3.1.77 \(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^7} \, dx\) [77]

Optimal. Leaf size=214 \[ -\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

1/48*d*e^3*(85*e*x+8*d)*(-e^2*x^2+d^2)^(3/2)/x^3-1/120*e^2*(12*e*x+85*d)*(-e^2*x^2+d^2)^(5/2)/x^4-1/6*d*(-e^2*
x^2+d^2)^(7/2)/x^6-3/5*e*(-e^2*x^2+d^2)^(7/2)/x^5-1/2*d^2*e^6*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-85/16*d^2*e^6*a
rctanh((-e^2*x^2+d^2)^(1/2)/d)-1/16*d*e^5*(-85*e*x+8*d)*(-e^2*x^2+d^2)^(1/2)/x

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Rubi [A]
time = 0.19, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1821, 827, 825, 858, 223, 209, 272, 65, 214} \begin {gather*} -\frac {1}{2} d^2 e^6 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {85}{16} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^7,x]

[Out]

-1/16*(d*e^5*(8*d - 85*e*x)*Sqrt[d^2 - e^2*x^2])/x + (d*e^3*(8*d + 85*e*x)*(d^2 - e^2*x^2)^(3/2))/(48*x^3) - (
e^2*(85*d + 12*e*x)*(d^2 - e^2*x^2)^(5/2))/(120*x^4) - (d*(d^2 - e^2*x^2)^(7/2))/(6*x^6) - (3*e*(d^2 - e^2*x^2
)^(7/2))/(5*x^5) - (d^2*e^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - (85*d^2*e^6*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])
/16

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 214

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

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 825

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/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps

\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-18 d^4 e-17 d^3 e^2 x-6 d^2 e^3 x^2\right )}{x^6} \, dx}{6 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {\int \frac {\left (85 d^5 e^2-6 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx}{30 d^4}\\ &=-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {\left (48 d^6 e^3+340 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx}{96 d^4}\\ &=\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {\int \frac {\left (192 d^8 e^5+2040 d^7 e^6 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{384 d^6}\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {-4080 d^9 e^6+384 d^8 e^7 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{768 d^6}\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {1}{16} \left (85 d^3 e^6\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (d^2 e^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac {1}{32} \left (85 d^3 e^6\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{2} \left (d^2 e^7\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{16} \left (85 d^3 e^4\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )\\ &=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {1}{2} d^2 e^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 192, normalized size = 0.90 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-40 d^7-144 d^6 e x-50 d^5 e^2 x^2+448 d^4 e^3 x^3+645 d^3 e^4 x^4-544 d^2 e^5 x^5+720 d e^6 x^6+120 e^7 x^7\right )}{240 x^6}+\frac {85}{8} d^2 e^6 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {1}{2} d^2 e \left (-e^2\right )^{5/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^7,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-40*d^7 - 144*d^6*e*x - 50*d^5*e^2*x^2 + 448*d^4*e^3*x^3 + 645*d^3*e^4*x^4 - 544*d^2*e^5
*x^5 + 720*d*e^6*x^6 + 120*e^7*x^7))/(240*x^6) + (85*d^2*e^6*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d]
)/8 - (d^2*e*(-e^2)^(5/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(186)=372\).
time = 0.07, size = 722, normalized size = 3.37

method	result	size
risch	\(-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (544 e^{5} x^{5}-645 d \,e^{4} x^{4}-448 d^{2} e^{3} x^{3}+50 x^{2} d^{3} e^{2}+144 d^{4} x e +40 d^{5}\right )}{240 x^{6}}+\frac {e^{7} x \sqrt {-e^{2} x^{2}+d^{2}}}{2}-\frac {e^{7} d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}+3 e^{6} d \sqrt {-e^{2} x^{2}+d^{2}}-\frac {85 e^{6} d^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}}\)	\(196\)
default	\(3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )+e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )\)	\(722\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 282, normalized size = 1.32 \begin {gather*} -\frac {1}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{6} - \frac {85}{16} \, d^{2} e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} x e^{7} + \frac {85}{16} \, \sqrt {-x^{2} e^{2} + d^{2}} d e^{6} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{7}}{3 \, d^{2}} + \frac {85 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}}{48 \, d} + \frac {17 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{16 \, d^{3}} - \frac {4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{15 \, d^{2} x} + \frac {17 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{16 \, d^{3} x^{2}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{15 \, d^{2} x^{3}} - \frac {17 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{24 \, d x^{4}} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{5 \, x^{5}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

