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

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

[Out]

-4/21*d^4*x^2*(-e^2*x^2+d^2)^(3/2)/e^4+5/24*d^3*x^3*(-e^2*x^2+d^2)^(3/2)/e^3-5/21*d^2*x^4*(-e^2*x^2+d^2)^(3/2)
/e^2+1/4*d*x^5*(-e^2*x^2+d^2)^(3/2)/e-1/9*x^6*(-e^2*x^2+d^2)^(3/2)-1/2016*d^5*(-315*e*x+256*d)*(-e^2*x^2+d^2)^
(3/2)/e^6-5/64*d^9*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6-5/64*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^5

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Rubi [A]  time = 0.31, antiderivative size = 229, normalized size of antiderivative = 1.00, 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 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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.21, size = 135, normalized size = 0.59 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-512 d^8+315 d^7 e x-256 d^6 e^2 x^2+210 d^5 e^3 x^3-192 d^4 e^4 x^4+168 d^3 e^5 x^5+512 d^2 e^6 x^6-1008 d e^7 x^7+448 e^8 x^8\right )-315 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4032 e^6} \]

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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fricas [A]  time = 0.88, size = 138, normalized size = 0.60 \[ \frac {630 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (448 \, e^{8} x^{8} - 1008 \, d e^{7} x^{7} + 512 \, d^{2} e^{6} x^{6} + 168 \, d^{3} e^{5} x^{5} - 192 \, d^{4} e^{4} x^{4} + 210 \, d^{5} e^{3} x^{3} - 256 \, d^{6} e^{2} x^{2} + 315 \, d^{7} e x - 512 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4032 \, e^{6}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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

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maple [A]  time = 0.02, size = 375, normalized size = 1.64 \[ \frac {5 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{4 \sqrt {e^{2}}\, e^{5}}-\frac {85 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{64 \sqrt {e^{2}}\, e^{5}}-\frac {85 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} x}{64 e^{5}}+\frac {5 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{7} x}{4 e^{5}}-\frac {85 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5} x}{96 e^{5}}+\frac {5 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{5} x}{6 e^{5}}-\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} x}{24 e^{5}}+\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{4}}{3 e^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} x^{2}}{9 e^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d x}{4 e^{5}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{4}}{3 \left (x +\frac {d}{e}\right )^{2} e^{8}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2}}{63 e^{6}} \]

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

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maxima [C]  time = 1.07, size = 299, normalized size = 1.31 \[ -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{4 \, {\left (e^{7} x + d e^{6}\right )}} - \frac {5 i \, d^{9} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{6}} - \frac {85 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{64 \, e^{6}} + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7} x}{4 \, e^{5}} - \frac {85 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{64 \, e^{5}} + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{8}}{2 \, e^{6}} + \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{96 \, e^{5}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{12 \, e^{6}} - \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{24 \, e^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{e^{6}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x}{4 \, e^{5}} - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e^{6}} \]

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)^2,x, algorithm="maxima")

[Out]

-1/4*(-e^2*x^2 + d^2)^(5/2)*d^5/(e^7*x + d*e^6) - 5/4*I*d^9*arcsin(e*x/d + 2)/e^6 - 85/64*d^9*arcsin(e*x/d)/e^
6 + 5/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^7*x/e^5 - 85/64*sqrt(-e^2*x^2 + d^2)*d^7*x/e^5 + 5/2*sqrt(e^2*x^2 +
4*d*e*x + 3*d^2)*d^8/e^6 + 35/96*(-e^2*x^2 + d^2)^(3/2)*d^5*x/e^5 - 5/12*(-e^2*x^2 + d^2)^(3/2)*d^6/e^6 - 17/2
4*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^5 - 1/9*(-e^2*x^2 + d^2)^(7/2)*x^2/e^4 + (-e^2*x^2 + d^2)^(5/2)*d^4/e^6 + 1/4
*(-e^2*x^2 + d^2)^(7/2)*d*x/e^5 - 29/63*(-e^2*x^2 + d^2)^(7/2)*d^2/e^6

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

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)^2,x)

[Out]

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

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sympy [A]  time = 17.48, size = 571, normalized size = 2.49 \[ d^{2} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {16 d^{8} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac {8 d^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac {2 d^{4} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac {x^{8} \sqrt {d^{2} - e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \]

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/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*sqrt(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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