3.1.24 \(\int \frac {x^2 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [24]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {810, 792, 197} \begin {gather*} \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 {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 197

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 792

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 810

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 = 0.31, 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]

Integrate[(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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Maple [A]
time = 0.07, size = 147, normalized size = 1.56

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (2 e^{4} x^{4}-2 d \,e^{3} x^{3}-3 d^{2} x^{2} e^{2}-2 d^{3} e x +2 d^{4}\right )}{15 d^{3} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(77\)
trager	\(-\frac {\left (2 e^{4} x^{4}-2 d \,e^{3} x^{3}-3 d^{2} x^{2} e^{2}-2 d^{3} e x +2 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} e^{3} \left (-e x +d \right )^{3} \left (e x +d \right )^{2}}\)	\(79\)
default	\(e \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+d \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\)	\(147\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 102, normalized size = 1.09 \begin {gather*} \frac {x^{2} e^{\left (-1\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, d^{2} e^{\left (-3\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {x e^{\left (-2\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} - \frac {2 \, x e^{\left (-2\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{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="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (79) = 158\).
time = 4.24, size = 161, normalized size = 1.71 \begin {gather*} -\frac {2 \, x^{5} e^{5} - 2 \, d x^{4} e^{4} - 4 \, d^{2} x^{3} e^{3} + 4 \, d^{3} x^{2} e^{2} + 2 \, d^{4} x e - 2 \, d^{5} - {\left (2 \, x^{4} e^{4} - 2 \, d x^{3} e^{3} - 3 \, d^{2} x^{2} e^{2} - 2 \, d^{3} x e + 2 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{5} e^{8} - d^{4} x^{4} e^{7} - 2 \, d^{5} x^{3} e^{6} + 2 \, d^{6} x^{2} e^{5} + d^{7} x e^{4} - 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*x^5*e^5 - 2*d*x^4*e^4 - 4*d^2*x^3*e^3 + 4*d^3*x^2*e^2 + 2*d^4*x*e - 2*d^5 - (2*x^4*e^4 - 2*d*x^3*e^3
- 3*d^2*x^2*e^2 - 2*d^3*x*e + 2*d^4)*sqrt(-x^2*e^2 + d^2))/(d^3*x^5*e^8 - d^4*x^4*e^7 - 2*d^5*x^3*e^6 + 2*d^6*
x^2*e^5 + d^7*x*e^4 - d^8*e^3)

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Sympy [C] Result contains complex when optimal does not.
time = 7.49, 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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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]

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

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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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