\(\int \frac {x^5 (d^2-e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\) [157]

Optimal result

Integrand size = 27, antiderivative size = 229 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {5 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^6} \]

[Out]

-4/21*d^4*x^2*(-e^2*x^2+d^2)^(3/2)/e^4+5/24*d^3*x^3*(-e^2*x^2+d^2)^(3/2)/e^3-5/21*d^2*x^4*(-e^2*x^2+d^2)^(3/2)
/e^2+1/4*d*x^5*(-e^2*x^2+d^2)^(3/2)/e-1/9*x^6*(-e^2*x^2+d^2)^(3/2)-1/2016*d^5*(-315*e*x+256*d)*(-e^2*x^2+d^2)^
(3/2)/e^6-5/64*d^9*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6-5/64*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1823, 847, 794, 201, 223, 209} \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {5 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^6}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}-\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3} \]

[In]

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

[Out]

(-5*d^7*x*Sqrt[d^2 - e^2*x^2])/(64*e^5) - (4*d^4*x^2*(d^2 - e^2*x^2)^(3/2))/(21*e^4) + (5*d^3*x^3*(d^2 - e^2*x
^2)^(3/2))/(24*e^3) - (5*d^2*x^4*(d^2 - e^2*x^2)^(3/2))/(21*e^2) + (d*x^5*(d^2 - e^2*x^2)^(3/2))/(4*e) - (x^6*
(d^2 - e^2*x^2)^(3/2))/9 - (d^5*(256*d - 315*e*x)*(d^2 - e^2*x^2)^(3/2))/(2016*e^6) - (5*d^9*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(64*e^6)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

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}& = \int x^5 (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx \\ & = -\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^5 \left (-15 d^2 e^2+18 d e^3 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{9 e^2} \\ & = \frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^4 \left (-90 d^3 e^3+120 d^2 e^4 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{72 e^4} \\ & = -\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^3 \left (-480 d^4 e^4+630 d^3 e^5 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{504 e^6} \\ & = \frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^2 \left (-1890 d^5 e^5+2880 d^4 e^6 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{3024 e^8} \\ & = -\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x \left (-5760 d^6 e^6+9450 d^5 e^7 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{15120 e^{10}} \\ & = -\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {\left (5 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{32 e^5} \\ & = -\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {\left (5 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{64 e^5} \\ & = -\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {\left (5 d^9\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^5} \\ & = -\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {5 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.68 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {e \sqrt {d^2-e^2 x^2} \left (-512 d^8+315 d^7 e x-256 d^6 e^2 x^2+210 d^5 e^3 x^3-192 d^4 e^4 x^4+168 d^3 e^5 x^5+512 d^2 e^6 x^6-1008 d e^7 x^7+448 e^8 x^8\right )-315 d^9 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{4032 e^7} \]

[In]

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

[Out]

(e*Sqrt[d^2 - e^2*x^2]*(-512*d^8 + 315*d^7*e*x - 256*d^6*e^2*x^2 + 210*d^5*e^3*x^3 - 192*d^4*e^4*x^4 + 168*d^3
*e^5*x^5 + 512*d^2*e^6*x^6 - 1008*d*e^7*x^7 + 448*e^8*x^8) - 315*d^9*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2
- e^2*x^2]])/(4032*e^7)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62

method	result	size
risch	\(-\frac {\left (-448 e^{8} x^{8}+1008 d \,e^{7} x^{7}-512 d^{2} e^{6} x^{6}-168 d^{3} e^{5} x^{5}+192 d^{4} x^{4} e^{4}-210 d^{5} e^{3} x^{3}+256 d^{6} e^{2} x^{2}-315 d^{7} e x +512 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{4032 e^{6}}-\frac {5 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{64 e^{5} \sqrt {e^{2}}}\)	\(141\)
default	\(\frac {-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}}{e^{2}}-\frac {4 d^{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 )}{e^{5}}-\frac {2 d \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 )}{e^{3}}-\frac {3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{6}}-\frac {d^{5} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{e^{7}}+\frac {5 d^{4} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{6}}\)	\(750\)

[In]

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

[Out]

-1/4032*(-448*e^8*x^8+1008*d*e^7*x^7-512*d^2*e^6*x^6-168*d^3*e^5*x^5+192*d^4*e^4*x^4-210*d^5*e^3*x^3+256*d^6*e
^2*x^2-315*d^7*e*x+512*d^8)/e^6*(-e^2*x^2+d^2)^(1/2)-5/64*d^9/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d
^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.60 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {630 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (448 \, e^{8} x^{8} - 1008 \, d e^{7} x^{7} + 512 \, d^{2} e^{6} x^{6} + 168 \, d^{3} e^{5} x^{5} - 192 \, d^{4} e^{4} x^{4} + 210 \, d^{5} e^{3} x^{3} - 256 \, d^{6} e^{2} x^{2} + 315 \, d^{7} e x - 512 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4032 \, e^{6}} \]

[In]

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

[Out]

