3.1.19 \(\int \frac {x^7 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [19]

Optimal. Leaf size=161 \[ \frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]

[Out]

1/5*x^6*(e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^4*(7*e*x+6*d)/e^4/(-e^2*x^2+d^2)^(3/2)-7/2*d^2*arctan(e*x/(-e^
2*x^2+d^2)^(1/2))/e^8+1/15*x^2*(35*e*x+24*d)/e^6/(-e^2*x^2+d^2)^(1/2)+1/10*(35*e*x+32*d)*(-e^2*x^2+d^2)^(1/2)/
e^8

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Rubi [A]
time = 0.09, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {833, 794, 223, 209} \begin {gather*} -\frac {7 d^2 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}+\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*
d + 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d + 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) - (7*d^2*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/(2*e^8)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 833

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])

Rubi steps

\begin {align*} \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (6 d^3+7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 \left (24 d^5+35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (48 d^7+105 d^6 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {\left (7 d^2\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^7}\\ &=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 149, normalized size = 0.93 \begin {gather*} \frac {\frac {e \sqrt {d^2-e^2 x^2} \left (-96 d^6-9 d^5 e x+249 d^4 e^2 x^2-4 d^3 e^3 x^3-176 d^2 e^4 x^4+15 d e^5 x^5+15 e^6 x^6\right )}{(-d+e x)^3 (d+e x)^2}-105 d^2 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{30 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*Sqrt[d^2 - e^2*x^2]*(-96*d^6 - 9*d^5*e*x + 249*d^4*e^2*x^2 - 4*d^3*e^3*x^3 - 176*d^2*e^4*x^4 + 15*d*e^5*x^
5 + 15*e^6*x^6))/((-d + e*x)^3*(d + e*x)^2) - 105*d^2*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(
30*e^9)

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Maple [A]
time = 0.11, size = 251, normalized size = 1.56

method	result	size
default	\(e \left (-\frac {x^{7}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )}{2 e^{2}}\right )+d \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )\)	\(251\)
risch	\(\frac {\left (e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{8}}-\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{7} \sqrt {e^{2}}}-\frac {31 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{48 e^{9} \left (x +\frac {d}{e}\right )}-\frac {773 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{240 e^{9} \left (x -\frac {d}{e}\right )}-\frac {7 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{15 e^{10} \left (x -\frac {d}{e}\right )^{2}}-\frac {d^{4} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{20 e^{11} \left (x -\frac {d}{e}\right )^{3}}+\frac {d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{24 e^{10} \left (x +\frac {d}{e}\right )^{2}}\)	\(297\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(-1/2*x^7/e^2/(-e^2*x^2+d^2)^(5/2)+7/2*d^2/e^2*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/e^2*(1/3*x^3/e^2/(-e^2*x^
2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))
)+d*(-x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^2*(x^4/e^2/(-e^2*x^2+d^2)^(5/2)-4*d^2/e^2*(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))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (136) = 272\).
time = 0.49, size = 286, normalized size = 1.78 \begin {gather*} -\frac {x^{7} e^{\left (-1\right )}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {d x^{6} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {6 \, d^{3} x^{4} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {8 \, d^{5} x^{2} e^{\left (-6\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {16 \, d^{7} e^{\left (-8\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7}{30} \, {\left (\frac {15 \, x^{4} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {20 \, d^{2} x^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, d^{4} e^{\left (-6\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}\right )} d^{2} x e^{\left (-1\right )} - \frac {7}{6} \, {\left (\frac {3 \, x^{2} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, d^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}\right )} d^{2} x e^{\left (-3\right )} + \frac {14 \, d^{4} x e^{\left (-7\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {7}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-8\right )} - \frac {49 \, d^{2} x e^{\left (-7\right )}}{30 \, \sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^7*e^(-1)/(-x^2*e^2 + d^2)^(5/2) - d*x^6*e^(-2)/(-x^2*e^2 + d^2)^(5/2) + 6*d^3*x^4*e^(-4)/(-x^2*e^2 + d^
2)^(5/2) - 8*d^5*x^2*e^(-6)/(-x^2*e^2 + d^2)^(5/2) + 16/5*d^7*e^(-8)/(-x^2*e^2 + d^2)^(5/2) + 7/30*(15*x^4*e^(
-2)/(-x^2*e^2 + d^2)^(5/2) - 20*d^2*x^2*e^(-4)/(-x^2*e^2 + d^2)^(5/2) + 8*d^4*e^(-6)/(-x^2*e^2 + d^2)^(5/2))*d
^2*x*e^(-1) - 7/6*(3*x^2*e^(-2)/(-x^2*e^2 + d^2)^(3/2) - 2*d^2*e^(-4)/(-x^2*e^2 + d^2)^(3/2))*d^2*x*e^(-3) + 1
4/15*d^4*x*e^(-7)/(-x^2*e^2 + d^2)^(3/2) - 7/2*d^2*arcsin(x*e/d)*e^(-8) - 49/30*d^2*x*e^(-7)/sqrt(-x^2*e^2 + d
^2)

