\(\int \frac {x^2 (d^2-e^2 x^2)^{5/2}}{d+e x} \, dx\) [105]

Optimal result

Integrand size = 27, antiderivative size = 140 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3} \]

[Out]

1/24*d^3*x*(-e^2*x^2+d^2)^(3/2)/e^2+1/30*d*(-5*e*x+6*d)*(-e^2*x^2+d^2)^(5/2)/e^3-1/7*(-e^2*x^2+d^2)^(7/2)/e^3+
1/16*d^7*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^3+1/16*d^5*x*(-e^2*x^2+d^2)^(1/2)/e^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1653, 12, 799, 794, 201, 223, 209} \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2} \]

[In]

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

[Out]

(d^5*x*Sqrt[d^2 - e^2*x^2])/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) + (d*(6*d - 5*e*x)*(d^2 - e^2*x^
2)^(5/2))/(30*e^3) - (d^2 - e^2*x^2)^(7/2)/(7*e^3) + (d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 799

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

Rule 1653

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\int \frac {7 d e^3 x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx}{7 e^4} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {d \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx}{e} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\int x \left (d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2} \\ & = \frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^3 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e^2} \\ & = \frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^5 \int \sqrt {d^2-e^2 x^2} \, dx}{8 e^2} \\ & = \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^2} \\ & = \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^2} \\ & = \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}+\frac {d (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^3}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (96 d^6-105 d^5 e x+48 d^4 e^2 x^2+490 d^3 e^3 x^3-384 d^2 e^4 x^4-280 d e^5 x^5+240 e^6 x^6\right )-210 d^7 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{1680 e^3} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(96*d^6 - 105*d^5*e*x + 48*d^4*e^2*x^2 + 490*d^3*e^3*x^3 - 384*d^2*e^4*x^4 - 280*d*e^5*x^
5 + 240*e^6*x^6) - 210*d^7*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(1680*e^3)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {\left (240 e^{6} x^{6}-280 d \,e^{5} x^{5}-384 d^{2} e^{4} x^{4}+490 d^{3} x^{3} e^{3}+48 d^{4} e^{2} x^{2}-105 d^{5} e x +96 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{1680 e^{3}}+\frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{2} \sqrt {e^{2}}}\)	\(119\)
default	\(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{3}}-\frac {d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{3}}\)	\(317\)

[In]

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

[Out]

1/1680*(240*e^6*x^6-280*d*e^5*x^5-384*d^2*e^4*x^4+490*d^3*e^3*x^3+48*d^4*e^2*x^2-105*d^5*e*x+96*d^6)/e^3*(-e^2
*x^2+d^2)^(1/2)+1/16*d^7/e^2/(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 = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (240 \, e^{6} x^{6} - 280 \, d e^{5} x^{5} - 384 \, d^{2} e^{4} x^{4} + 490 \, d^{3} e^{3} x^{3} + 48 \, d^{4} e^{2} x^{2} - 105 \, d^{5} e x + 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1680 \, e^{3}} \]

[In]

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

[Out]

-1/1680*(210*d^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (240*e^6*x^6 - 280*d*e^5*x^5 - 384*d^2*e^4*x^4 +
490*d^3*e^3*x^3 + 48*d^4*e^2*x^2 - 105*d^5*e*x + 96*d^6)*sqrt(-e^2*x^2 + d^2))/e^3

Sympy [A] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.65 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=d^{3} \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**3*Piecewise((d**4*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))/(8*e**2) + sqrt(d**2 - e**2*x**2)*(-d**2*x/(8*e**2) + x**3/4), Ne(e**2,
0)), (x**3*sqrt(d**2)/3, True)) - d**2*e*Piecewise((sqrt(d**2 - e**2*x**2)*(-2*d**4/(15*e**4) - d**2*x**2/(15*
e**2) + x**4/5), Ne(e**2, 0)), (x**4*sqrt(d**2)/4, True)) - d*e**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)) + e**3*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))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.41 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{3}} - \frac {5 \, d^{7} \arcsin \left (\frac {e x}{d}\right )}{16 \, e^{3}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x}{8 \, e^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{16 \, e^{2}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{4 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{24 \, e^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{6 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{5 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e^{3}} \]

[In]

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

[Out]

-3/8*I*d^7*arcsin(e*x/d + 2)/e^3 - 5/16*d^7*arcsin(e*x/d)/e^3 + 3/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^5*x/e^2
- 5/16*sqrt(-e^2*x^2 + d^2)*d^5*x/e^2 + 3/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^6/e^3 + 1/24*(-e^2*x^2 + d^2)^(3
/2)*d^3*x/e^2 - 1/6*(-e^2*x^2 + d^2)^(5/2)*d*x/e^2 + 1/5*(-e^2*x^2 + d^2)^(5/2)*d^2/e^3 - 1/7*(-e^2*x^2 + d^2)
^(7/2)/e^3

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^{7} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, e^{2} {\left | e \right |}} + \frac {1}{1680} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {96 \, d^{6}}{e^{3}} - {\left (\frac {105 \, d^{5}}{e^{2}} - 2 \, {\left (\frac {24 \, d^{4}}{e} + {\left (245 \, d^{3} - 4 \, {\left (48 \, d^{2} e - 5 \, {\left (6 \, e^{3} x - 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

[In]

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

[Out]

1/16*d^7*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^2*abs(e)) + 1/1680*sqrt(-e^2*x^2 + d^2)*(96*d^6/e^3 - (105*d^5/e^2 - 2
*(24*d^4/e + (245*d^3 - 4*(48*d^2*e - 5*(6*e^3*x - 7*d*e^2)*x)*x)*x)*x)*x)

Mupad [F(-1)]

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

[In]

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

[Out]

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