3.149 \(\int \frac{x^2}{(d+e x) (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=123 \[ -\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

-x^2/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (2*(d + 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105*d
^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0575934, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {855, 778, 192, 191} \[ -\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

-x^2/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (2*(d + 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105*d
^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*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 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{x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{x (2 d+4 e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d e}\\ &=-\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d e^2}\\ &=-\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 x}{105 d^3 e^2 \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}{105 d^3 e^2}\\ &=-\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0842236, size = 104, normalized size = 0.85 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-15 d^4 e^2 x^2+20 d^3 e^3 x^3+20 d^2 e^4 x^4+6 d^5 e x+6 d^6-8 d e^5 x^5-8 e^6 x^6\right )}{105 d^5 e^3 (d-e x)^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(6*d^6 + 6*d^5*e*x - 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 20*d^2*e^4*x^4 - 8*d*e^5*x^5 - 8*e
^6*x^6))/(105*d^5*e^3*(d - e*x)^3*(d + e*x)^4)

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Maple [A]  time = 0.049, size = 92, normalized size = 0.8 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( -8\,{e}^{6}{x}^{6}-8\,{e}^{5}{x}^{5}d+20\,{e}^{4}{x}^{4}{d}^{2}+20\,{x}^{3}{d}^{3}{e}^{3}-15\,{x}^{2}{d}^{4}{e}^{2}+6\,x{d}^{5}e+6\,{d}^{6} \right ) }{105\,{d}^{5}{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{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)^(7/2),x)

[Out]

1/105*(-e*x+d)*(-8*e^6*x^6-8*d*e^5*x^5+20*d^2*e^4*x^4+20*d^3*e^3*x^3-15*d^4*e^2*x^2+6*d^5*e*x+6*d^6)/d^5/e^3/(
-e^2*x^2+d^2)^(7/2)

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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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.06759, size = 485, normalized size = 3.94 \begin{align*} \frac{6 \, e^{7} x^{7} + 6 \, d e^{6} x^{6} - 18 \, d^{2} e^{5} x^{5} - 18 \, d^{3} e^{4} x^{4} + 18 \, d^{4} e^{3} x^{3} + 18 \, d^{5} e^{2} x^{2} - 6 \, d^{6} e x - 6 \, d^{7} +{\left (8 \, e^{6} x^{6} + 8 \, d e^{5} x^{5} - 20 \, d^{2} e^{4} x^{4} - 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} - 6 \, d^{5} e x - 6 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{105 \,{\left (d^{5} e^{10} x^{7} + d^{6} e^{9} x^{6} - 3 \, d^{7} e^{8} x^{5} - 3 \, d^{8} e^{7} x^{4} + 3 \, d^{9} e^{6} x^{3} + 3 \, d^{10} e^{5} x^{2} - d^{11} e^{4} x - d^{12} 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)^(7/2),x, algorithm="fricas")

[Out]

1/105*(6*e^7*x^7 + 6*d*e^6*x^6 - 18*d^2*e^5*x^5 - 18*d^3*e^4*x^4 + 18*d^4*e^3*x^3 + 18*d^5*e^2*x^2 - 6*d^6*e*x
- 6*d^7 + (8*e^6*x^6 + 8*d*e^5*x^5 - 20*d^2*e^4*x^4 - 20*d^3*e^3*x^3 + 15*d^4*e^2*x^2 - 6*d^5*e*x - 6*d^6)*sq
rt(-e^2*x^2 + d^2))/(d^5*e^10*x^7 + d^6*e^9*x^6 - 3*d^7*e^8*x^5 - 3*d^8*e^7*x^4 + 3*d^9*e^6*x^3 + 3*d^10*e^5*x
^2 - d^11*e^4*x - d^12*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{7}{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)**(7/2),x)

[Out]

Integral(x**2/((-(-d + e*x)*(d + e*x))**(7/2)*(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}, \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)^(7/2),x, algorithm="giac")

[Out]

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