\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^5} \, dx\) [75]

Optimal result

Integrand size = 27, antiderivative size = 209 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

15/8*d*e^3*(-e*x+2*d)*(-e^2*x^2+d^2)^(3/2)/x-3/8*e^2*(2*e*x+3*d)*(-e^2*x^2+d^2)^(5/2)/x^2-1/4*d*(-e^2*x^2+d^2)
^(7/2)/x^4-e*(-e^2*x^2+d^2)^(7/2)/x^3+45/8*d^4*e^4*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+45/8*d^4*e^4*arctanh((-e^2
*x^2+d^2)^(1/2)/d)-45/8*d^2*e^4*(-e*x+d)*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 209, 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} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {45}{8} d^4 e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x} \]

[In]

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

[Out]

(-45*d^2*e^4*(d - e*x)*Sqrt[d^2 - e^2*x^2])/8 + (15*d*e^3*(2*d - e*x)*(d^2 - e^2*x^2)^(3/2))/(8*x) - (3*e^2*(3
*d + 2*e*x)*(d^2 - e^2*x^2)^(5/2))/(8*x^2) - (d*(d^2 - e^2*x^2)^(7/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(7/2))/x^3
+ (45*d^4*e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/8 + (45*d^4*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/8

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}}{4 x^4}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-12 d^4 e-9 d^3 e^2 x-4 d^2 e^3 x^2\right )}{x^4} \, dx}{4 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {\int \frac {\left (27 d^5 e^2-36 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx}{12 d^4} \\ & = -\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {\left (144 d^6 e^3+216 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx}{192 d^4} \\ & = \frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {5 \int \frac {\left (-432 d^7 e^4+864 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{384 d^4} \\ & = -\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {864 d^9 e^6-864 d^8 e^7 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{768 d^4 e^2} \\ & = -\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{8} \left (45 d^5 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{8} \left (45 d^4 e^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{16} \left (45 d^5 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{8} \left (45 d^4 e^5\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{8} \left (45 d^5 e^2\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {1}{8} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^7-8 d^6 e x-3 d^5 e^2 x^2+48 d^4 e^3 x^3-48 d^3 e^4 x^4+3 d^2 e^5 x^5+8 d e^6 x^6+2 e^7 x^7\right )}{x^4}-90 d^4 e^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+45 d^3 \sqrt {d^2} e^4 \log (x)-45 d^3 \sqrt {d^2} e^4 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )\right ) \]

[In]

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-2*d^7 - 8*d^6*e*x - 3*d^5*e^2*x^2 + 48*d^4*e^3*x^3 - 48*d^3*e^4*x^4 + 3*d^2*e^5*x^5 +
8*d*e^6*x^6 + 2*e^7*x^7))/x^4 - 90*d^4*e^4*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] + 45*d^3*Sqrt[d^2]*
e^4*Log[x] - 45*d^3*Sqrt[d^2]*e^4*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/8

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {d^{4} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-48 e^{3} x^{3}+3 d \,e^{2} x^{2}+8 d^{2} e x +2 d^{3}\right )}{8 x^{4}}+\frac {e^{7} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4}+\frac {3 e^{5} d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{8}+\frac {45 e^{5} d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {45 e^{4} d^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}+e^{6} d \,x^{2} \sqrt {-e^{2} x^{2}+d^{2}}-6 e^{4} d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\)	\(223\)
default	\(d^{3} \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}}}{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 \,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 )+3 d^{2} e \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 )\)	\(598\)

[In]

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

[Out]

-1/8*d^4*(-e^2*x^2+d^2)^(1/2)*(-48*e^3*x^3+3*d*e^2*x^2+8*d^2*e*x+2*d^3)/x^4+1/4*e^7*x^3*(-e^2*x^2+d^2)^(1/2)+3
/8*e^5*d^2*x*(-e^2*x^2+d^2)^(1/2)+45/8*e^5*d^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+45/8*e^4
*d^5/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+e^6*d*x^2*(-e^2*x^2+d^2)^(1/2)-6*e^4*d^3*(-e
^2*x^2+d^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=-\frac {90 \, d^{4} e^{4} x^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 45 \, d^{4} e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 48 \, d^{4} e^{4} x^{4} - {\left (2 \, e^{7} x^{7} + 8 \, d e^{6} x^{6} + 3 \, d^{2} e^{5} x^{5} - 48 \, d^{3} e^{4} x^{4} + 48 \, d^{4} e^{3} x^{3} - 3 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x - 2 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8 \, x^{4}} \]

