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

Optimal. Leaf size=252 \[ \frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}+\frac {65 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3} \]

[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)

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Rubi [A]  time = 0.66, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {852, 1635, 1815, 641, 217, 203} \[ \frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}+\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {65 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6} \]

Antiderivative was successfully verified.

[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

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 641

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 852

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 1635

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 1815

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

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Mathematica [A]  time = 0.23, size = 131, normalized size = 0.52 \[ \frac {1365 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (2144 d^7+779 d^6 e x-293 d^5 e^2 x^2+162 d^4 e^3 x^3-106 d^3 e^4 x^4+76 d^2 e^5 x^5-44 d e^6 x^6+12 e^7 x^7\right )}{d+e x}}{84 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(84*e^6)

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fricas [A]  time = 0.81, size = 156, normalized size = 0.62 \[ \frac {2144 \, d^{7} e x + 2144 \, d^{8} - 2730 \, {\left (d^{7} e x + d^{8}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (12 \, e^{7} x^{7} - 44 \, d e^{6} x^{6} + 76 \, d^{2} e^{5} x^{5} - 106 \, d^{3} e^{4} x^{4} + 162 \, d^{4} e^{3} x^{3} - 293 \, d^{5} e^{2} x^{2} + 779 \, d^{6} e x + 2144 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{84 \, {\left (e^{7} x + d e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-162*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^12*exp(2)^2-36*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/
exp(2))^5*exp(1)^10*exp(2)^3+720*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^12*e
xp(2)^2+684*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^10*exp(2)^3+162*d^7*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^8*exp(2)^4-402*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d
^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^12*exp(2)^2+350*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)/x/exp(2))^3*exp(1)^10*exp(2)^3+507*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^
8*exp(2)^4+123*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^6*exp(2)^5+1476*d^7*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^10*exp(2)^3-864*d^7*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^6*exp(2)^5-252*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^5*exp(1)^4*exp(2)^6+1248*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(
1)^8*exp(2)^4+84*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^6*exp(2)^5-654*d^7*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^4*exp(2)^6-192*d^7*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(2)^8-1836*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^2*exp(1)^6*exp(2)^5-1620*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^4*exp
(2)^6-47*d^7*exp(1)^8*exp(2)^4-486*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^8-
1464*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^4*exp(2)^6+180*d^7*exp(1)^6*exp(
2)^5-1296*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^8+158*d^7*exp(1)^4*exp(2)^6
-972*d^7*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^8-486*d^7*exp(2)^8-188*d^7*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^14*exp(2)+552*d^7*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))*exp(2)^8/x/exp(2)+684*d^7*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^6/x/exp(
2)-825/2*d^7*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^5/x/exp(2)-459*d^7*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))*exp(1)^8*exp(2)^4/x/exp(2)+123*d^7*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1
)^10*exp(2)^3/x/exp(2))/((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)-(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))^3/(3*exp(1)^16+9*exp(1)^12*exp(2)^2+3*exp(1)^10*exp(2)^3+9*exp(1)^14*exp
(2))+1/2*(-300*d^7*exp(1)^10*exp(2)^2-82*d^7*exp(1)^8*exp(2)^3+1000*d^7*exp(1)^6*exp(2)^4+464*d^7*exp(1)^4*exp
(2)^5-1248*d^7*exp(2)^7+40*d^7*exp(1)^12*exp(2))*atan((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(
2))/sqrt(-exp(1)^4+exp(2)^2))/sqrt(-exp(1)^4+exp(2)^2)/(-exp(1)^18-3*exp(1)^14*exp(2)^2-exp(1)^12*exp(2)^3-3*e
xp(1)^16*exp(2))+65/4*d^7*sign(d)*asin(x*exp(2)/d/exp(1))/exp(1)^6+2*((((((720*exp(1)^26*1/10080/exp(1)^26*x-3
360*exp(1)^25*d*1/10080/exp(1)^26)*x+7920*exp(1)^24*d^2*1/10080/exp(1)^26)*x-14280*exp(1)^23*d^3*1/10080/exp(1
)^26)*x+24000*exp(1)^22*d^4*1/10080/exp(1)^26)*x-41580*exp(1)^21*d^5*1/10080/exp(1)^26)*x+88320*exp(1)^20*d^6*
1/10080/exp(1)^26)*sqrt(d^2-x^2*exp(2))

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maple [A]  time = 0.03, size = 416, normalized size = 1.65 \[ \frac {35 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{5}}-\frac {5 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 \sqrt {e^{2}}\, e^{5}}+\frac {35 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{5} x}{2 e^{5}}-\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} x}{4 e^{5}}+\frac {35 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{3} x}{3 e^{5}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} x}{6 e^{5}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d x}{3 e^{5}}+\frac {28 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{2}}{3 e^{6}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{4}}{\left (x +\frac {d}{e}\right )^{4} e^{10}}+\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{3}}{\left (x +\frac {d}{e}\right )^{3} e^{9}}+\frac {22 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{2}}{3 \left (x +\frac {d}{e}\right )^{2} e^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^4/e^10/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)+8*d^3/e^9/(x+d/e)^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)
+22/3*d^2/e^8/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)+35/3*d^3/e^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x
+35/2*d^5/e^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x+35/2*d^7/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e
-(x+d/e)^2*e^2)^(1/2)*x)-2/3/e^5*d*x*(-e^2*x^2+d^2)^(5/2)-5/6/e^5*d^3*x*(-e^2*x^2+d^2)^(3/2)-5/4*d^5*x*(-e^2*x
^2+d^2)^(1/2)/e^5-5/4/e^5*d^7/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)-1/7/e^6*(-e^2*x^2+d^2)^(7
/2)+28/3*d^2/e^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)

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maxima [C]  time = 1.06, size = 478, normalized size = 1.90 \[ -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{2 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7}}{e^{7} x + d e^{6}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{3 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{6 \, {\left (e^{7} x + d e^{6}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{2 \, {\left (e^{7} x + d e^{6}\right )}} + \frac {5 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{6}} + \frac {75 \, d^{7} \arcsin \left (\frac {e x}{d}\right )}{4 \, e^{6}} - \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x}{2 \, e^{5}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{4 \, e^{5}} - \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{e^{6}} + \frac {25 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}}{2 \, e^{6}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{3 \, e^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{6 \, e^{6}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{3 \, e^{5}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{e^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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