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

Optimal. Leaf size=204 \[ \frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.59, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \begin {gather*} \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(3/2)) + (10*d^2*
(d - e*x)^2)/(e^6*Sqrt[d^2 - e^2*x^2]) + (59*d^2*Sqrt[d^2 - e^2*x^2])/(3*e^6) - (2*d*x*Sqrt[d^2 - e^2*x^2])/e^
5 + (x^2*Sqrt[d^2 - e^2*x^2])/(3*e^4) + (18*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/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 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx &=\int \frac {x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^5}{e^5}+\frac {5 d^4 x}{e^4}-\frac {5 d^3 x^2}{e^3}+\frac {5 d^2 x^3}{e^2}-\frac {5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d-e x)^2 \left (-\frac {60 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {30 d^3 x^2}{e^3}+\frac {15 d^2 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x) \left (-\frac {240 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {15 d^3 x^2}{e^3}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\int \frac {\frac {720 d^6}{e^3}-\frac {885 d^5 x}{e^2}+\frac {180 d^4 x^2}{e}}{\sqrt {d^2-e^2 x^2}} \, dx}{45 d^3 e^2}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}-\frac {\int \frac {-\frac {1620 d^6}{e}+1770 d^5 x}{\sqrt {d^2-e^2 x^2}} \, dx}{90 d^3 e^4}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 109, normalized size = 0.53 \begin {gather*} \frac {270 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right )}{(d+e x)^3}}{15 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(424*d^5 + 1002*d^4*e*x + 674*d^3*e^2*x^2 + 70*d^2*e^3*x^3 - 15*d*e^4*x^4 + 5*e^5*x^5))/
(d + e*x)^3 + 270*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.76, size = 130, normalized size = 0.64 \begin {gather*} \frac {18 d^3 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{e^7}+\frac {\sqrt {d^2-e^2 x^2} \left (424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right )}{15 e^6 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(424*d^5 + 1002*d^4*e*x + 674*d^3*e^2*x^2 + 70*d^2*e^3*x^3 - 15*d*e^4*x^4 + 5*e^5*x^5))/(
15*e^6*(d + e*x)^3) + (18*d^3*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/e^7

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 200, normalized size = 0.98 \begin {gather*} \frac {424 \, d^{3} e^{3} x^{3} + 1272 \, d^{4} e^{2} x^{2} + 1272 \, d^{5} e x + 424 \, d^{6} - 540 \, {\left (d^{3} e^{3} x^{3} + 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x + d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (5 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} + 70 \, d^{2} e^{3} x^{3} + 674 \, d^{3} e^{2} x^{2} + 1002 \, d^{4} e x + 424 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/15*(424*d^3*e^3*x^3 + 1272*d^4*e^2*x^2 + 1272*d^5*e*x + 424*d^6 - 540*(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3*d^5*e
*x + d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (5*e^5*x^5 - 15*d*e^4*x^4 + 70*d^2*e^3*x^3 + 674*d^3*e^2
*x^2 + 1002*d^4*e*x + 424*d^5)*sqrt(-e^2*x^2 + d^2))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-162*d^3*(-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^3*(-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+240*d^3*(-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+228*d^3*(-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+54*d^3*(-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^3*(-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+158*d^3*(-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+339*d^3*(-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+87*d^3*(-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+492*d^3*(-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+192*d^3*(-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^8*exp(2)^4-96*d^3*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^4*exp(1)^6*exp(2)^5-36*d^3*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^4*e
xp(2)^6+840*d^3*(-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+420*d^3*(-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)^5-102*d^3*(-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)^6-48*d^3*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^5*exp(2)^8-228*d^3*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^6*exp(2)^5-25
2*d^3*(-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^3*exp(1)^8*exp(2)^4
-102*d^3*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^8-288*d^3*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^4*exp(2)^6+60*d^3*exp(1)^6*exp(2)^5-360*d^3*(-1/2*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^8+110*d^3*exp(1)^4*exp(2)^6-204*d^3*(-1/2*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^8-102*d^3*exp(2)^8-188*d^3*(-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)+156*d^3*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(2)^8/x/exp(2)
+108*d^3*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^6/x/exp(2)-573/2*d^3*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^5/x/exp(2)-153*d^3*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^
8*exp(2)^4/x/exp(2)+123*d^3*(-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*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp
(2))^3/(3*exp(1)^16-6*exp(1)^12*exp(2)^2-6*exp(1)^10*exp(2)^3+3*exp(1)^8*exp(2)^4+3*exp(1)^6*exp(2)^5+3*exp(1)
^14*exp(2))+1/2*(-100*d^3*exp(1)^10*exp(2)^2-170*d^3*exp(1)^8*exp(2)^3+152*d^3*exp(1)^6*exp(2)^4+208*d^3*exp(1
)^4*exp(2)^5-144*d^3*exp(2)^7+40*d^3*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+2*exp(1)^14*exp(2)^2+2*exp(1)^12*exp(
2)^3-exp(1)^10*exp(2)^4-exp(1)^8*exp(2)^5-exp(1)^16*exp(2))+18*d^3*sign(d)*asin(x*exp(2)/d/exp(1))/exp(1)^6+2*
((2*exp(1)^16*1/12/exp(1)^20*x-12*exp(1)^15*d*1/12/exp(1)^20)*x+58*exp(1)^14*d^2*1/12/exp(1)^20)*sqrt(d^2-x^2*
exp(2))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 297, normalized size = 1.46 \begin {gather*} \frac {20 d^{3} \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 )}{\sqrt {e^{2}}\, e^{5}}-\frac {2 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{5}}-\frac {2 \sqrt {-e^{2} x^{2}+d^{2}}\, d x}{e^{5}}+\frac {20 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2}}{e^{6}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{4}}{5 \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 {3}{2}} d^{3}}{5 \left (x +\frac {d}{e}\right )^{3} e^{9}}+\frac {10 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{2}}{\left (x +\frac {d}{e}\right )^{2} e^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/3/e^6*(-e^2*x^2+d^2)^(3/2)-2*d*x*(-e^2*x^2+d^2)^(1/2)/e^5-2/e^5*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^
2+d^2)^(1/2)*x)+1/5*d^4/e^10/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)-8/5/e^9*d^3/(x+d/e)^3*(2*(x+d/e)*d*
e-(x+d/e)^2*e^2)^(3/2)+20/e^6*d^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)+20/e^5*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2
)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x)+10/e^8*d^2/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(d^2 - e^2*x^2)^(1/2))/(d + e*x)^4,x)

[Out]

int((x^5*(d^2 - e^2*x^2)^(1/2))/(d + e*x)^4, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

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

________________________________________________________________________________________