[In]

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

[Out]

-1/8*(90*d^4*e^4*x^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 45*d^4*e^4*x^4*log(-(d - sqrt(-e^2*x^2 + d^2)
)/x) + 48*d^4*e^4*x^4 - (2*e^7*x^7 + 8*d*e^6*x^6 + 3*d^2*e^5*x^5 - 48*d^3*e^4*x^4 + 48*d^4*e^3*x^3 - 3*d^5*e^2
*x^2 - 8*d^6*e*x - 2*d^7)*sqrt(-e^2*x^2 + d^2))/x^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.95 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.59 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

d**7*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)) + 3*d**6*e*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), True)) + d**5*e**2*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)) - 5*d**4*e**3*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**3*e**4*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)) + d**2*e**5*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)) + 3*d*e**6*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)) + e**7*Piecewise((d**4*Piecewise((log(-2*e**2*x + 2*sq
rt(-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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {45 \, d^{4} e^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} + \frac {45}{8} \, d^{4} e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {45}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{5} x - \frac {45}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{4} + \frac {15}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} x - \frac {15}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{8 \, d} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{x} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{8 \, d x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{4 \, x^{4}} \]

[In]

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

[Out]

45/8*d^4*e^5*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 45/8*d^4*e^4*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/
abs(x)) + 45/8*sqrt(-e^2*x^2 + d^2)*d^2*e^5*x - 45/8*sqrt(-e^2*x^2 + d^2)*d^3*e^4 + 15/4*(-e^2*x^2 + d^2)^(3/2
)*e^5*x - 15/8*(-e^2*x^2 + d^2)^(3/2)*d*e^4 - 9/8*(-e^2*x^2 + d^2)^(5/2)*e^4/d + 3*(-e^2*x^2 + d^2)^(5/2)*e^3/
x - 9/8*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^2) - (-e^2*x^2 + d^2)^(7/2)*e/x^3 - 1/4*(-e^2*x^2 + d^2)^(7/2)*d/x^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (183) = 366\).

Time = 0.30 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {45 \, d^{4} e^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} + \frac {45 \, d^{4} e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, {\left | e \right |}} + \frac {{\left (d^{4} e^{5} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e^{3}}{x} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} e}{x^{2}} - \frac {184 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4}}{e x^{3}}\right )} e^{8} x^{4}}{64 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}} - \frac {1}{8} \, {\left (48 \, d^{3} e^{4} - {\left (3 \, d^{2} e^{5} + 2 \, {\left (e^{7} x + 4 \, d e^{6}\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} + \frac {\frac {184 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e^{5} {\left | e \right |}}{x} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} e^{3} {\left | e \right |}}{x^{2}} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4} e {\left | e \right |}}{x^{3}} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{4} {\left | e \right |}}{e x^{4}}}{64 \, e^{4}} \]

[In]

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

[Out]

45/8*d^4*e^5*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 45/8*d^4*e^5*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs
(e))/(e^2*abs(x)))/abs(e) + 1/64*(d^4*e^5 + 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*e^3/x + 8*(d*e + sqrt(-e
^2*x^2 + d^2)*abs(e))^2*d^4*e/x^2 - 184*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^4/(e*x^3))*e^8*x^4/((d*e + sqr
t(-e^2*x^2 + d^2)*abs(e))^4*abs(e)) - 1/8*(48*d^3*e^4 - (3*d^2*e^5 + 2*(e^7*x + 4*d*e^6)*x)*x)*sqrt(-e^2*x^2 +
d^2) + 1/64*(184*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*e^5*abs(e)/x - 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))
^2*d^4*e^3*abs(e)/x^2 - 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^4*e*abs(e)/x^3 - (d*e + sqrt(-e^2*x^2 + d^2)
*abs(e))^4*d^4*abs(e)/(e*x^4))/e^4

Mupad [F(-1)]

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

[In]

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

[Out]

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