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

Optimal. Leaf size=224 \[ \frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {239 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3} \]

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Rubi [A]  time = 0.53, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {239 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^3*(d - e*x)^4)/(e^5*Sqrt[d^2 - e^2*x^2])) - (337*d^5*Sqrt[d^2 - e^2*x^2])/(15*e^5) + (175*d^4*x*Sqrt[d^2
- e^2*x^2])/(16*e^4) - (71*d^3*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) + (47*d^2*x^3*Sqrt[d^2 - e^2*x^2])/(24*e^2) -
(4*d*x^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (x^5*Sqrt[d^2 - e^2*x^2])/6 - (239*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]
])/(16*e^5)

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^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac {x^4 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x)^3 \left (\frac {4 d^4}{e^4}-\frac {d^3 x}{e^3}+\frac {d^2 x^2}{e^2}-\frac {d x^3}{e}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {-\frac {24 d^7}{e^2}+\frac {78 d^6 x}{e}-96 d^5 x^2+66 d^4 e x^3-47 d^3 e^2 x^4+24 d^2 e^3 x^5}{\sqrt {d^2-e^2 x^2}} \, dx}{6 d e^2}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {120 d^7-390 d^6 e x+480 d^5 e^2 x^2-426 d^4 e^3 x^3+235 d^3 e^4 x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{30 d e^4}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {-480 d^7 e^2+1560 d^6 e^3 x-2625 d^5 e^4 x^2+1704 d^4 e^5 x^3}{\sqrt {d^2-e^2 x^2}} \, dx}{120 d e^6}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {1440 d^7 e^4-8088 d^6 e^5 x+7875 d^5 e^6 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{360 d e^8}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {-10755 d^7 e^6+16176 d^6 e^7 x}{\sqrt {d^2-e^2 x^2}} \, dx}{720 d e^{10}}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\left (239 d^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^4}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {\left (239 d^6\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^4}\\ &=-\frac {d^3 (d-e x)^4}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {337 d^5 \sqrt {d^2-e^2 x^2}}{15 e^5}+\frac {175 d^4 x \sqrt {d^2-e^2 x^2}}{16 e^4}-\frac {71 d^3 x^2 \sqrt {d^2-e^2 x^2}}{15 e^3}+\frac {47 d^2 x^3 \sqrt {d^2-e^2 x^2}}{24 e^2}-\frac {4 d x^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{6} x^5 \sqrt {d^2-e^2 x^2}-\frac {239 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 125, normalized size = 0.56 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5632 d^6-2047 d^5 e x+769 d^4 e^2 x^2-426 d^3 e^3 x^3+278 d^2 e^4 x^4-152 d e^5 x^5+40 e^6 x^6\right )-3585 d^6 (d+e x) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{240 e^5 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5632*d^6 - 2047*d^5*e*x + 769*d^4*e^2*x^2 - 426*d^3*e^3*x^3 + 278*d^2*e^4*x^4 - 152*d*e
^5*x^5 + 40*e^6*x^6) - 3585*d^6*(d + e*x)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^5*(d + e*x))

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IntegrateAlgebraic [A]  time = 0.65, size = 143, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5632 d^6-2047 d^5 e x+769 d^4 e^2 x^2-426 d^3 e^3 x^3+278 d^2 e^4 x^4-152 d e^5 x^5+40 e^6 x^6\right )}{240 e^5 (d+e x)}-\frac {239 d^6 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{16 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5632*d^6 - 2047*d^5*e*x + 769*d^4*e^2*x^2 - 426*d^3*e^3*x^3 + 278*d^2*e^4*x^4 - 152*d*e
^5*x^5 + 40*e^6*x^6))/(240*e^5*(d + e*x)) - (239*d^6*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(1
6*e^6)

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fricas [A]  time = 0.41, size = 146, normalized size = 0.65 \begin {gather*} -\frac {5632 \, d^{6} e x + 5632 \, d^{7} - 7170 \, {\left (d^{6} e x + d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (40 \, e^{6} x^{6} - 152 \, d e^{5} x^{5} + 278 \, d^{2} e^{4} x^{4} - 426 \, d^{3} e^{3} x^{3} + 769 \, d^{4} e^{2} x^{2} - 2047 \, d^{5} e x - 5632 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*(5632*d^6*e*x + 5632*d^7 - 7170*(d^6*e*x + d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (40*e^6*x^6
- 152*d*e^5*x^5 + 278*d^2*e^4*x^4 - 426*d^3*e^3*x^3 + 769*d^4*e^2*x^2 - 2047*d^5*e*x - 5632*d^6)*sqrt(-e^2*x^
2 + d^2))/(e^6*x + d*e^5)

