3.22 \(\int \frac{x^4 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0522609, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {805, 266, 43} \[ \frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

Rule 805

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[(m*(c*d*f + a*e*g))/(2*a*c*(p + 1)), Int[(d + e*
x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplif
y[m + 2*p + 3], 0] && LtQ[p, -1]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 \int \frac{x^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=\frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{x}{\left (d^2-e^2 x\right )^{5/2}} \, dx,x,x^2\right )}{5 e}\\ &=\frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 \operatorname{Subst}\left (\int \left (\frac{d^2}{e^2 \left (d^2-e^2 x\right )^{5/2}}-\frac{1}{e^2 \left (d^2-e^2 x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{5 e}\\ &=\frac{x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0274661, size = 82, normalized size = 0.98 \[ \frac{-12 d^2 e^2 x^2-8 d^3 e x+8 d^4+12 d e^3 x^3+3 e^4 x^4}{15 d e^5 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(8*d^4 - 8*d^3*e*x - 12*d^2*e^2*x^2 + 12*d*e^3*x^3 + 3*e^4*x^4)/(15*d*e^5*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2
*x^2])

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Maple [A]  time = 0.048, size = 77, normalized size = 0.9 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( 3\,{x}^{4}{e}^{4}+12\,{x}^{3}d{e}^{3}-12\,{d}^{2}{x}^{2}{e}^{2}-8\,{d}^{3}xe+8\,{d}^{4} \right ) }{15\,d{e}^{5}} \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^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/15*(-e*x+d)*(e*x+d)^2*(3*e^4*x^4+12*d*e^3*x^3-12*d^2*e^2*x^2-8*d^3*e*x+8*d^4)/d/e^5/(-e^2*x^2+d^2)^(7/2)

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Maxima [B]  time = 0.962339, size = 215, normalized size = 2.56 \begin{align*} \frac{x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{4 \, d^{2} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{3 \, d^{3} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{5}} + \frac{d x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

x^4/((-e^2*x^2 + d^2)^(5/2)*e) + 1/2*d*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4/3*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*
e^3) - 3/10*d^3*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8/15*d^4/((-e^2*x^2 + d^2)^(5/2)*e^5) + 1/10*d*x/((-e^2*x^2 +
d^2)^(3/2)*e^4) + 1/5*x/(sqrt(-e^2*x^2 + d^2)*d*e^4)

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Fricas [B]  time = 1.96115, size = 343, normalized size = 4.08 \begin{align*} \frac{8 \, e^{5} x^{5} - 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 8 \, d^{5} -{\left (3 \, e^{4} x^{4} + 12 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} - 8 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d e^{10} x^{5} - d^{2} e^{9} x^{4} - 2 \, d^{3} e^{8} x^{3} + 2 \, d^{4} e^{7} x^{2} + d^{5} e^{6} x - d^{6} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(8*e^5*x^5 - 8*d*e^4*x^4 - 16*d^2*e^3*x^3 + 16*d^3*e^2*x^2 + 8*d^4*e*x - 8*d^5 - (3*e^4*x^4 + 12*d*e^3*x^
3 - 12*d^2*e^2*x^2 - 8*d^3*e*x + 8*d^4)*sqrt(-e^2*x^2 + d^2))/(d*e^10*x^5 - d^2*e^9*x^4 - 2*d^3*e^8*x^3 + 2*d^
4*e^7*x^2 + d^5*e^6*x - d^6*e^5)

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Sympy [C]  time = 38.0299, size = 420, normalized size = 5. \begin{align*} d \left (\begin{cases} - \frac{i x^{5}}{5 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{x^{5}}{5 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \frac{8 d^{4}}{15 d^{4} e^{6} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{20 d^{2} e^{2} x^{2}}{15 d^{4} e^{6} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{15 e^{4} x^{4}}{15 d^{4} e^{6} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{6}}{6 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((-I*x**5/(5*d**7*sqrt(-1 + e**2*x**2/d**2) - 10*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 5*d**3*
e**4*x**4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (x**5/(5*d**7*sqrt(1 - e**2*x**2/d**2) -
10*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 5*d**3*e**4*x**4*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise(
(8*d**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d
**2 - e**2*x**2)) - 20*d**2*e**2*x**2/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**
2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)) + 15*e**4*x**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*
e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**6/(6*(d**2)**(7/2)),
True))

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Giac [A]  time = 1.2162, size = 86, normalized size = 1.02 \begin{align*} -\frac{{\left (8 \, d^{4} e^{\left (-5\right )} +{\left (3 \, x^{2}{\left (\frac{x}{d} + 5 \, e^{\left (-1\right )}\right )} - 20 \, d^{2} e^{\left (-3\right )}\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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