int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=41 \[ -\frac {c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

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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} \[ -\frac {c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^6,x]

[Out]

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

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 {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^6} \, dx &=c^3 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.66 \[ -\frac {\sqrt {c (d+e x)^2}}{4 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^6,x]

[Out]

-1/4*Sqrt[c*(d + e*x)^2]/(e*(d + e*x)^5)

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fricas [B]  time = 1.13, size = 79, normalized size = 1.93 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} \]

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)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/4*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*
x + d^5*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)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((105*c*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x
^2*exp(2))-sqrt(c*exp(2))*x)^9*exp(1)^8-60*c*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)
^9*exp(1)^6-945*c*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*exp(1)^7+540*c
*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)^8*d*exp(1)^5+490*c^2*(sqrt(c
*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^2*exp(1)^8+3010*c^2*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+
c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^2*exp(1)^6-1880*c^2*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^2*exp(1)^4-3430*c^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
)^6*d^3*exp(1)^7-3430*c^2*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)^6*d
^3*exp(1)^5+3320*c^2*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)^6*d^3*
exp(1)^3-240*c^2*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^3*exp(
1)+896*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^4*exp(1)^8+7986*c^3*exp(2)*(sqrt(c*d
^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^4*exp(1)^6-2364*c^3*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^4*exp(1)^4-944*c^3*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^4*exp(1)^2+96*c^3*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^4-4
480*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)^4*d^5*exp(1)^7-2590*c^3*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^5*exp(1)^5+1640*c^3*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^5*exp(1)^3-240*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)^4*d^5*exp(1)+790*c^4*(sqrt(c*d^2+2*c*d*x
*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^6*exp(1)^8+2710*c^4*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2)
)-sqrt(c*exp(2))*x)^3*d^6*exp(1)^6-200*c^4*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^6*exp(1)^4+480*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)^3*d^6*exp(1)^2-450
*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)^2*d^7*exp(1)^7-450*c^4*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)^2*d^7*exp(1)^5-720*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)^2*d^7*exp(1)^3-105*c^5*(sqrt(c*d^2+2*c*d*
x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^8*exp(1)^8-30*c^5*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-s
qrt(c*exp(2))*x)*d^8*exp(1)^6+540*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)*d^8*
exp(1)^4+105*c^5*sqrt(c*exp(2))*d^9*exp(1)^7-150*c^5*exp(2)*sqrt(c*exp(2))*d^9*exp(1)^5)/(-120*d^3*exp(1)^4+12
0*exp(2)*d^3*exp(1)^2)/(-(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))^5+(7*c*exp(1)^2-4*c*exp(2))/2/(4*d^
3*exp(1)^2-4*exp(2)*d^3)/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 \[ -\frac {\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}}{4 \left (e x +d \right )^{5} e} \]

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)^(1/2)/(e*x+d)^6,x)

[Out]

-1/4/(e*x+d)^5/e*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)

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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)^(1/2)/(e*x+d)^6,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 = 0.49, size = 34, normalized size = 0.83 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,e\,{\left (d+e\,x\right )}^5} \]

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)^(1/2)/(d + e*x)^6,x)

[Out]

-(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2)/(4*e*(d + e*x)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{6}}\, dx \]

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)**(1/2)/(e*x+d)**6,x)

[Out]

Integral(sqrt(c*(d + e*x)**2)/(d + e*x)**6, x)

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