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

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

[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 = 0.42, 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 = {866, 1649, 1829, 655, 223, 209} \begin {gather*} \frac {65 d^7 \text {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} \end {gather*}

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 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*} \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 ) \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*}

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Mathematica [A]
time = 0.37, size = 151, normalized size = 0.60 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1212\) vs. \(2(218)=436\).
time = 0.08, size = 1213, normalized size = 4.81

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

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.54, size = 440, normalized size = 1.75 \begin {gather*} \frac {5}{2} i \, d^{7} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-6\right )} + \frac {75}{4} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} - \frac {5}{2} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{5} x e^{\left (-5\right )} - \frac {5}{4} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5} x e^{\left (-5\right )} - 5 \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{6} e^{\left (-6\right )} + \frac {25}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6} e^{\left (-6\right )} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{2 \, {\left (x^{3} e^{9} + 3 \, d x^{2} e^{8} + 3 \, d^{2} x e^{7} + d^{3} e^{6}\right )}} - \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{2 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}} + \frac {15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}}{x e^{7} + d e^{6}} + \frac {5}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{\left (-5\right )} - \frac {25}{6} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{\left (-6\right )} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{3 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\right )}} + \frac {25 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{6 \, {\left (x e^{7} + d e^{6}\right )}} - \frac {2}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{\left (-5\right )} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{\left (-6\right )} - \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{2 \, {\left (x e^{7} + d e^{6}\right )}} - \frac {1}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{\left (-6\right )} \end {gather*}

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]

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

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Fricas [A]
time = 1.71, size = 148, normalized size = 0.59 \begin {gather*} \frac {2144 \, d^{7} x e + 2144 \, d^{8} - 2730 \, {\left (d^{7} x e + d^{8}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (12 \, x^{7} e^{7} - 44 \, d x^{6} e^{6} + 76 \, d^{2} x^{5} e^{5} - 106 \, d^{3} x^{4} e^{4} + 162 \, d^{4} x^{3} e^{3} - 293 \, d^{5} x^{2} e^{2} + 779 \, d^{6} x e + 2144 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{84 \, {\left (x e^{7} + d 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)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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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Giac [A]
time = 1.38, size = 135, normalized size = 0.54 \begin {gather*} \frac {65}{4} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, d^{7} e^{\left (-6\right )}}{\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1} + \frac {1}{84} \, {\left (1472 \, d^{6} e^{\left (-6\right )} - {\left (693 \, d^{5} e^{\left (-5\right )} - 2 \, {\left (200 \, d^{4} e^{\left (-4\right )} - {\left (119 \, d^{3} e^{\left (-3\right )} - 2 \, {\left (33 \, d^{2} e^{\left (-2\right )} - {\left (14 \, d e^{\left (-1\right )} - 3 \, x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}

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]

65/4*d^7*arcsin(x*e/d)*e^(-6)*sgn(d) - 16*d^7*e^(-6)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1) + 1/84*(147
2*d^6*e^(-6) - (693*d^5*e^(-5) - 2*(200*d^4*e^(-4) - (119*d^3*e^(-3) - 2*(33*d^2*e^(-2) - (14*d*e^(-1) - 3*x)*
x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\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^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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