3.23 \(\int \frac{x^3 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=90 \[ \frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0420948, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {819, 778, 191} \[ \frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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^3 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x \left (2 d^3+3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e^3}\\ &=\frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0257207, size = 82, normalized size = 0.91 \[ \frac{3 d^2 e^2 x^2+2 d^3 e x-2 d^4-3 d e^3 x^3+3 e^4 x^4}{15 d^2 e^4 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.046, size = 77, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( -3\,{x}^{4}{e}^{4}+3\,{x}^{3}d{e}^{3}-3\,{x}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}xe+2\,{d}^{4} \right ) }{15\,{d}^{2}{e}^{4}} \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^3*(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+3*d*e^3*x^3-3*d^2*e^2*x^2-2*d^3*e*x+2*d^4)/d^2/e^4/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.985996, size = 181, normalized size = 2.01 \begin{align*} \frac{x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{3 \, d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{2 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{3}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.96432, size = 340, normalized size = 3.78 \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 (3 \, e^{4} x^{4} - 3 \, 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^{2} e^{9} x^{5} - d^{3} e^{8} x^{4} - 2 \, d^{4} e^{7} x^{3} + 2 \, d^{5} e^{6} x^{2} + d^{6} e^{5} x - d^{7} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(-e^2*x^2+d^2)^(7/2),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 + (3*e^4*x^4 - 3*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^2*e^9*x^5 - d^3*e^8*x^4 - 2*d^4*e^7*x^3 + 2*d^5*
e^6*x^2 + d^6*e^5*x - d^7*e^4)

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Sympy [B]  time = 13.4521, size = 338, normalized size = 3.76 \begin{align*} d \left (\begin{cases} - \frac{2 d^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{5 e^{2} x^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) + e \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((-2*d**2/(15*d**4*e**4*sqrt(d**2 - e**2*x**2) - 30*d**2*e**6*x**2*sqrt(d**2 - e**2*x**2) + 15*e**8
*x**4*sqrt(d**2 - e**2*x**2)) + 5*e**2*x**2/(15*d**4*e**4*sqrt(d**2 - e**2*x**2) - 30*d**2*e**6*x**2*sqrt(d**2
- e**2*x**2) + 15*e**8*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**4/(4*(d**2)**(7/2)), True)) + e*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*s
qrt(-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))

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Giac [A]  time = 1.20696, size = 78, normalized size = 0.87 \begin{align*} \frac{{\left (2 \, d^{3} e^{\left (-4\right )} -{\left (\frac{3 \, x^{3} e}{d^{2}} + 5 \, d e^{\left (-2\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^3*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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