-1/2*d^2*arcsin(x*e/d)*e^6 - 85/16*d^2*e^6*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x)) - 1/2*sqrt(-x^2
*e^2 + d^2)*x*e^7 + 85/16*sqrt(-x^2*e^2 + d^2)*d*e^6 - 1/3*(-x^2*e^2 + d^2)^(3/2)*x*e^7/d^2 + 85/48*(-x^2*e^2
+ d^2)^(3/2)*e^6/d + 17/16*(-x^2*e^2 + d^2)^(5/2)*e^6/d^3 - 4/15*(-x^2*e^2 + d^2)^(5/2)*e^5/(d^2*x) + 17/16*(-
x^2*e^2 + d^2)^(7/2)*e^4/(d^3*x^2) + 1/15*(-x^2*e^2 + d^2)^(7/2)*e^3/(d^2*x^3) - 17/24*(-x^2*e^2 + d^2)^(7/2)*
e^2/(d*x^4) - 3/5*(-x^2*e^2 + d^2)^(7/2)*e/x^5 - 1/6*(-x^2*e^2 + d^2)^(7/2)*d/x^6

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Fricas [A]
time = 1.60, size = 167, normalized size = 0.78 \begin {gather*} \frac {240 \, d^{2} x^{6} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{6} + 1275 \, d^{2} x^{6} e^{6} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + 720 \, d^{2} x^{6} e^{6} + {\left (120 \, x^{7} e^{7} + 720 \, d x^{6} e^{6} - 544 \, d^{2} x^{5} e^{5} + 645 \, d^{3} x^{4} e^{4} + 448 \, d^{4} x^{3} e^{3} - 50 \, d^{5} x^{2} e^{2} - 144 \, d^{6} x e - 40 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{240 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^7,x, algorithm="fricas")

[Out]

1/240*(240*d^2*x^6*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x)*e^6 + 1275*d^2*x^6*e^6*log(-(d - sqrt(-x^2*e^2
+ d^2))/x) + 720*d^2*x^6*e^6 + (120*x^7*e^7 + 720*d*x^6*e^6 - 544*d^2*x^5*e^5 + 645*d^3*x^4*e^4 + 448*d^4*x^3
*e^3 - 50*d^5*x^2*e^2 - 144*d^6*x*e - 40*d^7)*sqrt(-x^2*e^2 + d^2))/x^6

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Sympy [C] Result contains complex when optimal does not.
time = 12.70, size = 1397, normalized size = 6.53 \begin {gather*} d^{7} \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) + 3 d^{6} e \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) + d^{5} e^{2} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) - 5 d^{4} e^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) - 5 d^{3} e^{4} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) + d^{2} e^{5} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + 3 d e^{6} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + e^{7} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**7,x)

[Out]

d**7*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/
(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d
**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2
*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + 3*d**6*e*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)
/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2
*I*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d
**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x*
*5 + 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(
1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 +
15*d*e**2*x**7), True)) + d**5*e**2*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(
d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(
e**2*x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1))
+ I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) - 5*d**4*e**3*Piecewis
e((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) >
1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True)) - 5*d**3*
e**4*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(2*x) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (
I*d**2/(2*e*x**3*sqrt(-d**2/(e**2*x**2) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**2*asin(d/(e*x))/(
2*d), True)) + d**2*e**5*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1
+ e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sq
rt(1 - e**2*x**2/d**2)), True)) + 3*d*e**6*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x))
- e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*
d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True)) + e**7*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d
*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**
2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (179) = 358\).
time = 1.11, size = 476, normalized size = 2.22 \begin {gather*} -\frac {1}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{6} \mathrm {sgn}\left (d\right ) - \frac {85}{16} \, d^{2} e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{4}}{16 \, x} + \frac {81 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{2}}{128 \, x^{2}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{\left (-2\right )}}{128 \, x^{4}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{2} e^{\left (-4\right )}}{160 \, x^{5}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{2} e^{\left (-6\right )}}{384 \, x^{6}} + \frac {17 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2}}{96 \, x^{3}} + \frac {{\left (5 \, d^{2} e^{6} + \frac {36 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{4}}{x} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{2}}{x^{2}} - \frac {1215 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{\left (-2\right )}}{x^{4}} + \frac {1800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{2} e^{\left (-4\right )}}{x^{5}} - \frac {340 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2}}{x^{3}}\right )} x^{6} e^{12}}{1920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6}} + \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e^{7} + 6 \, d e^{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^7,x, algorithm="giac")

[Out]

-1/2*d^2*arcsin(x*e/d)*e^6*sgn(d) - 85/16*d^2*e^6*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x)
) - 15/16*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*e^4/x + 81/128*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*e^2/x^2 - 3/1
28*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*e^(-2)/x^4 - 3/160*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^2*e^(-4)/x^5 - 1
/384*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^2*e^(-6)/x^6 + 17/96*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2/x^3 + 1/1920
*(5*d^2*e^6 + 36*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*e^4/x + 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*e^2/x^2 -
1215*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*e^(-2)/x^4 + 1800*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^2*e^(-4)/x^5 -
340*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2/x^3)*x^6*e^12/(d*e + sqrt(-x^2*e^2 + d^2)*e)^6 + 1/2*sqrt(-x^2*e^2 +
d^2)*(x*e^7 + 6*d*e^6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^7,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^7, x)

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