3.1673 \(\int \frac{1}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=239 \[ -\frac{693 e^5}{128 \sqrt{d+e x} (b d-a e)^6}-\frac{231 e^4}{128 (a+b x) \sqrt{d+e x} (b d-a e)^5}+\frac{231 e^3}{320 (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{33 e^2}{80 (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}+\frac{693 \sqrt{b} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{13/2}}+\frac{11 e}{40 (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 \sqrt{d+e x} (b d-a e)} \]

[Out]

(-693*e^5)/(128*(b*d - a*e)^6*Sqrt[d + e*x]) - 1/(5*(b*d - a*e)*(a + b*x)^5*Sqrt[d + e*x]) + (11*e)/(40*(b*d -
a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) - (33*e^2)/(80*(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) + (231*e^3)/(320*(b
*d - a*e)^4*(a + b*x)^2*Sqrt[d + e*x]) - (231*e^4)/(128*(b*d - a*e)^5*(a + b*x)*Sqrt[d + e*x]) + (693*Sqrt[b]*
e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(13/2))

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Rubi [A]  time = 0.16294, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{693 e^5}{128 \sqrt{d+e x} (b d-a e)^6}-\frac{231 e^4}{128 (a+b x) \sqrt{d+e x} (b d-a e)^5}+\frac{231 e^3}{320 (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{33 e^2}{80 (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}+\frac{693 \sqrt{b} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{13/2}}+\frac{11 e}{40 (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-693*e^5)/(128*(b*d - a*e)^6*Sqrt[d + e*x]) - 1/(5*(b*d - a*e)*(a + b*x)^5*Sqrt[d + e*x]) + (11*e)/(40*(b*d -
a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) - (33*e^2)/(80*(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) + (231*e^3)/(320*(b
*d - a*e)^4*(a + b*x)^2*Sqrt[d + e*x]) - (231*e^4)/(128*(b*d - a*e)^5*(a + b*x)*Sqrt[d + e*x]) + (693*Sqrt[b]*
e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(13/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^6 (d+e x)^{3/2}} \, dx\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{(11 e) \int \frac{1}{(a+b x)^5 (d+e x)^{3/2}} \, dx}{10 (b d-a e)}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{\left (99 e^2\right ) \int \frac{1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{80 (b d-a e)^2}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}-\frac{\left (231 e^3\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{160 (b d-a e)^3}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}+\frac{231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}+\frac{\left (231 e^4\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}+\frac{231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}-\frac{231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt{d+e x}}-\frac{\left (693 e^5\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac{693 e^5}{128 (b d-a e)^6 \sqrt{d+e x}}-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}+\frac{231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}-\frac{231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt{d+e x}}-\frac{\left (693 b e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^6}\\ &=-\frac{693 e^5}{128 (b d-a e)^6 \sqrt{d+e x}}-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}+\frac{231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}-\frac{231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt{d+e x}}-\frac{\left (693 b e^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{128 (b d-a e)^6}\\ &=-\frac{693 e^5}{128 (b d-a e)^6 \sqrt{d+e x}}-\frac{1}{5 (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}+\frac{231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}-\frac{231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt{d+e x}}+\frac{693 \sqrt{b} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{13/2}}\\ \end{align*}

Mathematica [C]  time = 0.0160358, size = 50, normalized size = 0.21 \[ -\frac{2 e^5 \, _2F_1\left (-\frac{1}{2},6;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{\sqrt{d+e x} (a e-b d)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-2*e^5*Hypergeometric2F1[-1/2, 6, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^6*Sqrt[d + e*x])

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Maple [B]  time = 0.219, size = 641, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2*e^5/(a*e-b*d)^6/(e*x+d)^(1/2)-437/128*e^5*b^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)-977/64*e^6*b^4/(a*e-b
*d)^6/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a+977/64*e^5*b^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d-131/5*e^7*b^3/(a*
e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2+262/5*e^6*b^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d-131/5*e^5*b
^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^2-1327/64*e^8*b^2/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3+398
1/64*e^7*b^3/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d-3981/64*e^6*b^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(
3/2)*a*d^2+1327/64*e^5*b^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3-843/128*e^9*b/(a*e-b*d)^6/(b*e*x+a*e)^5
*(e*x+d)^(1/2)*a^4+843/32*e^8*b^2/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d-2529/64*e^7*b^3/(a*e-b*d)^6/(b
*e*x+a*e)^5*(e*x+d)^(1/2)*d^2*a^2+843/32*e^6*b^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3-843/128*e^5*b^5
/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^4-693/128*e^5*b/(a*e-b*d)^6/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1
/2)/((a*e-b*d)*b)^(1/2))

