3.1.31 \(\int \frac {x^2 (d+e x)}{(d^2-e^2 x^2)^{11/2}} \, dx\) [31]

Optimal. Leaf size=148 \[ \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}} \]

[Out]

1/9*x^2*(e*x+d)/d/e/(-e^2*x^2+d^2)^(9/2)-2/63*(-3*e*x+d)/d/e^3/(-e^2*x^2+d^2)^(7/2)-2/105*x/d^3/e^2/(-e^2*x^2+
d^2)^(5/2)-8/315*x/d^5/e^2/(-e^2*x^2+d^2)^(3/2)-16/315*x/d^7/e^2/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, 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)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(11/2),x]

[Out]

(x^2*(d + e*x))/(9*d*e*(d^2 - e^2*x^2)^(9/2)) - (2*(d - 3*e*x))/(63*d*e^3*(d^2 - e^2*x^2)^(7/2)) - (2*x)/(105*
d^3*e^2*(d^2 - e^2*x^2)^(5/2)) - (8*x)/(315*d^5*e^2*(d^2 - e^2*x^2)^(3/2)) - (16*x)/(315*d^7*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 )^{11/2}} \, dx &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {\int \frac {x \left (2 d^2 e-6 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{9 d^2 e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{21 d e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{105 d^3 e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{315 d^5 e^2}\\ &=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 126, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-10 d^8+10 d^7 e x+35 d^6 e^2 x^2+70 d^5 e^3 x^3-70 d^4 e^4 x^4-56 d^3 e^5 x^5+56 d^2 e^6 x^6+16 d e^7 x^7-16 e^8 x^8\right )}{315 d^7 e^3 (d-e x)^5 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(11/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-10*d^8 + 10*d^7*e*x + 35*d^6*e^2*x^2 + 70*d^5*e^3*x^3 - 70*d^4*e^4*x^4 - 56*d^3*e^5*x^5
+ 56*d^2*e^6*x^6 + 16*d*e^7*x^7 - 16*e^8*x^8))/(315*d^7*e^3*(d - e*x)^5*(d + e*x)^4)

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Maple [A]
time = 0.06, size = 199, normalized size = 1.34

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (16 e^{8} x^{8}-16 e^{7} x^{7} d -56 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} x^{4} e^{4}-70 x^{3} d^{5} e^{3}-35 d^{6} e^{2} x^{2}-10 d^{7} e x +10 d^{8}\right )}{315 d^{7} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\)	\(121\)
trager	\(-\frac {\left (16 e^{8} x^{8}-16 e^{7} x^{7} d -56 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} x^{4} e^{4}-70 x^{3} d^{5} e^{3}-35 d^{6} e^{2} x^{2}-10 d^{7} e x +10 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{315 d^{7} \left (-e x +d \right )^{5} \left (e x +d \right )^{4} e^{3}}\)	\(123\)
default	\(e \left (\frac {x^{2}}{7 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {2 d^{2}}{63 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\right )+d \left (\frac {x}{8 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {d^{2} \left (\frac {x}{9 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )}{8 e^{2}}\right )\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x+d)/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)

[Out]

e*(1/7*x^2/e^2/(-e^2*x^2+d^2)^(9/2)-2/63*d^2/e^4/(-e^2*x^2+d^2)^(9/2))+d*(1/8*x/e^2/(-e^2*x^2+d^2)^(9/2)-1/8*d
^2/e^2*(1/9*x/d^2/(-e^2*x^2+d^2)^(9/2)+8/9/d^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 = 0.36, size = 144, normalized size = 0.97 \begin {gather*} \frac {x^{2} e^{\left (-1\right )}}{7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {9}{2}}} + \frac {d x e^{\left (-2\right )}}{9 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {9}{2}}} - \frac {2 \, d^{2} e^{\left (-3\right )}}{63 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {9}{2}}} - \frac {x e^{\left (-2\right )}}{63 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d} - \frac {2 \, x e^{\left (-2\right )}}{105 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} - \frac {8 \, x e^{\left (-2\right )}}{315 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} - \frac {16 \, x e^{\left (-2\right )}}{315 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \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)^(11/2),x, algorithm="maxima")

[Out]

