\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^4} \, dx\) [74]

Optimal result

Integrand size = 27, antiderivative size = 210 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {25}{8} d^5 e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

-1/12*d*e^3*(25*e*x+26*d)*(-e^2*x^2+d^2)^(3/2)-1/30*e^2*(39*e*x+50*d)*(-e^2*x^2+d^2)^(5/2)/x-1/3*d*(-e^2*x^2+d
^2)^(7/2)/x^3-3/2*e*(-e^2*x^2+d^2)^(7/2)/x^2-25/8*d^5*e^3*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+13/2*d^5*e^3*arctan
h((-e^2*x^2+d^2)^(1/2)/d)-1/8*d^3*e^3*(25*e*x+52*d)*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1821, 827, 829, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {25}{8} d^5 e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2} \]

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^4,x]

[Out]

-1/8*(d^3*e^3*(52*d + 25*e*x)*Sqrt[d^2 - e^2*x^2]) - (d*e^3*(26*d + 25*e*x)*(d^2 - e^2*x^2)^(3/2))/12 - (e^2*(
50*d + 39*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*x) - (d*(d^2 - e^2*x^2)^(7/2))/(3*x^3) - (3*e*(d^2 - e^2*x^2)^(7/2))
/(2*x^2) - (25*d^5*e^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/8 + (13*d^5*e^3*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 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 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}}{3 x^3}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-9 d^4 e-5 d^3 e^2 x-3 d^2 e^3 x^2\right )}{x^3} \, dx}{3 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}+\frac {\int \frac {\left (10 d^5 e^2-39 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^2} \, dx}{6 d^4} \\ & = -\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {\int \frac {\left (78 d^6 e^3+100 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{12 d^4} \\ & = -\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}+\frac {\int \frac {\left (-312 d^8 e^5-300 d^7 e^6 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{48 d^4 e^2} \\ & = -\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {\int \frac {624 d^{10} e^7+300 d^9 e^8 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{96 d^4 e^4} \\ & = -\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {1}{2} \left (13 d^6 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{8} \left (25 d^5 e^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {1}{4} \left (13 d^6 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{8} \left (25 d^5 e^4\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}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {25}{8} d^5 e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} \left (13 d^6 e\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 {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {25}{8} d^5 e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-40 d^7-180 d^6 e x-80 d^5 e^2 x^2-656 d^4 e^3 x^3-345 d^3 e^4 x^4+32 d^2 e^5 x^5+90 d e^6 x^6+24 e^7 x^7\right )}{120 x^3}-13 d^5 e^3 \text {arctanh}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {25}{8} d^5 \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right ) \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^4,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-40*d^7 - 180*d^6*e*x - 80*d^5*e^2*x^2 - 656*d^4*e^3*x^3 - 345*d^3*e^4*x^4 + 32*d^2*e^5*
x^5 + 90*d*e^6*x^6 + 24*e^7*x^7))/(120*x^3) - 13*d^5*e^3*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d] + (
25*d^5*(-e^2)^(3/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/8

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e^{2} x^{2}+9 d e x +2 d^{2}\right )}{6 x^{3}}+\frac {e^{7} x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5}+\frac {4 e^{5} d^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{15}-\frac {82 e^{3} d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{15}-\frac {25 e^{4} d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {13 e^{3} d^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}+\frac {3 e^{6} d \,x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4}-\frac {23 e^{4} d^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{8}\)	\(238\)
default	\(e^{3} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )\)	\(536\)

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/6*d^5*(-e^2*x^2+d^2)^(1/2)*(4*e^2*x^2+9*d*e*x+2*d^2)/x^3+1/5*e^7*x^4*(-e^2*x^2+d^2)^(1/2)+4/15*e^5*d^2*x^2*
(-e^2*x^2+d^2)^(1/2)-82/15*e^3*d^4*(-e^2*x^2+d^2)^(1/2)-25/8*e^4*d^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^
2+d^2)^(1/2))+13/2*e^3*d^6/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+3/4*e^6*d*x^3*(-e^2*x^
2+d^2)^(1/2)-23/8*e^4*d^3*x*(-e^2*x^2+d^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\frac {750 \, d^{5} e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 780 \, d^{5} e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 656 \, d^{5} e^{3} x^{3} + {\left (24 \, e^{7} x^{7} + 90 \, d e^{6} x^{6} + 32 \, d^{2} e^{5} x^{5} - 345 \, d^{3} e^{4} x^{4} - 656 \, d^{4} e^{3} x^{3} - 80 \, d^{5} e^{2} x^{2} - 180 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, x^{3}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^4,x, algorithm="fricas")

