3.1.30 \(\int \frac {x^2 (d+e x)}{(d^2-e^2 x^2)^{9/2}} \, dx\) [30]

Optimal. Leaf size=121 \[ \frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}} \]

[Out]

1/7*x^2*(e*x+d)/d/e/(-e^2*x^2+d^2)^(7/2)-2/35*(-2*e*x+d)/d/e^3/(-e^2*x^2+d^2)^(5/2)-4/105*x/d^3/e^2/(-e^2*x^2+
d^2)^(3/2)-8/105*x/d^5/e^2/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {810, 792, 198, 197} \begin {gather*} \frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(9/2),x]

[Out]

(x^2*(d + e*x))/(7*d*e*(d^2 - e^2*x^2)^(7/2)) - (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 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 198

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 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 )^{9/2}} \, dx &=\frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {\int \frac {x \left (2 d^2 e-4 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d^2 e^2}\\ &=\frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\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 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\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 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\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*}

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Mathematica [A]
time = 0.39, size = 104, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-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\right )}{105 d^5 e^3 (d-e x)^4 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(9/2),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)^4*(d + e*x)^3)

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Maple [A]
time = 0.07, size = 173, normalized size = 1.43

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (-8 e^{6} x^{6}+8 d \,e^{5} x^{5}+20 d^{2} e^{4} x^{4}-20 d^{3} x^{3} e^{3}-15 d^{4} e^{2} x^{2}-6 d^{5} e x +6 d^{6}\right )}{105 d^{5} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\)	\(99\)
trager	\(-\frac {\left (-8 e^{6} x^{6}+8 d \,e^{5} x^{5}+20 d^{2} e^{4} x^{4}-20 d^{3} x^{3} e^{3}-15 d^{4} e^{2} x^{2}-6 d^{5} e x +6 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{5} \left (-e x +d \right )^{4} \left (e x +d \right )^{3} e^{3}}\)	\(101\)
default	\(e \left (\frac {x^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}-\frac {2 d^{2}}{35 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\right )+d \left (\frac {x}{6 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}-\frac {d^{2} \left (\frac {x}{7 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}}{d^{2}}\right )}{6 e^{2}}\right )\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x+d)/(-e^2*x^2+d^2)^(9/2),x,method=_RETURNVERBOSE)

[Out]

e*(1/5*x^2/e^2/(-e^2*x^2+d^2)^(7/2)-2/35*d^2/e^4/(-e^2*x^2+d^2)^(7/2))+d*(1/6*x/e^2/(-e^2*x^2+d^2)^(7/2)-1/6*d
^2/e^2*(1/7*x/d^2/(-e^2*x^2+d^2)^(7/2)+6/7/d^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 = 0.28, size = 123, normalized size = 1.02 \begin {gather*} \frac {x^{2} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}} + \frac {d x e^{\left (-2\right )}}{7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}} - \frac {2 \, d^{2} e^{\left (-3\right )}}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}} - \frac {x e^{\left (-2\right )}}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d} - \frac {4 \, x e^{\left (-2\right )}}{105 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} - \frac {8 \, x e^{\left (-2\right )}}{105 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5}} \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)^(9/2),x, algorithm="maxima")

[Out]

1/5*x^2*e^(-1)/(-x^2*e^2 + d^2)^(7/2) + 1/7*d*x*e^(-2)/(-x^2*e^2 + d^2)^(7/2) - 2/35*d^2*e^(-3)/(-x^2*e^2 + d^
2)^(7/2) - 1/35*x*e^(-2)/((-x^2*e^2 + d^2)^(5/2)*d) - 4/105*x*e^(-2)/((-x^2*e^2 + d^2)^(3/2)*d^3) - 8/105*x*e^
(-2)/(sqrt(-x^2*e^2 + d^2)*d^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (101) = 202\).
time = 3.22, size = 221, normalized size = 1.83 \begin {gather*} -\frac {6 \, x^{7} e^{7} - 6 \, d x^{6} e^{6} - 18 \, d^{2} x^{5} e^{5} + 18 \, d^{3} x^{4} e^{4} + 18 \, d^{4} x^{3} e^{3} - 18 \, d^{5} x^{2} e^{2} - 6 \, d^{6} x e + 6 \, d^{7} - {\left (8 \, x^{6} e^{6} - 8 \, d x^{5} e^{5} - 20 \, d^{2} x^{4} e^{4} + 20 \, d^{3} x^{3} e^{3} + 15 \, d^{4} x^{2} e^{2} + 6 \, d^{5} x e - 6 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{105 \, {\left (d^{5} x^{7} e^{10} - d^{6} x^{6} e^{9} - 3 \, d^{7} x^{5} e^{8} + 3 \, d^{8} x^{4} e^{7} + 3 \, d^{9} x^{3} e^{6} - 3 \, d^{10} x^{2} e^{5} - d^{11} x e^{4} + d^{12} 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)^(9/2),x, algorithm="fricas")

[Out]