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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(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.33728, size = 4783, normalized size = 20.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(3465*(b^5*e^6*x^6 + a^5*d*e^5 + (b^5*d*e^5 + 5*a*b^4*e^6)*x^5 + 5*(a*b^4*d*e^5 + 2*a^2*b^3*e^6)*x^4 +
10*(a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 5*(2*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 + (5*a^4*b*d*e^5 + a^5*e^6)*x)*sqr
t(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(3
465*b^5*e^5*x^5 + 128*b^5*d^5 - 816*a*b^4*d^4*e + 2248*a^2*b^3*d^3*e^2 - 3590*a^3*b^2*d^2*e^3 + 4215*a^4*b*d*e
^4 + 1280*a^5*e^5 + 1155*(b^5*d*e^4 + 14*a*b^4*e^5)*x^4 - 462*(b^5*d^2*e^3 - 12*a*b^4*d*e^4 - 64*a^2*b^3*e^5)*
x^3 + 66*(4*b^5*d^3*e^2 - 33*a*b^4*d^2*e^3 + 159*a^2*b^3*d*e^4 + 395*a^3*b^2*e^5)*x^2 - 11*(16*b^5*d^4*e - 112
*a*b^4*d^3*e^2 + 366*a^2*b^3*d^2*e^3 - 880*a^3*b^2*d*e^4 - 965*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^5*b^6*d^7 - 6*a
^6*b^5*d^6*e + 15*a^7*b^4*d^5*e^2 - 20*a^8*b^3*d^4*e^3 + 15*a^9*b^2*d^3*e^4 - 6*a^10*b*d^2*e^5 + a^11*d*e^6 +
(b^11*d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6*a^5*b^6*d*e^
6 + a^6*b^5*e^7)*x^6 + (b^11*d^7 - a*b^10*d^6*e - 15*a^2*b^9*d^5*e^2 + 55*a^3*b^8*d^4*e^3 - 85*a^4*b^7*d^3*e^4
+ 69*a^5*b^6*d^2*e^5 - 29*a^6*b^5*d*e^6 + 5*a^7*b^4*e^7)*x^5 + 5*(a*b^10*d^7 - 4*a^2*b^9*d^6*e + 3*a^3*b^8*d^
5*e^2 + 10*a^4*b^7*d^4*e^3 - 25*a^5*b^6*d^3*e^4 + 24*a^6*b^5*d^2*e^5 - 11*a^7*b^4*d*e^6 + 2*a^8*b^3*e^7)*x^4 +
10*(a^2*b^9*d^7 - 5*a^3*b^8*d^6*e + 9*a^4*b^7*d^5*e^2 - 5*a^5*b^6*d^4*e^3 - 5*a^6*b^5*d^3*e^4 + 9*a^7*b^4*d^2
*e^5 - 5*a^8*b^3*d*e^6 + a^9*b^2*e^7)*x^3 + 5*(2*a^3*b^8*d^7 - 11*a^4*b^7*d^6*e + 24*a^5*b^6*d^5*e^2 - 25*a^6*
b^5*d^4*e^3 + 10*a^7*b^4*d^3*e^4 + 3*a^8*b^3*d^2*e^5 - 4*a^9*b^2*d*e^6 + a^10*b*e^7)*x^2 + (5*a^4*b^7*d^7 - 29
*a^5*b^6*d^6*e + 69*a^6*b^5*d^5*e^2 - 85*a^7*b^4*d^4*e^3 + 55*a^8*b^3*d^3*e^4 - 15*a^9*b^2*d^2*e^5 - a^10*b*d*
e^6 + a^11*e^7)*x), 1/640*(3465*(b^5*e^6*x^6 + a^5*d*e^5 + (b^5*d*e^5 + 5*a*b^4*e^6)*x^5 + 5*(a*b^4*d*e^5 + 2*
a^2*b^3*e^6)*x^4 + 10*(a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 5*(2*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 + (5*a^4*b*d*e^5
+ a^5*e^6)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (3
465*b^5*e^5*x^5 + 128*b^5*d^5 - 816*a*b^4*d^4*e + 2248*a^2*b^3*d^3*e^2 - 3590*a^3*b^2*d^2*e^3 + 4215*a^4*b*d*e
^4 + 1280*a^5*e^5 + 1155*(b^5*d*e^4 + 14*a*b^4*e^5)*x^4 - 462*(b^5*d^2*e^3 - 12*a*b^4*d*e^4 - 64*a^2*b^3*e^5)*
x^3 + 66*(4*b^5*d^3*e^2 - 33*a*b^4*d^2*e^3 + 159*a^2*b^3*d*e^4 + 395*a^3*b^2*e^5)*x^2 - 11*(16*b^5*d^4*e - 112
