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

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

[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))

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Rubi [A]  time = 0.0376555, antiderivative size = 100, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.039272, 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]

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

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Maple [A]  time = 0.043, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{2}{x}^{2}+6\,dex+7\,{d}^{2} \right ) }{15\,e{d}^{3} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.11496, size = 216, normalized size = 2.16 \begin{align*} -\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 )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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