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

Optimal. Leaf size=229 \[ -\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{5 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{64 e^6} \]

[Out]

(-5*d^7*x*Sqrt[d^2 - e^2*x^2])/(64*e^5) - (4*d^4*x^2*(d^2 - e^2*x^2)^(3/2))/(21*e^4) + (5*d^3*x^3*(d^2 - e^2*x
^2)^(3/2))/(24*e^3) - (5*d^2*x^4*(d^2 - e^2*x^2)^(3/2))/(21*e^2) + (d*x^5*(d^2 - e^2*x^2)^(3/2))/(4*e) - (x^6*
(d^2 - e^2*x^2)^(3/2))/9 - (d^5*(256*d - 315*e*x)*(d^2 - e^2*x^2)^(3/2))/(2016*e^6) - (5*d^9*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(64*e^6)

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Rubi [A]  time = 0.314594, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {852, 1809, 833, 780, 195, 217, 203} \[ -\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{5 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{64 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*d^7*x*Sqrt[d^2 - e^2*x^2])/(64*e^5) - (4*d^4*x^2*(d^2 - e^2*x^2)^(3/2))/(21*e^4) + (5*d^3*x^3*(d^2 - e^2*x
^2)^(3/2))/(24*e^3) - (5*d^2*x^4*(d^2 - e^2*x^2)^(3/2))/(21*e^2) + (d*x^5*(d^2 - e^2*x^2)^(3/2))/(4*e) - (x^6*
(d^2 - e^2*x^2)^(3/2))/9 - (d^5*(256*d - 315*e*x)*(d^2 - e^2*x^2)^(3/2))/(2016*e^6) - (5*d^9*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(64*e^6)

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 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 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])

Rubi steps

\begin{align*} \int \frac{x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int x^5 (d-e x)^2 \sqrt{d^2-e^2 x^2} \, dx\\ &=-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x^5 \left (-15 d^2 e^2+18 d e^3 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{9 e^2}\\ &=\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int x^4 \left (-90 d^3 e^3+120 d^2 e^4 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{72 e^4}\\ &=-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x^3 \left (-480 d^4 e^4+630 d^3 e^5 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{504 e^6}\\ &=\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int x^2 \left (-1890 d^5 e^5+2880 d^4 e^6 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{3024 e^8}\\ &=-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x \left (-5760 d^6 e^6+9450 d^5 e^7 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{15120 e^{10}}\\ &=-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{\left (5 d^7\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{32 e^5}\\ &=-\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{\left (5 d^9\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{64 e^5}\\ &=-\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{\left (5 d^9\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{64 e^5}\\ &=-\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac{5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac{d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac{1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac{5 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{64 e^6}\\ \end{align*}

Mathematica [A]  time = 0.230254, size = 135, normalized size = 0.59 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-256 d^6 e^2 x^2+210 d^5 e^3 x^3-192 d^4 e^4 x^4+168 d^3 e^5 x^5+512 d^2 e^6 x^6+315 d^7 e x-512 d^8-1008 d e^7 x^7+448 e^8 x^8\right )-315 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4032 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-512*d^8 + 315*d^7*e*x - 256*d^6*e^2*x^2 + 210*d^5*e^3*x^3 - 192*d^4*e^4*x^4 + 168*d^3*e
^5*x^5 + 512*d^2*e^6*x^6 - 1008*d*e^7*x^7 + 448*e^8*x^8) - 315*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4032*e^
6)

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Maple [A]  time = 0.076, size = 375, normalized size = 1.6 \begin{align*} -{\frac{{x}^{2}}{9\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{29\,{d}^{2}}{63\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{dx}{4\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{17\,{d}^{3}x}{24\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{85\,{d}^{5}x}{96\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{85\,{d}^{7}x}{64\,{e}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{85\,{d}^{9}}{64\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{2\,{d}^{4}}{3\,{e}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{5}x}{6\,{e}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{7}x}{4\,{e}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{5\,{d}^{9}}{4\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{4}}{3\,{e}^{8}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}} \end{align*}