-1/105*(6*x^7*e^7 - 6*d*x^6*e^6 - 18*d^2*x^5*e^5 + 18*d^3*x^4*e^4 + 18*d^4*x^3*e^3 - 18*d^5*x^2*e^2 - 6*d^6*x*
e + 6*d^7 - (8*x^6*e^6 - 8*d*x^5*e^5 - 20*d^2*x^4*e^4 + 20*d^3*x^3*e^3 + 15*d^4*x^2*e^2 + 6*d^5*x*e - 6*d^6)*s
qrt(-x^2*e^2 + d^2))/(d^5*x^7*e^10 - d^6*x^6*e^9 - 3*d^7*x^5*e^8 + 3*d^8*x^4*e^7 + 3*d^9*x^3*e^6 - 3*d^10*x^2*
e^5 - d^11*x*e^4 + d^12*e^3)

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Sympy [C] Result contains complex when optimal does not.
time = 7.37, size = 903, normalized size = 7.46 \begin {gather*} d \left (\begin {cases} \frac {35 i d^{4} x^{3}}{- 105 d^{13} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {28 i d^{2} e^{2} x^{5}}{- 105 d^{13} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {8 i e^{4} x^{7}}{- 105 d^{13} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {35 d^{4} x^{3}}{- 105 d^{13} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {28 d^{2} e^{2} x^{5}}{- 105 d^{13} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {8 e^{4} x^{7}}{- 105 d^{13} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {2 d^{2}}{- 35 d^{6} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 105 d^{4} e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}} - 105 d^{2} e^{8} x^{4} \sqrt {d^{2} - e^{2} x^{2}} + 35 e^{10} x^{6} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {7 e^{2} x^{2}}{- 35 d^{6} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 105 d^{4} e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}} - 105 d^{2} e^{8} x^{4} \sqrt {d^{2} - e^{2} x^{2}} + 35 e^{10} x^{6} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {9}{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)**(9/2),x)

[Out]

d*Piecewise((35*I*d**4*x**3/(-105*d**13*sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(-1 + e**2*x**2/d*
*2) - 315*d**9*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)) - 28*I*d**2
*e**2*x**5/(-105*d**13*sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) - 315*d**9*e*
*4*x**4*sqrt(-1 + e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)) + 8*I*e**4*x**7/(-105*d**13*
sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(-1 + e**2*
x**2/d**2) + 105*d**7*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-35*d**4*x**3/(-105*d**
13*sqrt(1 - e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(1 - e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(1 - e**2*
x**2/d**2) + 105*d**7*e**6*x**6*sqrt(1 - e**2*x**2/d**2)) + 28*d**2*e**2*x**5/(-105*d**13*sqrt(1 - e**2*x**2/d
**2) + 315*d**11*e**2*x**2*sqrt(1 - e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(1 - e**2*x**2/d**2) + 105*d**7*e
**6*x**6*sqrt(1 - e**2*x**2/d**2)) - 8*e**4*x**7/(-105*d**13*sqrt(1 - e**2*x**2/d**2) + 315*d**11*e**2*x**2*sq
rt(1 - e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(1 - e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(1 - e**2*x**2/d
**2)), True)) + e*Piecewise((2*d**2/(-35*d**6*e**4*sqrt(d**2 - e**2*x**2) + 105*d**4*e**6*x**2*sqrt(d**2 - e**
2*x**2) - 105*d**2*e**8*x**4*sqrt(d**2 - e**2*x**2) + 35*e**10*x**6*sqrt(d**2 - e**2*x**2)) - 7*e**2*x**2/(-35
*d**6*e**4*sqrt(d**2 - e**2*x**2) + 105*d**4*e**6*x**2*sqrt(d**2 - e**2*x**2) - 105*d**2*e**8*x**4*sqrt(d**2 -
e**2*x**2) + 35*e**10*x**6*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**4/(4*(d**2)**(9/2)), 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)^(9/2),x, algorithm="giac")

[Out]

integrate((x*e + d)*x^2/(-x^2*e^2 + d^2)^(9/2), x)

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Mupad [B]
time = 2.69, size = 164, normalized size = 1.36 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}}{56\,d^2\,e^3\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {2}{35\,e^3}-\frac {3\,x}{70\,d\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{56\,d^2\,e^3}+\frac {4\,x}{105\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{105\,d^5\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \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)^(9/2),x)

[Out]

(d^2 - e^2*x^2)^(1/2)/(56*d^2*e^3*(d - e*x)^4) - ((d^2 - e^2*x^2)^(1/2)*(2/(35*e^3) - (3*x)/(70*d*e^2)))/((d +
e*x)^3*(d - e*x)^3) - ((d^2 - e^2*x^2)^(1/2)*(1/(56*d^2*e^3) + (4*x)/(105*d^3*e^2)))/((d + e*x)^2*(d - e*x)^2
) - (8*x*(d^2 - e^2*x^2)^(1/2))/(105*d^5*e^2*(d + e*x)*(d - e*x))

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