\(\int \frac {x^4 (d^2-e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\) [158]

Optimal result

Integrand size = 27, antiderivative size = 200 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {13 d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]

[Out]

8/35*d^3*x^2*(-e^2*x^2+d^2)^(3/2)/e^3-13/48*d^2*x^3*(-e^2*x^2+d^2)^(3/2)/e^2+2/7*d*x^4*(-e^2*x^2+d^2)^(3/2)/e-
1/8*x^5*(-e^2*x^2+d^2)^(3/2)+1/6720*d^4*(-1365*e*x+1024*d)*(-e^2*x^2+d^2)^(3/2)/e^5+13/128*d^8*arctan(e*x/(-e^
2*x^2+d^2)^(1/2))/e^5+13/128*d^6*x*(-e^2*x^2+d^2)^(1/2)/e^4

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1823, 847, 794, 201, 223, 209} \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {13 d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3} \]

[In]

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

[Out]

(13*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (8*d^3*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^3) - (13*d^2*x^3*(d^2 - e^2
*x^2)^(3/2))/(48*e^2) + (2*d*x^4*(d^2 - e^2*x^2)^(3/2))/(7*e) - (x^5*(d^2 - e^2*x^2)^(3/2))/8 + (d^4*(1024*d -
1365*e*x)*(d^2 - e^2*x^2)^(3/2))/(6720*e^5) + (13*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

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^4 (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx \\ & = -\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^4 \left (-13 d^2 e^2+16 d e^3 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{8 e^2} \\ & = \frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^3 \left (-64 d^3 e^3+91 d^2 e^4 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{56 e^4} \\ & = -\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^2 \left (-273 d^4 e^4+384 d^3 e^5 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{336 e^6} \\ & = \frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x \left (-768 d^5 e^5+1365 d^4 e^6 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{1680 e^8} \\ & = \frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {\left (13 d^6\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^4} \\ & = \frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {\left (13 d^8\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^4} \\ & = \frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {\left (13 d^8\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4} \\ & = \frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {13 d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.68 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (2048 d^7-1365 d^6 e x+1024 d^5 e^2 x^2-910 d^4 e^3 x^3+768 d^3 e^4 x^4+1960 d^2 e^5 x^5-3840 d e^6 x^6+1680 e^7 x^7\right )-2730 d^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{13440 e^5} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(2048*d^7 - 1365*d^6*e*x + 1024*d^5*e^2*x^2 - 910*d^4*e^3*x^3 + 768*d^3*e^4*x^4 + 1960*d^
2*e^5*x^5 - 3840*d*e^6*x^6 + 1680*e^7*x^7) - 2730*d^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(13440*
e^5)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {\left (1680 e^{7} x^{7}-3840 d \,e^{6} x^{6}+1960 d^{2} e^{5} x^{5}+768 d^{3} e^{4} x^{4}-910 d^{4} e^{3} x^{3}+1024 d^{5} e^{2} x^{2}-1365 d^{6} e x +2048 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{13440 e^{5}}+\frac {13 d^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\)	\(130\)
default	\(\frac {-\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}}}{e^{2}}+\frac {3 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 )}{e^{4}}+\frac {2 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{5}}+\frac {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 {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^{6}}-\frac {4 d^{3} \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^{5}}\)	\(696\)

[In]

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

[Out]

1/13440*(1680*e^7*x^7-3840*d*e^6*x^6+1960*d^2*e^5*x^5+768*d^3*e^4*x^4-910*d^4*e^3*x^3+1024*d^5*e^2*x^2-1365*d^
6*e*x+2048*d^7)/e^5*(-e^2*x^2+d^2)^(1/2)+13/128*d^8/e^4/(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 = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {2730 \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (1680 \, e^{7} x^{7} - 3840 \, d e^{6} x^{6} + 1960 \, d^{2} e^{5} x^{5} + 768 \, d^{3} e^{4} x^{4} - 910 \, d^{4} e^{3} x^{3} + 1024 \, d^{5} e^{2} x^{2} - 1365 \, d^{6} e x + 2048 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{13440 \, e^{5}} \]

[In]

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

[Out]

