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

Optimal. Leaf size=95 \[ -\frac{x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0533926, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {855, 778, 191} \[ -\frac{x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 855

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(d*(f + g*x)
^n*(a + c*x^2)^(p + 1))/(2*a*e*p*(d + e*x)), x] - Dist[1/(2*d*e*p), Int[(f + g*x)^(n - 1)*(a + c*x^2)^p*Simp[d
*g*n - e*f*(2*p + 1) - e*g*(n + 2*p + 1)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[n, 0] && ILtQ[n + 2*p, 0]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && 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{x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac{x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{x (2 d+2 e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=-\frac{x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e^2}\\ &=-\frac{x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0689794, size = 82, normalized size = 0.86 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-3 d^2 e^2 x^2+2 d^3 e x+2 d^4+2 d e^3 x^3+2 e^4 x^4\right )}{15 d^3 e^3 (d-e x)^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(2*d^4 + 2*d^3*e*x - 3*d^2*e^2*x^2 + 2*d*e^3*x^3 + 2*e^4*x^4))/(15*d^3*e^3*(d - e*x)^2*(d
+ e*x)^3)

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Maple [A]  time = 0.048, size = 70, normalized size = 0.7 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{4}{x}^{4}+2\,{x}^{3}d{e}^{3}-3\,{x}^{2}{d}^{2}{e}^{2}+2\,x{d}^{3}e+2\,{d}^{4} \right ) }{15\,{d}^{3}{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(-e*x+d)*(2*e^4*x^4+2*d*e^3*x^3-3*d^2*e^2*x^2+2*d^3*e*x+2*d^4)/d^3/e^3/(-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(x^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.74563, size = 339, normalized size = 3.57 \begin{align*} \frac{2 \, e^{5} x^{5} + 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x + 2 \, d^{5} +{\left (2 \, e^{4} x^{4} + 2 \, d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 2 \, d^{3} e x + 2 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{8} x^{5} + d^{4} e^{7} x^{4} - 2 \, d^{5} e^{6} x^{3} - 2 \, d^{6} e^{5} x^{2} + d^{7} e^{4} x + d^{8} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(2*e^5*x^5 + 2*d*e^4*x^4 - 4*d^2*e^3*x^3 - 4*d^3*e^2*x^2 + 2*d^4*e*x + 2*d^5 + (2*e^4*x^4 + 2*d*e^3*x^3 -
3*d^2*e^2*x^2 + 2*d^3*e*x + 2*d^4)*sqrt(-e^2*x^2 + d^2))/(d^3*e^8*x^5 + d^4*e^7*x^4 - 2*d^5*e^6*x^3 - 2*d^6*e
^5*x^2 + d^7*e^4*x + d^8*e^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[undef, undef, undef, undef, undef, 1]