\(\int \frac {x^5 (d^2-e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\) [198]

Optimal result

Integrand size = 27, antiderivative size = 252 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {65 d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6} \]

[Out]

65/4*d^7*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6+d^4*(-e*x+d)^4/e^6/(-e^2*x^2+d^2)^(1/2)+515/21*d^6*(-e^2*x^2+d^2
)^(1/2)/e^6-49/4*d^5*x*(-e^2*x^2+d^2)^(1/2)/e^5+121/21*d^4*x^2*(-e^2*x^2+d^2)^(1/2)/e^4-17/6*d^3*x^3*(-e^2*x^2
+d^2)^(1/2)/e^3+11/7*d^2*x^4*(-e^2*x^2+d^2)^(1/2)/e^2-2/3*d*x^5*(-e^2*x^2+d^2)^(1/2)/e+1/7*x^6*(-e^2*x^2+d^2)^
(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1649, 1829, 655, 223, 209} \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {65 d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3} \]

[In]

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

[Out]

(d^4*(d - e*x)^4)/(e^6*Sqrt[d^2 - e^2*x^2]) + (515*d^6*Sqrt[d^2 - e^2*x^2])/(21*e^6) - (49*d^5*x*Sqrt[d^2 - e^
2*x^2])/(4*e^5) + (121*d^4*x^2*Sqrt[d^2 - e^2*x^2])/(21*e^4) - (17*d^3*x^3*Sqrt[d^2 - e^2*x^2])/(6*e^3) + (11*
d^2*x^4*Sqrt[d^2 - e^2*x^2])/(7*e^2) - (2*d*x^5*Sqrt[d^2 - e^2*x^2])/(3*e) + (x^6*Sqrt[d^2 - e^2*x^2])/7 + (65
*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^6)

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 655

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

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 1649

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

Rule 1829

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*(q + 2*p + 1))), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^5}{e^5}+\frac {d^4 x}{e^4}-\frac {d^3 x^2}{e^3}+\frac {d^2 x^3}{e^2}-\frac {d x^4}{e}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {\frac {28 d^8}{e^3}-\frac {91 d^7 x}{e^2}+\frac {112 d^6 x^2}{e}-77 d^5 x^3+56 d^4 e x^4-55 d^3 e^2 x^5+28 d^2 e^3 x^6}{\sqrt {d^2-e^2 x^2}} \, dx}{7 d e^2} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-\frac {168 d^8}{e}+546 d^7 x-672 d^6 e x^2+462 d^5 e^2 x^3-476 d^4 e^3 x^4+330 d^3 e^4 x^5}{\sqrt {d^2-e^2 x^2}} \, dx}{42 d e^4} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {840 d^8 e-2730 d^7 e^2 x+3360 d^6 e^3 x^2-3630 d^5 e^4 x^3+2380 d^4 e^5 x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{210 d e^6} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-3360 d^8 e^3+10920 d^7 e^4 x-20580 d^6 e^5 x^2+14520 d^5 e^6 x^3}{\sqrt {d^2-e^2 x^2}} \, dx}{840 d e^8} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {10080 d^8 e^5-61800 d^7 e^6 x+61740 d^6 e^7 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{2520 d e^{10}} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-81900 d^8 e^7+123600 d^7 e^8 x}{\sqrt {d^2-e^2 x^2}} \, dx}{5040 d e^{12}} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\left (65 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^5} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\left (65 d^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 )}{4 e^5} \\ & = \frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {65 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.60 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {\frac {e \sqrt {d^2-e^2 x^2} \left (2144 d^7+779 d^6 e x-293 d^5 e^2 x^2+162 d^4 e^3 x^3-106 d^3 e^4 x^4+76 d^2 e^5 x^5-44 d e^6 x^6+12 e^7 x^7\right )}{d+e x}+1365 d^7 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{84 e^7} \]

[In]

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

[Out]

((e*Sqrt[d^2 - e^2*x^2]*(2144*d^7 + 779*d^6*e*x - 293*d^5*e^2*x^2 + 162*d^4*e^3*x^3 - 106*d^3*e^4*x^4 + 76*d^2
*e^5*x^5 - 44*d*e^6*x^6 + 12*e^7*x^7))/(d + e*x) + 1365*d^7*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^
2]])/(84*e^7)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {\left (12 e^{6} x^{6}-56 d \,e^{5} x^{5}+132 d^{2} e^{4} x^{4}-238 d^{3} x^{3} e^{3}+400 d^{4} e^{2} x^{2}-693 d^{5} e x +1472 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{84 e^{6}}+\frac {65 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 e^{5} \sqrt {e^{2}}}+\frac {8 d^{7} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{7} \left (x +\frac {d}{e}\right )}\)	\(164\)
default	\(\text {Expression too large to display}\)	\(1213\)

[In]

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

[Out]

