### 3.840 $$\int \frac{(d+e x)^2}{(d^2-e^2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=53 $\frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0133077, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {653, 191} $\frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[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{(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{3} \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0281359, size = 47, normalized size = 0.89 $\frac{(2 d-e x) (d+e x)}{3 d^2 e (d-e x) \sqrt{d^2-e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 44, normalized size = 0.8 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3} \left ( -ex+d \right ) \left ( -ex+2\,d \right ) }{3\,{d}^{2}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((e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.14806, size = 78, normalized size = 1.47 \begin{align*} \frac{2 \, x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, d}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.03883, size = 143, normalized size = 2.7 \begin{align*} \frac{2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} - \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x - 2 \, d\right )}}{3 \,{\left (d^{2} e^{3} x^{2} - 2 \, d^{3} e^{2} x + d^{4} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.45495, size = 65, normalized size = 1.23 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (x{\left (\frac{x^{2} e^{2}}{d^{2}} - 3\right )} - 2 \, d e^{\left (-1\right )}\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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