-1/13440*(2730*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (1680*e^7*x^7 - 3840*d*e^6*x^6 + 1960*d^2*e^5*x
^5 + 768*d^3*e^4*x^4 - 910*d^4*e^3*x^3 + 1024*d^5*e^2*x^2 - 1365*d^6*e*x + 2048*d^7)*sqrt(-e^2*x^2 + d^2))/e^5

Sympy [A] (verification not implemented)

Time = 2.03 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.70 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=d^{2} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {5 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{6}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{6} x}{128 e^{6}} - \frac {5 d^{4} x^{3}}{192 e^{4}} - \frac {d^{2} x^{5}}{48 e^{2}} + \frac {x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{7} \sqrt {d^{2}}}{7} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**2*Piecewise((d**6*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))/(16*e**4) + sqrt(d**2 - e**2*x**2)*(-d**4*x/(16*e**4) - d**2*x**3/(24*e*
*2) + x**5/6), Ne(e**2, 0)), (x**5*sqrt(d**2)/5, True)) - 2*d*e*Piecewise((sqrt(d**2 - e**2*x**2)*(-8*d**6/(10
5*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)) + e
**2*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))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{4 \, {\left (e^{6} x + d e^{5}\right )}} + \frac {7 i \, d^{8} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{5}} + \frac {125 \, d^{8} \arcsin \left (\frac {e x}{d}\right )}{128 \, e^{5}} - \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6} x}{8 \, e^{4}} + \frac {125 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} x}{128 \, e^{4}} - \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7}}{4 \, e^{5}} - \frac {67 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x}{192 \, e^{4}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{12 \, e^{5}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x}{48 \, e^{4}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{5 \, e^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x}{8 \, e^{4}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{7 \, e^{5}} \]

[In]

integrate(x^4*(-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^4/(e^6*x + d*e^5) + 7/8*I*d^8*arcsin(e*x/d + 2)/e^5 + 125/128*d^8*arcsin(e*x/d)/e
^5 - 7/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^6*x/e^4 + 125/128*sqrt(-e^2*x^2 + d^2)*d^6*x/e^4 - 7/4*sqrt(e^2*x^2
+ 4*d*e*x + 3*d^2)*d^7/e^5 - 67/192*(-e^2*x^2 + d^2)^(3/2)*d^4*x/e^4 + 5/12*(-e^2*x^2 + d^2)^(3/2)*d^5/e^5 +
25/48*(-e^2*x^2 + d^2)^(5/2)*d^2*x/e^4 - 4/5*(-e^2*x^2 + d^2)^(5/2)*d^3/e^5 - 1/8*(-e^2*x^2 + d^2)^(7/2)*x/e^4
+ 2/7*(-e^2*x^2 + d^2)^(7/2)*d/e^5

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.56 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {{\left (349440 \, d^{9} e^{9} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (1365 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {15}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 61215 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 20517 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 141159 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 34969 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 34853 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 10465 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1365 \, d^{9} e^{9} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{8}}{d^{8}}\right )} {\left | e \right |}}{1720320 \, d e^{15}} \]

[In]

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

[Out]

-1/1720320*(349440*d^9*e^9*arctan(sqrt(2*d/(e*x + d) - 1))*sgn(1/(e*x + d))*sgn(e) + (1365*d^9*e^9*(2*d/(e*x +
d) - 1)^(15/2)*sgn(1/(e*x + d))*sgn(e) - 61215*d^9*e^9*(2*d/(e*x + d) - 1)^(13/2)*sgn(1/(e*x + d))*sgn(e) + 2
0517*d^9*e^9*(2*d/(e*x + d) - 1)^(11/2)*sgn(1/(e*x + d))*sgn(e) - 141159*d^9*e^9*(2*d/(e*x + d) - 1)^(9/2)*sgn
(1/(e*x + d))*sgn(e) - 34969*d^9*e^9*(2*d/(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 34853*d^9*e^9*(2*d/(e
*x + d) - 1)^(5/2)*sgn(1/(e*x + d))*sgn(e) - 10465*d^9*e^9*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))*sgn(e) -
1365*d^9*e^9*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn(e))*(e*x + d)^8/d^8)*abs(e)/(d*e^15)

Mupad [F(-1)]

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

[In]

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

[Out]

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