1/7*x^2*e^(-1)/(-x^2*e^2 + d^2)^(9/2) + 1/9*d*x*e^(-2)/(-x^2*e^2 + d^2)^(9/2) - 2/63*d^2*e^(-3)/(-x^2*e^2 + d^
2)^(9/2) - 1/63*x*e^(-2)/((-x^2*e^2 + d^2)^(7/2)*d) - 2/105*x*e^(-2)/((-x^2*e^2 + d^2)^(5/2)*d^3) - 8/315*x*e^
(-2)/((-x^2*e^2 + d^2)^(3/2)*d^5) - 16/315*x*e^(-2)/(sqrt(-x^2*e^2 + d^2)*d^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (122) = 244\).
time = 4.18, size = 281, normalized size = 1.90 \begin {gather*} -\frac {10 \, x^{9} e^{9} - 10 \, d x^{8} e^{8} - 40 \, d^{2} x^{7} e^{7} + 40 \, d^{3} x^{6} e^{6} + 60 \, d^{4} x^{5} e^{5} - 60 \, d^{5} x^{4} e^{4} - 40 \, d^{6} x^{3} e^{3} + 40 \, d^{7} x^{2} e^{2} + 10 \, d^{8} x e - 10 \, d^{9} - {\left (16 \, x^{8} e^{8} - 16 \, d x^{7} e^{7} - 56 \, d^{2} x^{6} e^{6} + 56 \, d^{3} x^{5} e^{5} + 70 \, d^{4} x^{4} e^{4} - 70 \, d^{5} x^{3} e^{3} - 35 \, d^{6} x^{2} e^{2} - 10 \, d^{7} x e + 10 \, d^{8}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{315 \, {\left (d^{7} x^{9} e^{12} - d^{8} x^{8} e^{11} - 4 \, d^{9} x^{7} e^{10} + 4 \, d^{10} x^{6} e^{9} + 6 \, d^{11} x^{5} e^{8} - 6 \, d^{12} x^{4} e^{7} - 4 \, d^{13} x^{3} e^{6} + 4 \, d^{14} x^{2} e^{5} + d^{15} x e^{4} - d^{16} 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)^(11/2),x, algorithm="fricas")

[Out]

-1/315*(10*x^9*e^9 - 10*d*x^8*e^8 - 40*d^2*x^7*e^7 + 40*d^3*x^6*e^6 + 60*d^4*x^5*e^5 - 60*d^5*x^4*e^4 - 40*d^6
*x^3*e^3 + 40*d^7*x^2*e^2 + 10*d^8*x*e - 10*d^9 - (16*x^8*e^8 - 16*d*x^7*e^7 - 56*d^2*x^6*e^6 + 56*d^3*x^5*e^5
+ 70*d^4*x^4*e^4 - 70*d^5*x^3*e^3 - 35*d^6*x^2*e^2 - 10*d^7*x*e + 10*d^8)*sqrt(-x^2*e^2 + d^2))/(d^7*x^9*e^12
- d^8*x^8*e^11 - 4*d^9*x^7*e^10 + 4*d^10*x^6*e^9 + 6*d^11*x^5*e^8 - 6*d^12*x^4*e^7 - 4*d^13*x^3*e^6 + 4*d^14*
x^2*e^5 + d^15*x*e^4 - d^16*e^3)

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Sympy [C] Result contains complex when optimal does not.
time = 15.58, size = 1401, normalized size = 9.47 \begin {gather*} d \left (\begin {cases} - \frac {105 i d^{6} x^{3}}{315 d^{17} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {126 i d^{4} e^{2} x^{5}}{315 d^{17} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {72 i d^{2} e^{4} x^{7}}{315 d^{17} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {16 i e^{6} x^{9}}{315 d^{17} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {105 d^{6} x^{3}}{315 d^{17} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {126 d^{4} e^{2} x^{5}}{315 d^{17} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {72 d^{2} e^{4} x^{7}}{315 d^{17} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {16 e^{6} x^{9}}{315 d^{17} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{15} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 1890 d^{13} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 1260 d^{11} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{9} e^{8} x^{8} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} - \frac {2 d^{2}}{63 d^{8} e^{4} \sqrt {d^{2} - e^{2} x^{2}} - 252 d^{6} e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 378 d^{4} e^{8} x^{4} \sqrt {d^{2} - e^{2} x^{2}} - 252 d^{2} e^{10} x^{6} \sqrt {d^{2} - e^{2} x^{2}} + 63 e^{12} x^{8} \sqrt {d^{2} - e^{2} x^{2}}} + \frac {9 e^{2} x^{2}}{63 d^{8} e^{4} \sqrt {d^{2} - e^{2} x^{2}} - 252 d^{6} e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 378 d^{4} e^{8} x^{4} \sqrt {d^{2} - e^{2} x^{2}} - 252 d^{2} e^{10} x^{6} \sqrt {d^{2} - e^{2} x^{2}} + 63 e^{12} x^{8} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {11}{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)**(11/2),x)

