3.1.20 \(\int \frac {x^6 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [20]

Optimal. Leaf size=147 \[ \frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7} \]

[Out]

1/5*x^5*(e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^3*(6*e*x+5*d)/e^4/(-e^2*x^2+d^2)^(3/2)-d*arctan(e*x/(-e^2*x^2+
d^2)^(1/2))/e^7+1/5*x*(8*e*x+5*d)/e^6/(-e^2*x^2+d^2)^(1/2)+16/5*(-e^2*x^2+d^2)^(1/2)/e^7

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 147, 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, 655, 223, 209} \begin {gather*} -\frac {d \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7}+\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(x^5*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^3*(5*d + 6*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x*(5*d +
8*e*x))/(5*e^6*Sqrt[d^2 - e^2*x^2]) + (16*Sqrt[d^2 - e^2*x^2])/(5*e^7) - (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]
)/e^7

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[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^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^4 \left (5 d^3+6 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^2 \left (15 d^5+24 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^7+48 d^6 e x}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^6}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}\\ &=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.40, size = 136, normalized size = 0.93 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-48 d^5+33 d^4 e x+87 d^3 e^2 x^2-52 d^2 e^3 x^3-38 d e^4 x^4+15 e^5 x^5\right )}{15 e^7 (-d+e x)^3 (d+e x)^2}+\frac {d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^6 \sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-48*d^5 + 33*d^4*e*x + 87*d^3*e^2*x^2 - 52*d^2*e^3*x^3 - 38*d*e^4*x^4 + 15*e^5*x^5))/(15
*e^7*(-d + e*x)^3*(d + e*x)^2) + (d*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(e^6*Sqrt[-e^2])

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 220, normalized size = 1.50

method	result	size
default	\(e \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+d \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )\)	\(220\)
risch	\(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{7}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6} \sqrt {e^{2}}}+\frac {25 d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{48 e^{8} \left (x +\frac {d}{e}\right )}-\frac {493 d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{240 e^{8} \left (x -\frac {d}{e}\right )}-\frac {23 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{60 e^{9} \left (x -\frac {d}{e}\right )^{2}}-\frac {d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{20 e^{10} \left (x -\frac {d}{e}\right )^{3}}-\frac {d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{24 e^{9} \left (x +\frac {d}{e}\right )^{2}}\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

e*(-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))))+d*(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)))))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (123) = 246\).
time = 0.55, size = 255, normalized size = 1.73 \begin {gather*} -\frac {x^{6} e^{\left (-1\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {6 \, d^{2} x^{4} e^{\left (-3\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {8 \, d^{4} x^{2} e^{\left (-5\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {16 \, d^{6} e^{\left (-7\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {1}{3} \, {\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 x e^{\left (-2\right )} + \frac {4 \, d^{3} x e^{\left (-6\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {1}{15} \, {\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 x - d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-7\right )} - \frac {7 \, d x e^{\left (-6\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 2.47, size = 245, normalized size = 1.67 \begin {gather*} \frac {48 \, d x^{5} e^{5} - 48 \, d^{2} x^{4} e^{4} - 96 \, d^{3} x^{3} e^{3} + 96 \, d^{4} x^{2} e^{2} + 48 \, d^{5} x e - 48 \, d^{6} + 30 \, {\left (d x^{5} e^{5} - d^{2} x^{4} e^{4} - 2 \, d^{3} x^{3} e^{3} + 2 \, d^{4} x^{2} e^{2} + d^{5} x e - d^{6}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (15 \, x^{5} e^{5} - 38 \, d x^{4} e^{4} - 52 \, d^{2} x^{3} e^{3} + 87 \, d^{3} x^{2} e^{2} + 33 \, d^{4} x e - 48 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{5} e^{12} - d x^{4} e^{11} - 2 \, d^{2} x^{3} e^{10} + 2 \, d^{3} x^{2} e^{9} + d^{4} x e^{8} - d^{5} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(48*d*x^5*e^5 - 48*d^2*x^4*e^4 - 96*d^3*x^3*e^3 + 96*d^4*x^2*e^2 + 48*d^5*x*e - 48*d^6 + 30*(d*x^5*e^5 -
d^2*x^4*e^4 - 2*d^3*x^3*e^3 + 2*d^4*x^2*e^2 + d^5*x*e - d^6)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (1
5*x^5*e^5 - 38*d*x^4*e^4 - 52*d^2*x^3*e^3 + 87*d^3*x^2*e^2 + 33*d^4*x*e - 48*d^5)*sqrt(-x^2*e^2 + d^2))/(x^5*e
^12 - d*x^4*e^11 - 2*d^2*x^3*e^10 + 2*d^3*x^2*e^9 + d^4*x*e^8 - d^5*e^7)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 22.82, size = 1821, normalized size = 12.39 \begin {gather*} d \left (\begin {cases} \frac {30 i d^{5} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {15 \pi d^{5} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {30 i d^{4} e x}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {60 i d^{3} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {30 \pi d^{3} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {70 i d^{2} e^{3} x^{3}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {30 i d e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {15 \pi d e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {46 i e^{5} x^{5}}{30 d^{5} e^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 60 d^{3} e^{9} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 30 d e^{11} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {15 d^{5} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{15 d^{5} e^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{3} e^{9} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d e^{11} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {15 d^{4} e x}{15 d^{5} e^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{3} e^{9} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d e^{11} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {30 d^{3} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{15 d^{5} e^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{3} e^{9} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d e^{11} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {35 d^{2} e^{3} x^{3}}{15 d^{5} e^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{3} e^{9} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d e^{11} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {15 d e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{15 d^{5} e^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{3} e^{9} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d e^{11} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {23 e^{5} x^{5}}{15 d^{5} e^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{3} e^{9} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d e^{11} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((30*I*d**5*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**
3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 15*pi*d**5*sqrt(-1 + e**2
*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11
*x**4*sqrt(-1 + e**2*x**2/d**2)) - 30*I*d**4*e*x/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*s
qrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 60*I*d**3*e**2*x**2*sqrt(-1 + e**2*x**
2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 3
0*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*pi*d**3*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(30*d**5*e**7*sqrt(
-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)
) + 70*I*d**2*e**3*x**3/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2)
+ 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*I*d*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**
5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**
2*x**2/d**2)) - 15*pi*d*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*
e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 46*I*e**5*x**5/(30*d**5*e**
7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**
2/d**2)), Abs(e**2*x**2/d**2) > 1), (-15*d**5*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(15*d**5*e**7*sqrt(1 - e**2
*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) + 15*d**4
*e*x/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqr
t(1 - e**2*x**2/d**2)) + 30*d**3*e**2*x**2*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(15*d**5*e**7*sqrt(1 - e**2*x*
*2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) - 35*d**2*e*
*3*x**3/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*
sqrt(1 - e**2*x**2/d**2)) - 15*d*e**4*x**4*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(15*d**5*e**7*sqrt(1 - e**2*x*
*2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) + 23*e**5*x*
*5/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(
1 - e**2*x**2/d**2)), True)) + e*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*s
qrt(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*sqrt(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))

________________________________________________________________________________________

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^6*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate((x*e + d)*x^6/(-x^2*e^2 + d^2)^(7/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6\,\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^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int((x^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2), x)

________________________________________________________________________________________