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Fricas [A]
time = 2.38, size = 259, normalized size = 1.61 \begin {gather*} \frac {96 \, d^{2} x^{5} e^{5} - 96 \, d^{3} x^{4} e^{4} - 192 \, d^{4} x^{3} e^{3} + 192 \, d^{5} x^{2} e^{2} + 96 \, d^{6} x e - 96 \, d^{7} + 210 \, {\left (d^{2} x^{5} e^{5} - d^{3} x^{4} e^{4} - 2 \, d^{4} x^{3} e^{3} + 2 \, d^{5} x^{2} e^{2} + d^{6} x e - d^{7}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (15 \, x^{6} e^{6} + 15 \, d x^{5} e^{5} - 176 \, d^{2} x^{4} e^{4} - 4 \, d^{3} x^{3} e^{3} + 249 \, d^{4} x^{2} e^{2} - 9 \, d^{5} x e - 96 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{5} e^{13} - d x^{4} e^{12} - 2 \, d^{2} x^{3} e^{11} + 2 \, d^{3} x^{2} e^{10} + d^{4} x e^{9} - d^{5} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(96*d^2*x^5*e^5 - 96*d^3*x^4*e^4 - 192*d^4*x^3*e^3 + 192*d^5*x^2*e^2 + 96*d^6*x*e - 96*d^7 + 210*(d^2*x^5
*e^5 - d^3*x^4*e^4 - 2*d^4*x^3*e^3 + 2*d^5*x^2*e^2 + d^6*x*e - d^7)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/
x) + (15*x^6*e^6 + 15*d*x^5*e^5 - 176*d^2*x^4*e^4 - 4*d^3*x^3*e^3 + 249*d^4*x^2*e^2 - 9*d^5*x*e - 96*d^6)*sqrt
(-x^2*e^2 + d^2))/(x^5*e^13 - d*x^4*e^12 - 2*d^2*x^3*e^11 + 2*d^3*x^2*e^10 + d^4*x*e^9 - d^5*e^8)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (144) = 288\).
time = 26.02, size = 2004, normalized size = 12.45 \begin {gather*} d \left (\begin {cases} \frac {16 d^{6}}{5 d^{4} e^{8} \sqrt {d^{2} - e^{2} x^{2}} - 10 d^{2} e^{10} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 5 e^{12} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {40 d^{4} e^{2} x^{2}}{5 d^{4} e^{8} \sqrt {d^{2} - e^{2} x^{2}} - 10 d^{2} e^{10} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 5 e^{12} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} + \frac {30 d^{2} e^{4} x^{4}}{5 d^{4} e^{8} \sqrt {d^{2} - e^{2} x^{2}} - 10 d^{2} e^{10} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 5 e^{12} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {5 e^{6} x^{6}}{5 d^{4} e^{8} \sqrt {d^{2} - e^{2} x^{2}} - 10 d^{2} e^{10} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 5 e^{12} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{8}}{8 \left (d^{2}\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {210 i d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {105 \pi d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {210 i d^{6} e x}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {420 i d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {210 \pi d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {490 i d^{4} e^{3} x^{3}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {210 i d^{3} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {105 \pi d^{3} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {322 i d^{2} e^{5} x^{5}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {30 i e^{7} x^{7}}{60 d^{5} e^{9} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 120 d^{3} e^{11} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 60 d e^{13} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {105 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {105 d^{6} e x}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {210 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {245 d^{4} e^{3} x^{3}}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {105 d^{3} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {161 d^{2} e^{5} x^{5}}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {15 e^{7} x^{7}}{30 d^{5} e^{9} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{11} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{13} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12
*x**4*sqrt(d**2 - e**2*x**2)) - 40*d**4*e**2*x**2/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqr
t(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) + 30*d**2*e**4*x**4/(5*d**4*e**8*sqrt(d**2 - e**2*x
**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 5*e**6*x**6/(5*d**4*
e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2))
, Ne(e, 0)), (x**8/(8*(d**2)**(7/2)), True)) + e*Piecewise((210*I*d**7*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/
(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt
(-1 + e**2*x**2/d**2)) - 105*pi*d**7*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d
**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 210*I*d**6*e*x/(60*d**
5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e
**2*x**2/d**2)) - 420*I*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1 + e**2*x**
2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 210*pi*
d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1
+ e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 490*I*d**4*e**3*x**3/(60*d**5*e**9*sqrt(-1 +
e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2))
+ 210*I*d**3*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d*
*3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 105*pi*d**3*e**4*x**4*s
qrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**
2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 322*I*d**2*e**5*x**5/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2)
- 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*I*e**7*x**7
/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqr
t(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-105*d**7*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(30*d**5*e*
*9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2
/d**2)) + 105*d**6*e*x/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) +
30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) + 210*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(30*d**5*e
**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**
2/d**2)) - 245*d**4*e**3*x**3/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d
**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 105*d**3*e**4*x**4*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(30
*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e
**2*x**2/d**2)) + 161*d**2*e**5*x**5/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2
*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 15*e**7*x**7/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2)
- 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**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^7*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^7\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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