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)^2,x)

[Out]

-1/9/e^4*x^2*(-e^2*x^2+d^2)^(7/2)-29/63*d^2/e^6*(-e^2*x^2+d^2)^(7/2)+1/4*d/e^5*x*(-e^2*x^2+d^2)^(7/2)-17/24*d^
3/e^5*x*(-e^2*x^2+d^2)^(5/2)-85/96*d^5/e^5*x*(-e^2*x^2+d^2)^(3/2)-85/64*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^5-85/64*d
^9/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+2/3/e^6*d^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)
+5/6/e^5*d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+5/4/e^5*d^7*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+5/4/e^5
*d^9/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))-1/3*d^4/e^8/(d/e+x)^2*(-(d/e+x)^2*
e^2+2*d*e*(d/e+x))^(7/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6114, size = 313, normalized size = 1.37 \begin{align*} \frac{630 \, d^{9} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (448 \, e^{8} x^{8} - 1008 \, d e^{7} x^{7} + 512 \, d^{2} e^{6} x^{6} + 168 \, d^{3} e^{5} x^{5} - 192 \, d^{4} e^{4} x^{4} + 210 \, d^{5} e^{3} x^{3} - 256 \, d^{6} e^{2} x^{2} + 315 \, d^{7} e x - 512 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4032 \, e^{6}} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

1/4032*(630*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (448*e^8*x^8 - 1008*d*e^7*x^7 + 512*d^2*e^6*x^6 +
168*d^3*e^5*x^5 - 192*d^4*e^4*x^4 + 210*d^5*e^3*x^3 - 256*d^6*e^2*x^2 + 315*d^7*e*x - 512*d^8)*sqrt(-e^2*x^2 +
d^2))/e^6

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Sympy [A]  time = 21.4452, size = 573, normalized size = 2.5 \begin{align*} d^{2} \left (\begin{cases} - \frac{8 d^{6} \sqrt{d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac{4 d^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac{d^{2} x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac{x^{6} \sqrt{d^{2} - e^{2} x^{2}}}{7} & \text{for}\: e \neq 0 \\\frac{x^{6} \sqrt{d^{2}}}{6} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} - \frac{5 i d^{8} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{128 e^{7}} + \frac{5 i d^{7} x}{128 e^{6} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{5 i d^{5} x^{3}}{384 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d^{3} x^{5}}{192 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{7 i d x^{7}}{48 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{9}}{8 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{5 d^{8} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{128 e^{7}} - \frac{5 d^{7} x}{128 e^{6} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{5 d^{5} x^{3}}{384 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d^{3} x^{5}}{192 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{7 d x^{7}}{48 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{9}}{8 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{16 d^{8} \sqrt{d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac{8 d^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac{2 d^{4} x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac{d^{2} x^{6} \sqrt{d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac{x^{8} \sqrt{d^{2} - e^{2} x^{2}}}{9} & \text{for}\: e \neq 0 \\\frac{x^{8} \sqrt{d^{2}}}{8} & \text{otherwise} \end{cases}\right ) \end{align*}

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)**2,x)

[Out]

d**2*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d*
*2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True)
) - 2*d*e*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(128*e**6*sqrt(-1 + e**2*x**2/d**2)) - 5*I
*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/(192*e**2*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**
7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1),
(5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2/d**2)) + 5*d**5*x**3/(384*e**4*sqrt(1
- e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2)) + 7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) -
e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True)) + e**2*Piecewise((-16*d**8*sqrt(d**2 - e**2*x**2)/(315*e**8)
- 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d**4*x**4*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**6*sq
rt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2 - e**2*x**2)/9, Ne(e, 0)), (x**8*sqrt(d**2)/8, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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)^2,x, algorithm="giac")

[Out]

sage0*x