1/84*(12*e^6*x^6-56*d*e^5*x^5+132*d^2*e^4*x^4-238*d^3*e^3*x^3+400*d^4*e^2*x^2-693*d^5*e*x+1472*d^6)/e^6*(-e^2*
x^2+d^2)^(1/2)+65/4*d^7/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+8*d^7/e^7/(x+d/e)*(-(x+d/e)
^2*e^2+2*d*e*(x+d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.62 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {2144 \, d^{7} e x + 2144 \, d^{8} - 2730 \, {\left (d^{7} e x + d^{8}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (12 \, e^{7} x^{7} - 44 \, d e^{6} x^{6} + 76 \, d^{2} e^{5} x^{5} - 106 \, d^{3} e^{4} x^{4} + 162 \, d^{4} e^{3} x^{3} - 293 \, d^{5} e^{2} x^{2} + 779 \, d^{6} e x + 2144 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{84 \, {\left (e^{7} x + d e^{6}\right )}} \]

[In]

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

[Out]

1/84*(2144*d^7*e*x + 2144*d^8 - 2730*(d^7*e*x + d^8)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (12*e^7*x^7 -
44*d*e^6*x^6 + 76*d^2*e^5*x^5 - 106*d^3*e^4*x^4 + 162*d^4*e^3*x^3 - 293*d^5*e^2*x^2 + 779*d^6*e*x + 2144*d^7)
*sqrt(-e^2*x^2 + d^2))/(e^7*x + d*e^6)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.90 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{2 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7}}{e^{7} x + d e^{6}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{3 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{6 \, {\left (e^{7} x + d e^{6}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{2 \, {\left (e^{7} x + d e^{6}\right )}} + \frac {5 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{6}} + \frac {75 \, d^{7} \arcsin \left (\frac {e x}{d}\right )}{4 \, e^{6}} - \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x}{2 \, e^{5}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{4 \, e^{5}} - \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{e^{6}} + \frac {25 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}}{2 \, e^{6}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{3 \, e^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{6 \, e^{6}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{3 \, e^{5}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{e^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e^{6}} \]

[In]

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

[Out]

-1/2*(-e^2*x^2 + d^2)^(5/2)*d^5/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) - 5/2*(-e^2*x^2 + d^2)^(3/2)*d
^6/(e^8*x^2 + 2*d*e^7*x + d^2*e^6) + 15*sqrt(-e^2*x^2 + d^2)*d^7/(e^7*x + d*e^6) + 5/3*(-e^2*x^2 + d^2)^(5/2)*
d^4/(e^8*x^2 + 2*d*e^7*x + d^2*e^6) + 25/6*(-e^2*x^2 + d^2)^(3/2)*d^5/(e^7*x + d*e^6) - 5/2*(-e^2*x^2 + d^2)^(
5/2)*d^3/(e^7*x + d*e^6) + 5/2*I*d^7*arcsin(e*x/d + 2)/e^6 + 75/4*d^7*arcsin(e*x/d)/e^6 - 5/2*sqrt(e^2*x^2 + 4
*d*e*x + 3*d^2)*d^5*x/e^5 - 5/4*sqrt(-e^2*x^2 + d^2)*d^5*x/e^5 - 5*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^6/e^6 + 2
5/2*sqrt(-e^2*x^2 + d^2)*d^6/e^6 + 5/3*(-e^2*x^2 + d^2)^(3/2)*d^3*x/e^5 - 25/6*(-e^2*x^2 + d^2)^(3/2)*d^4/e^6
- 2/3*(-e^2*x^2 + d^2)^(5/2)*d*x/e^5 + 2*(-e^2*x^2 + d^2)^(5/2)*d^2/e^6 - 1/7*(-e^2*x^2 + d^2)^(7/2)/e^6

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.60 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {65 \, d^{7} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{4 \, e^{5} {\left | e \right |}} + \frac {1}{84} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, {\left ({\left (2 \, {\left ({\left (3 \, x - \frac {14 \, d}{e}\right )} x + \frac {33 \, d^{2}}{e^{2}}\right )} x - \frac {119 \, d^{3}}{e^{3}}\right )} x + \frac {200 \, d^{4}}{e^{4}}\right )} x - \frac {693 \, d^{5}}{e^{5}}\right )} x + \frac {1472 \, d^{6}}{e^{6}}\right )} - \frac {16 \, d^{7}}{e^{5} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} \]

[In]

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

[Out]

65/4*d^7*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^5*abs(e)) + 1/84*sqrt(-e^2*x^2 + d^2)*((2*((2*((3*x - 14*d/e)*x + 33*d
^2/e^2)*x - 119*d^3/e^3)*x + 200*d^4/e^4)*x - 693*d^5/e^5)*x + 1472*d^6/e^6) - 16*d^7/(e^5*((d*e + sqrt(-e^2*x
^2 + d^2)*abs(e))/(e^2*x) + 1)*abs(e))

Mupad [F(-1)]

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

[In]

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

[Out]

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