3.843 \(\int \frac{1}{(d+e x)^2 (d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(4*x)/(21*d^4*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d^2*e*(d + e*x)*(d^2
- e^2*x^2)^(3/2)) + (8*x)/(21*d^6*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0353934, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(4*x)/(21*d^4*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d^2*e*(d + e*x)*(d^2
- e^2*x^2)^(3/2)) + (8*x)/(21*d^6*Sqrt[d^2 - e^2*x^2])

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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{5 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d}\\ &=-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d^2}\\ &=\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{21 d^4}\\ &=\frac{4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{21 d^6 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0612647, size = 93, normalized size = 0.81 \[ \frac{\sqrt{d^2-e^2 x^2} \left (24 d^3 e^2 x^2+4 d^2 e^3 x^3+9 d^4 e x-6 d^5-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^5 + 9*d^4*e*x + 24*d^3*e^2*x^2 + 4*d^2*e^3*x^3 - 16*d*e^4*x^4 - 8*e^5*x^5))/(21*d^6
*e*(d - e*x)^2*(d + e*x)^4)

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Maple [A]  time = 0.046, size = 88, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{5}{x}^{5}+16\,{e}^{4}{x}^{4}d-4\,{e}^{3}{x}^{3}{d}^{2}-24\,{e}^{2}{x}^{2}{d}^{3}-9\,x{d}^{4}e+6\,{d}^{5} \right ) }{ \left ( 21\,ex+21\,d \right ){d}^{6}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/21*(-e*x+d)*(8*e^5*x^5+16*d*e^4*x^4-4*d^2*e^3*x^3-24*d^3*e^2*x^2-9*d^4*e*x+6*d^5)/(e*x+d)/d^6/e/(-e^2*x^2+d
^2)^(5/2)

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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)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.35041, size = 413, normalized size = 3.59 \begin{align*} -\frac{6 \, e^{6} x^{6} + 12 \, d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} - 24 \, d^{3} e^{3} x^{3} - 6 \, d^{4} e^{2} x^{2} + 12 \, d^{5} e x + 6 \, d^{6} +{\left (8 \, e^{5} x^{5} + 16 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 24 \, d^{3} e^{2} x^{2} - 9 \, d^{4} e x + 6 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{21 \,{\left (d^{6} e^{7} x^{6} + 2 \, d^{7} e^{6} x^{5} - d^{8} e^{5} x^{4} - 4 \, d^{9} e^{4} x^{3} - d^{10} e^{3} x^{2} + 2 \, d^{11} e^{2} x + d^{12} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/21*(6*e^6*x^6 + 12*d*e^5*x^5 - 6*d^2*e^4*x^4 - 24*d^3*e^3*x^3 - 6*d^4*e^2*x^2 + 12*d^5*e*x + 6*d^6 + (8*e^5
*x^5 + 16*d*e^4*x^4 - 4*d^2*e^3*x^3 - 24*d^3*e^2*x^2 - 9*d^4*e*x + 6*d^5)*sqrt(-e^2*x^2 + d^2))/(d^6*e^7*x^6 +
2*d^7*e^6*x^5 - d^8*e^5*x^4 - 4*d^9*e^4*x^3 - d^10*e^3*x^2 + 2*d^11*e^2*x + d^12*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError