3.9.60 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^6} \, dx\)

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

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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} \begin {gather*} -\frac {c^2}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

-c^2/(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 )^{3/2}}{(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 )^{3/2}} \, dx\\ &=-\frac {c^2}{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 \begin {gather*} -\frac {\left (c (d+e x)^2\right )^{3/2}}{2 e (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

-1/2*(c*(d + e*x)^2)^(3/2)/(e*(d + e*x)^5)

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IntegrateAlgebraic [B]  time = 0.93, size = 213, normalized size = 5.20 \begin {gather*} \frac {c^3 \left (c d^4 e-63 c d^2 e^3 x^2-128 c d e^4 x^3-64 c e^5 x^4\right )+c^3 \sqrt {c e^2} \left (d^3-d^2 e x+64 d e^2 x^2+64 e^3 x^3\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{d^2 e x^2 \sqrt {c e^2} \left (2 c^2 d^2 e^2+4 c^2 d e^3 x+2 c^2 e^4 x^2\right )+d^2 e x^2 \left (-2 c^2 d e^3-2 c^2 e^4 x\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(c^3*Sqrt[c*e^2]*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]*(d^3 - d^2*e*x + 64*d*e^2*x^2 + 64*e^3*x^3) + c^3*(c*d^4*
e - 63*c*d^2*e^3*x^2 - 128*c*d*e^4*x^3 - 64*c*e^5*x^4))/(d^2*e*x^2*(-2*c^2*d*e^3 - 2*c^2*e^4*x)*Sqrt[c*d^2 + 2
*c*d*e*x + c*e^2*x^2] + d^2*e*Sqrt[c*e^2]*x^2*(2*c^2*d^2*e^2 + 4*c^2*d*e^3*x + 2*c^2*e^4*x^2))

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fricas [A]  time = 0.40, size = 58, normalized size = 1.41 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{2 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end {gather*}

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)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c/(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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

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)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(-(15*c^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c
*x^2*exp(2))-sqrt(c*exp(2))*x)^9*exp(1)^8-135*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)^8*d*exp(1)^7+40*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)^8*d*exp(1)^3+70*c^3*(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+470*c^3*e
xp(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)^6-240*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)^7*d^2*exp(1)^4-80*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)^7*d^2*exp(1)^2-490*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)^6*d^3*exp(1)^7-130*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)^6*d^3*exp(1)^5+400*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)^6*d^3*exp(1)^3+80*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)^6*
d^3*exp(1)+128*c^4*(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+318*c^4*exp(2)*(s
qrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^4*exp(1)^6-492*c^4*exp(2)^2*(sqrt(c*d^2+2*c*d*x*e
xp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^4*exp(1)^4-272*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)^5*d^4*exp(1)^2-32*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)^5
*d^4+430*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)^4*d^5*exp(1)^5+4
00*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)^4*d^5*exp(1)^3+80*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)^4*d^5*exp(1)-70*c^5*(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)^8-430*c^5*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)^6-400*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)^3*d^6*exp(1)^4-80*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)^3*d^6*
exp(1)^2+210*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)^2*d^7*exp(1)^7+330*
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*exp(2))*x)^2*d^7*exp(1)^5+40*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)^2*d^7*exp(1)^3-15*c^6*(sqrt(c*
d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^8*exp(1)^8-150*c^6*exp(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)^6-20*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)*d^8*exp(1)^4+15*c^6*sqrt(c*exp(2))*d^9*exp(1)^7+10*c^6*exp(2)*sqrt(c*exp(2))*d^9*exp(1)^5)/40/d/exp(1)^4
/(-(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*ex
p(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d-c*d^2*exp(1))^5+3*c^2/4/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 \begin {gather*} -\frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{5} e} \end {gather*}

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)^(3/2)/(e*x+d)^6,x)

[Out]

-1/2/(e*x+d)^5/e*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

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

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)^(3/2)/(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 = 0.44, size = 35, normalized size = 0.85 \begin {gather*} -\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,e\,{\left (d+e\,x\right )}^3} \end {gather*}

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)^(3/2)/(d + e*x)^6,x)

[Out]

-(c*(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 \begin {gather*} \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}

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)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((c*(d + e*x)**2)**(3/2)/(d + e*x)**6, x)

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