[Out]

1/120*(750*d^5*e^3*x^3*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 780*d^5*e^3*x^3*log(-(d - sqrt(-e^2*x^2 + d
^2))/x) - 656*d^5*e^3*x^3 + (24*e^7*x^7 + 90*d*e^6*x^6 + 32*d^2*e^5*x^5 - 345*d^3*e^4*x^4 - 656*d^4*e^3*x^3 -
80*d^5*e^2*x^2 - 180*d^6*e*x - 40*d^7)*sqrt(-e^2*x^2 + d^2))/x^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.18 (sec) , antiderivative size = 843, normalized size of antiderivative = 4.01 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**4,x)

[Out]

d**7*Piecewise((-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), Tru
e)) + 3*d**6*e*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**5*e**2*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*sqrt(1 - e**2*x**2/d**2)), True)) - 5*d**4*e**3*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*a
cosh(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**3*e**4*Piecewise((d**2*Piecewi
se((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)) + d**2*e**5*Piecewise((-d**2*sqr
t(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)) + 3*d*e
**6*Piecewise((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)) + e**7*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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {25 \, d^{5} e^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} + \frac {13}{2} \, d^{5} e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {25}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{4} x - \frac {13}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{3} - \frac {25}{12} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} x - \frac {13}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} - \frac {13}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2}}{3 \, x} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{2 \, x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{3 \, x^{3}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^4,x, algorithm="maxima")

[Out]

-25/8*d^5*e^4*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 13/2*d^5*e^3*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d
/abs(x)) - 25/8*sqrt(-e^2*x^2 + d^2)*d^3*e^4*x - 13/2*sqrt(-e^2*x^2 + d^2)*d^4*e^3 - 25/12*(-e^2*x^2 + d^2)^(3
/2)*d*e^4*x - 13/6*(-e^2*x^2 + d^2)^(3/2)*d^2*e^3 - 13/10*(-e^2*x^2 + d^2)^(5/2)*e^3 - 5/3*(-e^2*x^2 + d^2)^(5
/2)*d*e^2/x - 3/2*(-e^2*x^2 + d^2)^(7/2)*e/x^2 - 1/3*(-e^2*x^2 + d^2)^(7/2)*d/x^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.65 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=-\frac {25 \, d^{5} e^{4} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} + \frac {13 \, d^{5} e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} + \frac {{\left (d^{5} e^{4} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{5} e^{2}}{x} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5}}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} {\left | e \right |}} - \frac {1}{120} \, {\left (656 \, d^{4} e^{3} + {\left (345 \, d^{3} e^{4} - 2 \, {\left (16 \, d^{2} e^{5} + 3 \, {\left (4 \, e^{7} x + 15 \, d e^{6}\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} - \frac {\frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{5} e^{4}}{x} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5} e^{2}}{x^{2}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{5}}{x^{3}}}{24 \, e^{2} {\left | e \right |}} \]

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^4,x, algorithm="giac")

[Out]

-25/8*d^5*e^4*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 13/2*d^5*e^4*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*ab
s(e))/(e^2*abs(x)))/abs(e) + 1/24*(d^5*e^4 + 9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^5*e^2/x + 9*(d*e + sqrt(-
e^2*x^2 + d^2)*abs(e))^2*d^5/x^2)*e^6*x^3/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*abs(e)) - 1/120*(656*d^4*e^3
+ (345*d^3*e^4 - 2*(16*d^2*e^5 + 3*(4*e^7*x + 15*d*e^6)*x)*x)*x)*sqrt(-e^2*x^2 + d^2) - 1/24*(9*(d*e + sqrt(-e
^2*x^2 + d^2)*abs(e))*d^5*e^4/x + 9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^5*e^2/x^2 + (d*e + sqrt(-e^2*x^2 +
d^2)*abs(e))^3*d^5/x^3)/(e^2*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^4} \,d x \]

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^4,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^4, x)