3.9.82 \(\int \frac {1}{(d+e x)^4 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac {c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, 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}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(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 {1}{(d+e x)^4 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c^2 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c}{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 = 0.69 \begin {gather*} -\frac {1}{4 e (d+e x)^3 \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 1.28, size = 338, normalized size = 8.67 \begin {gather*} \frac {2 c^3 \left (c d^8 e-175 c d^4 e^5 x^4-704 c d^3 e^6 x^5-1056 c d^2 e^7 x^6-704 c d e^8 x^7-176 c e^9 x^8\right )+2 c^3 \sqrt {c e^2} \left (d^7-d^6 e x+d^5 e^2 x^2-d^4 e^3 x^3+176 d^3 e^4 x^4+528 d^2 e^5 x^5+528 d e^6 x^6+176 e^7 x^7\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{d^4 e x^4 \sqrt {c d^2+2 c d e x+c e^2 x^2} \left (-8 c^4 d^3 e^5-24 c^4 d^2 e^6 x-24 c^4 d e^7 x^2-8 c^4 e^8 x^3\right )+d^4 e x^4 \sqrt {c e^2} \left (8 c^4 d^4 e^4+32 c^4 d^3 e^5 x+48 c^4 d^2 e^6 x^2+32 c^4 d e^7 x^3+8 c^4 e^8 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c^3*Sqrt[c*e^2]*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]*(d^7 - d^6*e*x + d^5*e^2*x^2 - d^4*e^3*x^3 + 176*d^3*e^
4*x^4 + 528*d^2*e^5*x^5 + 528*d*e^6*x^6 + 176*e^7*x^7) + 2*c^3*(c*d^8*e - 175*c*d^4*e^5*x^4 - 704*c*d^3*e^6*x^
5 - 1056*c*d^2*e^7*x^6 - 704*c*d*e^8*x^7 - 176*c*e^9*x^8))/(d^4*e*x^4*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]*(-8*
c^4*d^3*e^5 - 24*c^4*d^2*e^6*x - 24*c^4*d*e^7*x^2 - 8*c^4*e^8*x^3) + d^4*e*Sqrt[c*e^2]*x^4*(8*c^4*d^4*e^4 + 32
*c^4*d^3*e^5*x + 48*c^4*d^2*e^6*x^2 + 32*c^4*d*e^7*x^3 + 8*c^4*e^8*x^4))

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fricas [B]  time = 0.39, size = 85, normalized size = 2.18 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (c e^{6} x^{5} + 5 \, c d e^{5} x^{4} + 10 \, c d^{2} e^{4} x^{3} + 10 \, c d^{3} e^{3} x^{2} + 5 \, c d^{4} e^{2} x + c d^{5} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-15*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*
2+c*d^2)-sqrt(c*exp(2))*x)^5*exp(1)^4+6*exp(2)*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^5*ex
p(1)^2+75*sqrt(c*exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^4*d*exp(1)^3-30*exp(2)*sqr
t(c*exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^4*d*exp(1)-40*c*(sqrt(c*exp(2)*x^2+c*d*
exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^3*d^2*exp(1)^4-94*c*exp(2)*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c
*exp(2))*x)^3*d^2*exp(1)^2+44*c*exp(2)^2*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^3*d^2+120*
c*sqrt(c*exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^2*d^3*exp(1)^3-30*c*exp(2)*sqrt(c*
exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^2*d^3*exp(1)-33*c^2*(sqrt(c*exp(2)*x^2+c*d*
exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)*d^4*exp(1)^4-12*c^2*exp(2)*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c
*exp(2))*x)*d^4*exp(1)^2+9*c^2*sqrt(c*exp(2))*d^5*exp(1)^3)/(6*d^3*exp(1)^2-6*exp(2)*d^3)/(-(sqrt(c*exp(2)*x^2
+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^2*exp(1)+2*sqrt(c*exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sq
rt(c*exp(2))*x)*d-c*d^2*exp(1))^3+(5*exp(1)^2-2*exp(2))/2/(d^3*exp(1)^2-exp(2)*d^3)/d/sqrt(c*exp(1)^2-c*exp(2)
)*atan((-d*sqrt(c*exp(2))+(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^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 = 0.90 \begin {gather*} -\frac {1}{4 \left (e x +d \right )^{3} \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.42, size = 61, normalized size = 1.56 \begin {gather*} -\frac {1}{4 \, {\left (\sqrt {c} e^{5} x^{4} + 4 \, \sqrt {c} d e^{4} x^{3} + 6 \, \sqrt {c} d^{2} e^{3} x^{2} + 4 \, \sqrt {c} d^{3} e^{2} x + \sqrt {c} d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

-1/4/(sqrt(c)*e^5*x^4 + 4*sqrt(c)*d*e^4*x^3 + 6*sqrt(c)*d^2*e^3*x^2 + 4*sqrt(c)*d^3*e^2*x + sqrt(c)*d^4*e)

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mupad [B]  time = 0.46, size = 37, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,c\,e\,{\left (d+e\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^4*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(c*(d + e*x)**2)*(d + e*x)**4), x)

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