\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^3} \, dx\) [73]

Optimal result

Integrand size = 27, antiderivative size = 207 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=\frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {85}{16} d^6 e^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^6 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

1/24*d^2*e^2*(-85*e*x+4*d)*(-e^2*x^2+d^2)^(3/2)+1/30*e^2*(-85*e*x+3*d)*(-e^2*x^2+d^2)^(5/2)-1/2*d*(-e^2*x^2+d^
2)^(7/2)/x^2-3*e*(-e^2*x^2+d^2)^(7/2)/x-85/16*d^6*e^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-1/2*d^6*e^2*arctanh((-e
^2*x^2+d^2)^(1/2)/d)+1/16*d^4*e^2*(-85*e*x+8*d)*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1821, 829, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=-\frac {85}{16} d^6 e^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^6 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2} \]

[In]

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

[Out]

(d^4*e^2*(8*d - 85*e*x)*Sqrt[d^2 - e^2*x^2])/16 + (d^2*e^2*(4*d - 85*e*x)*(d^2 - e^2*x^2)^(3/2))/24 + (e^2*(3*
d - 85*e*x)*(d^2 - e^2*x^2)^(5/2))/30 - (d*(d^2 - e^2*x^2)^(7/2))/(2*x^2) - (3*e*(d^2 - e^2*x^2)^(7/2))/x - (8
5*d^6*e^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/16 - (d^6*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/2

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 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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*} \text {integral}& = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-6 d^4 e-d^3 e^2 x-2 d^2 e^3 x^2\right )}{x^2} \, dx}{2 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {\int \frac {\left (d^5 e^2-34 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx}{2 d^4} \\ & = \frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {\left (-6 d^7 e^4+170 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{12 d^4 e^2} \\ & = \frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {\int \frac {\left (24 d^9 e^6-510 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{48 d^4 e^4} \\ & = \frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {-48 d^{11} e^8+510 d^{10} e^9 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{96 d^4 e^6} \\ & = \frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{2} \left (d^7 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{16} \left (85 d^6 e^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{4} \left (d^7 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{16} \left (85 d^6 e^3\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 {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {85}{16} d^6 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^7 \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 {1}{16} d^4 e^2 (8 d-85 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {85}{16} d^6 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^6 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-120 d^7-720 d^6 e x+544 d^5 e^2 x^2-645 d^4 e^3 x^3-448 d^3 e^4 x^4+50 d^2 e^5 x^5+144 d e^6 x^6+40 e^7 x^7\right )}{240 x^2}+\frac {85}{8} d^6 e^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{2} d^5 \sqrt {d^2} e^2 \log (x)+\frac {1}{2} d^5 \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-40 e^{7} x^{7}-144 d \,e^{6} x^{6}-50 d^{2} e^{5} x^{5}+448 d^{3} e^{4} x^{4}+645 d^{4} e^{3} x^{3}-544 d^{5} e^{2} x^{2}+720 d^{6} e x +120 d^{7}\right )}{240 x^{2}}-\frac {85 d^{6} e^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}-\frac {d^{7} e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}\)	\(175\)
default	\(e^{3} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )+3 d \,e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )\)	\(474\)

[In]

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

[Out]

-1/240*(-e^2*x^2+d^2)^(1/2)*(-40*e^7*x^7-144*d*e^6*x^6-50*d^2*e^5*x^5+448*d^3*e^4*x^4+645*d^4*e^3*x^3-544*d^5*
e^2*x^2+720*d^6*e*x+120*d^7)/x^2-85/16*d^6*e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2*d^7*
e^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=\frac {2550 \, d^{6} e^{2} x^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 120 \, d^{6} e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 544 \, d^{6} e^{2} x^{2} + {\left (40 \, e^{7} x^{7} + 144 \, d e^{6} x^{6} + 50 \, d^{2} e^{5} x^{5} - 448 \, d^{3} e^{4} x^{4} - 645 \, d^{4} e^{3} x^{3} + 544 \, d^{5} e^{2} x^{2} - 720 \, d^{6} e x - 120 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, x^{2}} \]

[In]

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

[Out]