[Out]

d*Piecewise((-105*I*d**6*x**3/(315*d**17*sqrt(-1 + e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(-1 + e**2*x**2/
d**2) + 1890*d**13*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*
d**9*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) + 126*I*d**4*e**2*x**5/(315*d**17*sqrt(-1 + e**2*x**2/d**2) - 1260*d
**15*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**11*e**6*x*
*6*sqrt(-1 + e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) - 72*I*d**2*e**4*x**7/(315*d**17*
sqrt(-1 + e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(-1 + e*
*2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(-1 + e**2*x**2/d**2))
+ 16*I*e**6*x**9/(315*d**17*sqrt(-1 + e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890
*d**13*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**9*e**8*x*
*8*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (105*d**6*x**3/(315*d**17*sqrt(1 - e**2*x**2/d**2) -
1260*d**15*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**11*e**
6*x**6*sqrt(1 - e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(1 - e**2*x**2/d**2)) - 126*d**4*e**2*x**5/(315*d**17
*sqrt(1 - e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(1 - e**2
*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(1 - e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(1 - e**2*x**2/d**2)) + 7
2*d**2*e**4*x**7/(315*d**17*sqrt(1 - e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 1890*d*
*13*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(1 - e**2*x**2/d**2) + 315*d**9*e**8*x**8*sq
rt(1 - e**2*x**2/d**2)) - 16*e**6*x**9/(315*d**17*sqrt(1 - e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(1 - e**
2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(1 - e**2*x**2/d**2) +
315*d**9*e**8*x**8*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise((-2*d**2/(63*d**8*e**4*sqrt(d**2 - e**2*x*
*2) - 252*d**6*e**6*x**2*sqrt(d**2 - e**2*x**2) + 378*d**4*e**8*x**4*sqrt(d**2 - e**2*x**2) - 252*d**2*e**10*x
**6*sqrt(d**2 - e**2*x**2) + 63*e**12*x**8*sqrt(d**2 - e**2*x**2)) + 9*e**2*x**2/(63*d**8*e**4*sqrt(d**2 - e**
2*x**2) - 252*d**6*e**6*x**2*sqrt(d**2 - e**2*x**2) + 378*d**4*e**8*x**4*sqrt(d**2 - e**2*x**2) - 252*d**2*e**
10*x**6*sqrt(d**2 - e**2*x**2) + 63*e**12*x**8*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**4/(4*(d**2)**(11/2)), T
rue))

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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)^(11/2),x, algorithm="giac")

[Out]

integrate((x*e + d)*x^2/(-x^2*e^2 + d^2)^(11/2), x)

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Mupad [B]
time = 2.74, size = 202, normalized size = 1.36 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}}{144\,d^3\,e^3\,{\left (d-e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{252\,e^3}-\frac {17\,x}{252\,d\,e^2}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {5}{144\,d^2\,e^3}+\frac {131\,x}{5040\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e^2\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^7\,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)^(11/2),x)

[Out]

(d^2 - e^2*x^2)^(1/2)/(144*d^3*e^3*(d - e*x)^5) - ((d^2 - e^2*x^2)^(1/2)*(1/(252*e^3) - (17*x)/(252*d*e^2)))/(
(d + e*x)^4*(d - e*x)^4) - ((d^2 - e^2*x^2)^(1/2)*(5/(144*d^2*e^3) + (131*x)/(5040*d^3*e^2)))/((d + e*x)^3*(d
- e*x)^3) - (8*x*(d^2 - e^2*x^2)^(1/2))/(315*d^5*e^2*(d + e*x)^2*(d - e*x)^2) - (16*x*(d^2 - e^2*x^2)^(1/2))/(
315*d^7*e^2*(d + e*x)*(d - e*x))

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