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

Optimal result

Integrand size = 27, antiderivative size = 214 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {1}{2} d^2 e^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 214, 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, 825, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=-\frac {1}{2} d^2 e^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d^2 e^6 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac {d e^5 (8 d-85 e x) \sqrt {d^2-e^2 x^2}}{16 x}+\frac {d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-40 d^7-144 d^6 e x-50 d^5 e^2 x^2+448 d^4 e^3 x^3+645 d^3 e^4 x^4-544 d^2 e^5 x^5+720 d e^6 x^6+120 e^7 x^7\right )}{240 x^6}+d^2 e^6 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {85}{16} d \sqrt {d^2} e^6 \log (x)+\frac {85}{16} d \sqrt {d^2} e^6 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^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) + d^2*e^6*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] - (85
*d*Sqrt[d^2]*e^6*Log[x])/16 + (85*d*Sqrt[d^2]*e^6*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/16

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=\frac {240 \, d^{2} e^{6} x^{6} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 1275 \, d^{2} e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 720 \, d^{2} e^{6} x^{6} + {\left (120 \, e^{7} x^{7} + 720 \, d e^{6} x^{6} - 544 \, d^{2} e^{5} x^{5} + 645 \, d^{3} e^{4} x^{4} + 448 \, d^{4} e^{3} x^{3} - 50 \, d^{5} e^{2} x^{2} - 144 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, x^{6}} \]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.65 (sec) , antiderivative size = 1380, normalized size of antiderivative = 6.45 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=\text {Too large to display} \]

[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((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))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=-\frac {d^{2} e^{7} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} - \frac {85}{16} \, d^{2} e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7} x + \frac {85}{16} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{6} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7} x}{3 \, d^{2}} + \frac {85 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}}{48 \, d} + \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{16 \, d^{3}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{15 \, d^{2} x} + \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{16 \, d^{3} x^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{15 \, d^{2} x^{3}} - \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{24 \, d x^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{5 \, x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{6 \, x^{6}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (186) = 372\).

Time = 0.31 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.46 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=\frac {{\left (5 \, d^{2} e^{7} + \frac {36 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e^{5}}{x} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} e^{3}}{x^{2}} - \frac {340 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2} e}{x^{3}} - \frac {1215 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2}}{e x^{4}} + \frac {1800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{2}}{e^{3} x^{5}}\right )} e^{12} x^{6}}{1920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} {\left | e \right |}} - \frac {d^{2} e^{7} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} - \frac {85 \, d^{2} e^{7} \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 )}{16 \, {\left | e \right |}} + \frac {1}{2} \, {\left (e^{7} x + 6 \, d e^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}} - \frac {\frac {1800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e^{9} {\left | e \right |}}{x} - \frac {1215 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} e^{7} {\left | e \right |}}{x^{2}} - \frac {340 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2} e^{5} {\left | e \right |}}{x^{3}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2} e^{3} {\left | e \right |}}{x^{4}} + \frac {36 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{2} e {\left | e \right |}}{x^{5}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{2} {\left | e \right |}}{e x^{6}}}{1920 \, e^{6}} \]

[In]

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

[Out]

1/1920*(5*d^2*e^7 + 36*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e^5/x + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^
2*d^2*e^3/x^2 - 340*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2*e/x^3 - 1215*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))
^4*d^2/(e*x^4) + 1800*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^2/(e^3*x^5))*e^12*x^6/((d*e + sqrt(-e^2*x^2 + d^
2)*abs(e))^6*abs(e)) - 1/2*d^2*e^7*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 85/16*d^2*e^7*log(1/2*abs(-2*d*e - 2*s
qrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/2*(e^7*x + 6*d*e^6)*sqrt(-e^2*x^2 + d^2) - 1/1920*(1800*(
d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e^9*abs(e)/x - 1215*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*e^7*abs(e
)/x^2 - 340*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2*e^5*abs(e)/x^3 + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^
4*d^2*e^3*abs(e)/x^4 + 36*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^2*e*abs(e)/x^5 + 5*(d*e + sqrt(-e^2*x^2 + d^
2)*abs(e))^6*d^2*abs(e)/(e*x^6))/e^6

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^7} \,d x \]

[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)