3.184 \(\int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]

[Out]

-1/5*(-e^2*x^2+d^2)^(1/2)/d/e/(e*x+d)^3-2/15*(-e^2*x^2+d^2)^(1/2)/d^2/e/(e*x+d)^2-2/15*(-e^2*x^2+d^2)^(1/2)/d^
3/e/(e*x+d)

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Rubi [A]  time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[d^2 - e^2*x^2]/(5*d*e*(d + e*x)^3) - (2*Sqrt[d^2 - e^2*x^2])/(15*d^2*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^2*
x^2])/(15*d^3*e*(d + e*x))

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}+\frac {2 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{5 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}+\frac {2 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{15 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 52, normalized size = 0.52 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (7 d^2+6 d e x+2 e^2 x^2\right )}{15 d^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(Sqrt[d^2 - e^2*x^2]*(7*d^2 + 6*d*e*x + 2*e^2*x^2))/(d^3*e*(d + e*x)^3)

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fricas [A]  time = 0.91, size = 104, normalized size = 1.04 \[ -\frac {7 \, e^{3} x^{3} + 21 \, d e^{2} x^{2} + 21 \, d^{2} e x + 7 \, d^{3} + {\left (2 \, e^{2} x^{2} + 6 \, d e x + 7 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/15*(7*e^3*x^3 + 21*d*e^2*x^2 + 21*d^2*e*x + 7*d^3 + (2*e^2*x^2 + 6*d*e*x + 7*d^2)*sqrt(-e^2*x^2 + d^2))/(d^
3*e^4*x^3 + 3*d^4*e^3*x^2 + 3*d^5*e^2*x + d^6*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(1/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^2*exp(1)^6*exp(2)^3-2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp
(1)^10*exp(2)-2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^8*exp(2)^2+5*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^4*exp(2)^4-exp(1)^6*exp(2)^3+4*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^6+4*exp(2)^6-11/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*e
xp(1)^4*exp(2)^4/x/exp(2)+(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^8*exp(2)^2/x/exp(2))/((-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)-(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2)
)^2/(-d^3*exp(1)^9+2*d^3*exp(1)^5*exp(2)^2-d^3*exp(1)*exp(2)^4)+1/2*(-2*exp(1)^4*exp(2)^3-4*exp(2)^5)*atan((-1
/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/sqrt(-exp(1)^4+exp(2)^2)/(d
^3*exp(1)^9-2*d^3*exp(1)^5*exp(2)^2+d^3*exp(1)*exp(2)^4)

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maple [A]  time = 0.01, size = 55, normalized size = 0.55 \[ -\frac {\left (-e x +d \right ) \left (2 e^{2} x^{2}+6 d e x +7 d^{2}\right )}{15 \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(-e*x+d)*(2*e^2*x^2+6*d*e*x+7*d^2)/(e*x+d)^2/d^3/e/(-e^2*x^2+d^2)^(1/2)

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maxima [A]  time = 0.98, size = 128, normalized size = 1.28 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{2} x + d^{4} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

-1/5*sqrt(-e^2*x^2 + d^2)/(d*e^4*x^3 + 3*d^2*e^3*x^2 + 3*d^3*e^2*x + d^4*e) - 2/15*sqrt(-e^2*x^2 + d^2)/(d^2*e
^3*x^2 + 2*d^3*e^2*x + d^4*e) - 2/15*sqrt(-e^2*x^2 + d^2)/(d^3*e^2*x + d^4*e)

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mupad [B]  time = 2.62, size = 48, normalized size = 0.48 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (7\,d^2+6\,d\,e\,x+2\,e^2\,x^2\right )}{15\,d^3\,e\,{\left (d+e\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d^2 - e^2*x^2)^(1/2)*(7*d^2 + 2*e^2*x^2 + 6*d*e*x))/(15*d^3*e*(d + e*x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**3), x)

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