1/240*(2550*d^6*e^2*x^2*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 120*d^6*e^2*x^2*log(-(d - sqrt(-e^2*x^2 +
d^2))/x) + 544*d^6*e^2*x^2 + (40*e^7*x^7 + 144*d*e^6*x^6 + 50*d^2*e^5*x^5 - 448*d^3*e^4*x^4 - 645*d^4*e^3*x^3
+ 544*d^5*e^2*x^2 - 720*d^6*e*x - 120*d^7)*sqrt(-e^2*x^2 + d^2))/x^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.96 (sec) , antiderivative size = 887, normalized size of antiderivative = 4.29 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

d**7*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)) + 3*d**6*e*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*sqr
t(1 - e**2*x**2/d**2)), True)) + d**5*e**2*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)) - 5*d**4*e**3*Piecewise((d**2*Piecewise((log(-2*e*
*2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/2 +
x*sqrt(d**2 - e**2*x**2)/2, Ne(e**2, 0)), (x*sqrt(d**2), True)) - 5*d**3*e**4*Piecewise((-d**2*sqrt(d**2 - e*
*2*x**2)/(3*e**2) + x**2*sqrt(d**2 - e**2*x**2)/3, Ne(e**2, 0)), (x**2*sqrt(d**2)/2, True)) + d**2*e**5*Piecew
ise((d**4*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x
)/sqrt(-e**2*x**2), True))/(8*e**2) - d**2*x*sqrt(d**2 - e**2*x**2)/(8*e**2) + x**3*sqrt(d**2 - e**2*x**2)/4,
Ne(e**2, 0)), (x**3*sqrt(d**2)/3, True)) + 3*d*e**6*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2
*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e**2, 0)), (x**4*sqrt(d**2)/4, True
)) + e**7*Piecewise((d**6*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**
2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(16*e**4) - d**4*x*sqrt(d**2 - e**2*x**2)/(16*e**4) - d**2*x**3*sqr
t(d**2 - e**2*x**2)/(24*e**2) + x**5*sqrt(d**2 - e**2*x**2)/6, Ne(e**2, 0)), (x**5*sqrt(d**2)/5, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=-\frac {85 \, d^{6} e^{3} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{16 \, \sqrt {e^{2}}} - \frac {1}{2} \, d^{6} e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {85}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{3} x + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} e^{2} - \frac {85}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} x + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2} + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3} x + \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e}{x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{2 \, x^{2}} \]

[In]

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

[Out]

-85/16*d^6*e^3*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - 1/2*d^6*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d
/abs(x)) - 85/16*sqrt(-e^2*x^2 + d^2)*d^4*e^3*x + 1/2*sqrt(-e^2*x^2 + d^2)*d^5*e^2 - 85/24*(-e^2*x^2 + d^2)^(3
/2)*d^2*e^3*x + 1/6*(-e^2*x^2 + d^2)^(3/2)*d^3*e^2 + 1/6*(-e^2*x^2 + d^2)^(5/2)*e^3*x + 1/10*(-e^2*x^2 + d^2)^
(5/2)*d*e^2 - 3*(-e^2*x^2 + d^2)^(5/2)*d^2*e/x - 1/2*(-e^2*x^2 + d^2)^(7/2)*d/x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=-\frac {85 \, d^{6} e^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, {\left | e \right |}} - \frac {d^{6} e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} + \frac {{\left (d^{6} e^{3} + \frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{6} e}{x}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} {\left | e \right |}} + \frac {1}{240} \, {\left (544 \, d^{5} e^{2} - {\left (645 \, d^{4} e^{3} + 2 \, {\left (224 \, d^{3} e^{4} - {\left (25 \, d^{2} e^{5} + 4 \, {\left (5 \, e^{7} x + 18 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} - \frac {\frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{6} e {\left | e \right |}}{x} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} {\left | e \right |}}{e x^{2}}}{8 \, e^{2}} \]

[In]

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

[Out]

-85/16*d^6*e^3*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/2*d^6*e^3*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*ab
s(e))/(e^2*abs(x)))/abs(e) + 1/8*(d^6*e^3 + 12*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^6*e/x)*e^4*x^2/((d*e + sq
rt(-e^2*x^2 + d^2)*abs(e))^2*abs(e)) + 1/240*(544*d^5*e^2 - (645*d^4*e^3 + 2*(224*d^3*e^4 - (25*d^2*e^5 + 4*(5
*e^7*x + 18*d*e^6)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2) - 1/8*(12*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^6*e*abs(e)
/x + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^6*abs(e)/(e*x^2))/e^2

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^3} \,d x \]

[In]

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

[Out]

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