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (84*d^6*(-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+18*d^6*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/ex
p(2))^5*exp(1)^10*exp(2)^3-576*d^6*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^12*exp
(2)^2-540*d^6*(-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)^3-126*d^6*(-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+228*d^6*(-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)^2-200*d^6*(-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)^3-264*d^6*(-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)^4-60*d^6*(-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)^5-1188*d^6*(-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+72*d^6*(-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+756*d^6*(-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+216*d^6*(-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)^6-726*d^6*(-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+138*d^6*(-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+537*d^6*(-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+147*d^6*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^5*exp(2)^8+1620*d^6*(-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
+1404*d^6*(-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+26*d^6*exp(1)^8*exp(
2)^4+402*d^6*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^8+1212*d^6*(-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)^6-144*d^6*exp(1)^6*exp(2)^5+1008*d^6*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^8-89*d^6*exp(1)^4*exp(2)^6+804*d^6*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^8+402*d^6*exp(2)^8+104*d^6*(-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)-861/2*d^6*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(2)^8/
x/exp(2)-594*d^6*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^6/x/exp(2)+237*d^6*(-2*d*exp(1)-2
*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^5/x/exp(2)+369*d^6*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*e
xp(1)^8*exp(2)^4/x/exp(2)-69*d^6*(-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)^15+9*exp(1)^11*exp(2)^2+3*exp(1)^9*exp(2)^3+9*exp(1)^13*exp(2))+1/2*(-192*d^6*exp(1)^10*
exp(2)^2+32*d^6*exp(1)^8*exp(2)^3+712*d^6*exp(1)^6*exp(2)^4+230*d^6*exp(1)^4*exp(2)^5-924*d^6*exp(2)^7+16*d^6*
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))/s
qrt(-exp(1)^4+exp(2)^2)/(exp(1)^17+3*exp(1)^13*exp(2)^2+exp(1)^11*exp(2)^3+3*exp(1)^15*exp(2))-239/16*d^6*sign
(d)*asin(x*exp(2)/d/exp(1))/exp(1)^5+2*(((((240*exp(1)^19*1/2880/exp(1)^19*x-1152*exp(1)^18*d*1/2880/exp(1)^19
)*x+2820*exp(1)^17*d^2*1/2880/exp(1)^19)*x-5376*exp(1)^16*d^3*1/2880/exp(1)^19)*x+9990*exp(1)^15*d^4*1/2880/ex
p(1)^19)*x-22272*exp(1)^14*d^5*1/2880/exp(1)^19)*sqrt(d^2-x^2*exp(2))

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maple [B]  time = 0.02, size = 393, normalized size = 1.75 \begin {gather*} -\frac {61 d^{6} \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 )}{4 \sqrt {e^{2}}\, e^{4}}+\frac {5 d^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}\, e^{4}}-\frac {61 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4} x}{4 e^{4}}+\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} x}{16 e^{4}}-\frac {61 \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} x}{6 e^{4}}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2} x}{24 e^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} x}{6 e^{4}}-\frac {122 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d}{15 e^{5}}-\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^{3}}{\left (x +\frac {d}{e}\right )^{4} e^{9}}-\frac {7 \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}}{\left (x +\frac {d}{e}\right )^{3} e^{8}}-\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}{3 \left (x +\frac {d}{e}\right )^{2} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^3/e^9/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)-7*d^2/e^8/(x+d/e)^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)
-22/3*d/e^7/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)-61/6*d^2/e^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x-6
1/4*d^4/e^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x-61/4*d^6/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(
x+d/e)^2*e^2)^(1/2)*x)+5/24/e^4*d^2*x*(-e^2*x^2+d^2)^(3/2)+5/16/(e^2)^(1/2)*d^6/e^4*arctan((e^2)^(1/2)/(-e^2*x
^2+d^2)^(1/2)*x)+5/16*(-e^2*x^2+d^2)^(1/2)*d^4/e^4*x-122/15*d/e^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)+1/6/e^4*
x*(-e^2*x^2+d^2)^(5/2)

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maxima [C]  time = 1.04, size = 456, normalized size = 2.04 \begin {gather*} \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{2 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} - \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}}{e^{6} x + d e^{5}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{3 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} - \frac {10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{3 \, {\left (e^{6} x + d e^{5}\right )}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{2 \, {\left (e^{6} x + d e^{5}\right )}} - \frac {9 i \, d^{6} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{5}} - \frac {275 \, d^{6} \arcsin \left (\frac {e x}{d}\right )}{16 \, e^{5}} + \frac {9 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x}{4 \, e^{4}} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} x}{16 \, e^{4}} + \frac {9 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{2 \, e^{5}} - \frac {10 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}}{e^{5}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x}{24 \, e^{4}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}}{2 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x}{6 \, e^{4}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-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^4/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) + 5/2*(-e^2*x^2 + d^2)^(3/2)*d^
5/(e^7*x^2 + 2*d*e^6*x + d^2*e^5) - 15*sqrt(-e^2*x^2 + d^2)*d^6/(e^6*x + d*e^5) - 4/3*(-e^2*x^2 + d^2)^(5/2)*d
^3/(e^7*x^2 + 2*d*e^6*x + d^2*e^5) - 10/3*(-e^2*x^2 + d^2)^(3/2)*d^4/(e^6*x + d*e^5) + 3/2*(-e^2*x^2 + d^2)^(5
/2)*d^2/(e^6*x + d*e^5) - 9/4*I*d^6*arcsin(e*x/d + 2)/e^5 - 275/16*d^6*arcsin(e*x/d)/e^5 + 9/4*sqrt(e^2*x^2 +
4*d*e*x + 3*d^2)*d^4*x/e^4 + 5/16*sqrt(-e^2*x^2 + d^2)*d^4*x/e^4 + 9/2*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^5/e^5
- 10*sqrt(-e^2*x^2 + d^2)*d^5/e^5 - 19/24*(-e^2*x^2 + d^2)^(3/2)*d^2*x/e^4 + 5/2*(-e^2*x^2 + d^2)^(3/2)*d^3/e
^5 + 1/6*(-e^2*x^2 + d^2)^(5/2)*x/e^4 - 4/5*(-e^2*x^2 + d^2)^(5/2)*d/e^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^4*(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 \begin {gather*} \int \frac {x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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