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

Optimal. Leaf size=94 \[ \frac {x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}} \]

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Rubi [A]  time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {796, 778, 191} \begin {gather*} \frac {x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 796

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(x^2*(a*g - c*f*x)*(a + c*x^2)^(p
+ 1))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1
), x], x] /; FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -2]

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 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x \left (2 d^2 e-2 d e^2 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 d e \left (d^2-e^2 x^2\right )^{5/2}}-\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 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\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*}

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Mathematica [A]  time = 0.02, size = 82, normalized size = 0.87 \begin {gather*} \frac {-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) \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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IntegrateAlgebraic [A]  time = 0.52, size = 82, normalized size = 0.87 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-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\right )}{15 d^3 e^3 (d-e x)^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 0.42, size = 173, normalized size = 1.84 \begin {gather*} -\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 {gather*}

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]

-1/15*(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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giac [A]  time = 0.27, size = 64, normalized size = 0.68 \begin {gather*} \frac {{\left ({\left (x {\left (\frac {2 \, x^{2} e^{2}}{d^{3}} - \frac {5}{d}\right )} - 5 \, e^{\left (-1\right )}\right )} x^{2} + 2 \, d^{2} e^{\left (-3\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end {gather*}

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]

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

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maple [A]  time = 0.01, size = 77, normalized size = 0.82 \begin {gather*} -\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (2 x^{4} e^{4}-2 x^{3} d \,e^{3}-3 d^{2} x^{2} e^{2}-2 d^{3} x e +2 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} e^{3}} \end {gather*}

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]

-1/15*(-e*x+d)*(e*x+d)^2*(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)^(7/2)

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maxima [A]  time = 0.44, size = 112, normalized size = 1.19 \begin {gather*} \frac {x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{2}} - \frac {2 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{2}} \end {gather*}

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]

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

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mupad [B]  time = 2.61, size = 78, normalized size = 0.83 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (-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\right )}{15\,d^3\,e^3\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x)

[Out]

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

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sympy [C]  time = 21.31, size = 513, normalized size = 5.46 \begin {gather*} d \left (\begin {cases} - \frac {5 i d^{2} x^{3}}{15 d^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {2 i e^{2} x^{5}}{15 d^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{2} x^{3}}{15 d^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {2 e^{2} x^{5}}{15 d^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \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 ) \end {gather*}

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]

d*Piecewise((-5*I*d**2*x**3/(15*d**9*sqrt(-1 + e**2*x**2/d**2) - 30*d**7*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) +
15*d**5*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) + 2*I*e**2*x**5/(15*d**9*sqrt(-1 + e**2*x**2/d**2) - 30*d**7*e**
2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**5*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (5*
d**2*x**3/(15*d**9*sqrt(1 - e**2*x**2/d**2) - 30*d**7*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**5*e**4*x**4*s
qrt(1 - e**2*x**2/d**2)) - 2*e**2*x**5/(15*d**9*sqrt(1 - e**2*x**2/d**2) - 30*d**7*e**2*x**2*sqrt(1 - e**2*x**
2/d**2) + 15*d**5*e**4*x**4*sqrt(1 - e**2*x**2/d**2)), True)) + e*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/(1
5*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))

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