1/4032*(630*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (448*e^8*x^8 - 1008*d*e^7*x^7 + 512*d^2*e^6*x^6 +
168*d^3*e^5*x^5 - 192*d^4*e^4*x^4 + 210*d^5*e^3*x^3 - 256*d^6*e^2*x^2 + 315*d^7*e*x - 512*d^8)*sqrt(-e^2*x^2 +
d^2))/e^6

Sympy [A] (verification not implemented)

Time = 2.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.31 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=d^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {5 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{6}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{6} x}{128 e^{6}} - \frac {5 d^{4} x^{3}}{192 e^{4}} - \frac {d^{2} x^{5}}{48 e^{2}} + \frac {x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{7} \sqrt {d^{2}}}{7} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {16 d^{8}}{315 e^{8}} - \frac {8 d^{6} x^{2}}{315 e^{6}} - \frac {2 d^{4} x^{4}}{105 e^{4}} - \frac {d^{2} x^{6}}{63 e^{2}} + \frac {x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**2*Piecewise((sqrt(d**2 - e**2*x**2)*(-8*d**6/(105*e**6) - 4*d**4*x**2/(105*e**4) - d**2*x**4/(35*e**2) + x*
*6/7), Ne(e**2, 0)), (x**6*sqrt(d**2)/6, True)) - 2*d*e*Piecewise((5*d**8*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))/(128*e**6) + sqrt(d
**2 - e**2*x**2)*(-5*d**6*x/(128*e**6) - 5*d**4*x**3/(192*e**4) - d**2*x**5/(48*e**2) + x**7/8), Ne(e**2, 0)),
(x**7*sqrt(d**2)/7, True)) + e**2*Piecewise((sqrt(d**2 - e**2*x**2)*(-16*d**8/(315*e**8) - 8*d**6*x**2/(315*e
**6) - 2*d**4*x**4/(105*e**4) - d**2*x**6/(63*e**2) + x**8/9), Ne(e**2, 0)), (x**8*sqrt(d**2)/8, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.31 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{4 \, {\left (e^{7} x + d e^{6}\right )}} - \frac {5 i \, d^{9} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{6}} - \frac {85 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{64 \, e^{6}} + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7} x}{4 \, e^{5}} - \frac {85 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{64 \, e^{5}} + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{8}}{2 \, e^{6}} + \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{96 \, e^{5}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{12 \, e^{6}} - \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{24 \, e^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{e^{6}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x}{4 \, e^{5}} - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e^{6}} \]

[In]

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

[Out]

-1/4*(-e^2*x^2 + d^2)^(5/2)*d^5/(e^7*x + d*e^6) - 5/4*I*d^9*arcsin(e*x/d + 2)/e^6 - 85/64*d^9*arcsin(e*x/d)/e^
6 + 5/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^7*x/e^5 - 85/64*sqrt(-e^2*x^2 + d^2)*d^7*x/e^5 + 5/2*sqrt(e^2*x^2 +
4*d*e*x + 3*d^2)*d^8/e^6 + 35/96*(-e^2*x^2 + d^2)^(3/2)*d^5*x/e^5 - 5/12*(-e^2*x^2 + d^2)^(3/2)*d^6/e^6 - 17/2
4*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^5 - 1/9*(-e^2*x^2 + d^2)^(7/2)*x^2/e^4 + (-e^2*x^2 + d^2)^(5/2)*d^4/e^6 + 1/4
*(-e^2*x^2 + d^2)^(7/2)*d*x/e^5 - 29/63*(-e^2*x^2 + d^2)^(7/2)*d^2/e^6

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.50 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (161280 \, d^{10} e^{10} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (315 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {17}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 18774 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {15}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 10458 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 68958 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 8192 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 32418 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 10458 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2730 \, d^{10} e^{10} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 315 \, d^{10} e^{10} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{9}}{d^{9}}\right )} {\left | e \right |}}{1032192 \, d e^{17}} \]

[In]

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

[Out]

1/1032192*(161280*d^10*e^10*arctan(sqrt(2*d/(e*x + d) - 1))*sgn(1/(e*x + d))*sgn(e) + (315*d^10*e^10*(2*d/(e*x
+ d) - 1)^(17/2)*sgn(1/(e*x + d))*sgn(e) - 18774*d^10*e^10*(2*d/(e*x + d) - 1)^(15/2)*sgn(1/(e*x + d))*sgn(e)
+ 10458*d^10*e^10*(2*d/(e*x + d) - 1)^(13/2)*sgn(1/(e*x + d))*sgn(e) - 68958*d^10*e^10*(2*d/(e*x + d) - 1)^(1
1/2)*sgn(1/(e*x + d))*sgn(e) - 8192*d^10*e^10*(2*d/(e*x + d) - 1)^(9/2)*sgn(1/(e*x + d))*sgn(e) - 32418*d^10*e
^10*(2*d/(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 10458*d^10*e^10*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x +
d))*sgn(e) - 2730*d^10*e^10*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))*sgn(e) - 315*d^10*e^10*sqrt(2*d/(e*x +
d) - 1)*sgn(1/(e*x + d))*sgn(e))*(e*x + d)^9/d^9)*abs(e)/(d*e^17)

Mupad [F(-1)]

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

[In]

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

[Out]

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