\(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x} \, dx\) [71]

Optimal result

Integrand size = 27, antiderivative size = 190 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

1/192*d^4*(125*e*x+64*d)*(-e^2*x^2+d^2)^(3/2)+1/240*d^2*(125*e*x+48*d)*(-e^2*x^2+d^2)^(5/2)-3/7*d*(-e^2*x^2+d^
2)^(7/2)-1/8*e*x*(-e^2*x^2+d^2)^(7/2)+125/128*d^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-d^8*arctanh((-e^2*x^2+d^2)^
(1/2)/d)+1/128*d^6*(125*e*x+128*d)*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 190, 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 = {1823, 829, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {125}{128} d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2} \]

[In]

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

[Out]

(d^6*(128*d + 125*e*x)*Sqrt[d^2 - e^2*x^2])/128 + (d^4*(64*d + 125*e*x)*(d^2 - e^2*x^2)^(3/2))/192 + (d^2*(48*
d + 125*e*x)*(d^2 - e^2*x^2)^(5/2))/240 - (3*d*(d^2 - e^2*x^2)^(7/2))/7 - (e*x*(d^2 - e^2*x^2)^(7/2))/8 + (125
*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/128 - d^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

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 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-8 d^3 e^2-25 d^2 e^3 x-24 d e^4 x^2\right )}{x} \, dx}{8 e^2} \\ & = -\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int \frac {\left (56 d^3 e^4+175 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx}{56 e^4} \\ & = \frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int \frac {\left (-336 d^5 e^6-875 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{336 e^6} \\ & = \frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int \frac {\left (1344 d^7 e^8+2625 d^6 e^9 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{1344 e^8} \\ & = \frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int \frac {-2688 d^9 e^{10}-2625 d^8 e^{11} x}{x \sqrt {d^2-e^2 x^2}} \, dx}{2688 e^{10}} \\ & = \frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+d^9 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{128} \left (125 d^8 e\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {1}{2} d^9 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{128} \left (125 d^8 e\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}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^9 \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 )}{e^2} \\ & = \frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (14848 d^7+27195 d^6 e x+7424 d^5 e^2 x^2-17710 d^4 e^3 x^3-14592 d^3 e^4 x^4+1960 d^2 e^5 x^5+5760 d e^6 x^6+1680 e^7 x^7\right )}{13440}-\frac {125}{64} d^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-d^7 \sqrt {d^2} \log (x)+d^7 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(14848*d^7 + 27195*d^6*e*x + 7424*d^5*e^2*x^2 - 17710*d^4*e^3*x^3 - 14592*d^3*e^4*x^4 + 1
960*d^2*e^5*x^5 + 5760*d*e^6*x^6 + 1680*e^7*x^7))/13440 - (125*d^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^
2])])/64 - d^7*Sqrt[d^2]*Log[x] + d^7*Sqrt[d^2]*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(352\) vs. \(2(164)=328\).

Time = 0.38 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.86

method	result	size
default	\(e^{3} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )+d^{3} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )+3 d^{2} e \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )-\frac {3 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7}\)	\(353\)

[In]

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

[Out]

e^3*(-1/8*x*(-e^2*x^2+d^2)^(7/2)/e^2+1/8*d^2/e^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*
(1/5*(-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))))+3*d^2*e*(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)))))-3/7*d*
(-e^2*x^2+d^2)^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=-\frac {125}{64} \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \frac {1}{13440} \, {\left (1680 \, e^{7} x^{7} + 5760 \, d e^{6} x^{6} + 1960 \, d^{2} e^{5} x^{5} - 14592 \, d^{3} e^{4} x^{4} - 17710 \, d^{4} e^{3} x^{3} + 7424 \, d^{5} e^{2} x^{2} + 27195 \, d^{6} e x + 14848 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

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

[Out]

-125/64*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + d^8*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + 1/13440*(1680
*e^7*x^7 + 5760*d*e^6*x^6 + 1960*d^2*e^5*x^5 - 14592*d^3*e^4*x^4 - 17710*d^4*e^3*x^3 + 7424*d^5*e^2*x^2 + 2719
5*d^6*e*x + 14848*d^7)*sqrt(-e^2*x^2 + d^2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.89 (sec) , antiderivative size = 954, normalized size of antiderivative = 5.02 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

d**7*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)) + 3*d**6*e*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)) + d**5*e**2*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)) - 5*d**4*e**3*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))
- 5*d**3*e**4*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)) + d**2*e**5*Piecewise((d**6*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))/(16*e**4) - d**4*x*sqrt(d**2 - e**2*x**2)/(16*e**4) - d**2*x**3*sqrt(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)) + 3*d*e**6*Piecewise((-8*d**6*sqrt(d**2
- e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(3
5*e**2) + x**6*sqrt(d**2 - e**2*x**2)/7, Ne(e**2, 0)), (x**6*sqrt(d**2)/6, True)) + e**7*Piecewise((5*d**8*Pie
cewise((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))/(128*e**6) - 5*d**6*x*sqrt(d**2 - e**2*x**2)/(128*e**6) - 5*d**4*x**3*sqrt(d**2 - e**2*x**2)/(19
2*e**4) - d**2*x**5*sqrt(d**2 - e**2*x**2)/(48*e**2) + x**7*sqrt(d**2 - e**2*x**2)/8, Ne(e**2, 0)), (x**7*sqrt
(d**2)/7, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {125 \, d^{8} e \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}}} - d^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {125}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{7} + \frac {125}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e x + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} + \frac {25}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e x + \frac {1}{5} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} - \frac {1}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x - \frac {3}{7} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d \]

[In]

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

[Out]

125/128*d^8*e*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - d^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))
+ 125/128*sqrt(-e^2*x^2 + d^2)*d^6*e*x + sqrt(-e^2*x^2 + d^2)*d^7 + 125/192*(-e^2*x^2 + d^2)^(3/2)*d^4*e*x + 1
/3*(-e^2*x^2 + d^2)^(3/2)*d^5 + 25/48*(-e^2*x^2 + d^2)^(5/2)*d^2*e*x + 1/5*(-e^2*x^2 + d^2)^(5/2)*d^3 - 1/8*(-
e^2*x^2 + d^2)^(7/2)*e*x - 3/7*(-e^2*x^2 + d^2)^(7/2)*d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx=\frac {125 \, d^{8} e \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, {\left | e \right |}} - \frac {d^{8} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} + \frac {1}{13440} \, {\left (14848 \, d^{7} + {\left (27195 \, d^{6} e + 2 \, {\left (3712 \, d^{5} e^{2} - {\left (8855 \, d^{4} e^{3} + 4 \, {\left (1824 \, d^{3} e^{4} - 5 \, {\left (49 \, d^{2} e^{5} + 6 \, {\left (7 \, e^{7} x + 24 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

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

[Out]

125/128*d^8*e*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - d^8*e*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(
e^2*abs(x)))/abs(e) + 1/13440*(14848*d^7 + (27195*d^6*e + 2*(3712*d^5*e^2 - (8855*d^4*e^3 + 4*(1824*d^3*e^4 -
5*(49*d^2*e^5 + 6*(7*e^7*x + 24*d*e^6)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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