*a*b^4*d^3*e^2 + 366*a^2*b^3*d^2*e^3 - 880*a^3*b^2*d*e^4 - 965*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^5*b^6*d^7 - 6*a
^6*b^5*d^6*e + 15*a^7*b^4*d^5*e^2 - 20*a^8*b^3*d^4*e^3 + 15*a^9*b^2*d^3*e^4 - 6*a^10*b*d^2*e^5 + a^11*d*e^6 +
(b^11*d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6*a^5*b^6*d*e^
6 + a^6*b^5*e^7)*x^6 + (b^11*d^7 - a*b^10*d^6*e - 15*a^2*b^9*d^5*e^2 + 55*a^3*b^8*d^4*e^3 - 85*a^4*b^7*d^3*e^4
+ 69*a^5*b^6*d^2*e^5 - 29*a^6*b^5*d*e^6 + 5*a^7*b^4*e^7)*x^5 + 5*(a*b^10*d^7 - 4*a^2*b^9*d^6*e + 3*a^3*b^8*d^
5*e^2 + 10*a^4*b^7*d^4*e^3 - 25*a^5*b^6*d^3*e^4 + 24*a^6*b^5*d^2*e^5 - 11*a^7*b^4*d*e^6 + 2*a^8*b^3*e^7)*x^4 +
10*(a^2*b^9*d^7 - 5*a^3*b^8*d^6*e + 9*a^4*b^7*d^5*e^2 - 5*a^5*b^6*d^4*e^3 - 5*a^6*b^5*d^3*e^4 + 9*a^7*b^4*d^2
*e^5 - 5*a^8*b^3*d*e^6 + a^9*b^2*e^7)*x^3 + 5*(2*a^3*b^8*d^7 - 11*a^4*b^7*d^6*e + 24*a^5*b^6*d^5*e^2 - 25*a^6*
b^5*d^4*e^3 + 10*a^7*b^4*d^3*e^4 + 3*a^8*b^3*d^2*e^5 - 4*a^9*b^2*d*e^6 + a^10*b*e^7)*x^2 + (5*a^4*b^7*d^7 - 29
*a^5*b^6*d^6*e + 69*a^6*b^5*d^5*e^2 - 85*a^7*b^4*d^4*e^3 + 55*a^8*b^3*d^3*e^4 - 15*a^9*b^2*d^2*e^5 - a^10*b*d*
e^6 + a^11*e^7)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.20611, size = 771, normalized size = 3.23 \begin{align*} -\frac{693 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \, e^{5}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{x e + d}} - \frac{2185 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} e^{5} - 9770 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d e^{5} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{2} e^{5} - 13270 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{3} e^{5} + 4215 \, \sqrt{x e + d} b^{5} d^{4} e^{5} + 9770 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} e^{6} - 33536 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d e^{6} + 39810 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{2} e^{6} - 16860 \, \sqrt{x e + d} a b^{4} d^{3} e^{6} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} e^{7} - 39810 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d e^{7} + 25290 \, \sqrt{x e + d} a^{2} b^{3} d^{2} e^{7} + 13270 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} e^{8} - 16860 \, \sqrt{x e + d} a^{3} b^{2} d e^{8} + 4215 \, \sqrt{x e + d} a^{4} b e^{9}}{640 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-693/128*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 2
0*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(-b^2*d + a*b*e)) - 2*e^5/((b^6*d^6 - 6*
a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(x*e
+ d)) - 1/640*(2185*(x*e + d)^(9/2)*b^5*e^5 - 9770*(x*e + d)^(7/2)*b^5*d*e^5 + 16768*(x*e + d)^(5/2)*b^5*d^2*
e^5 - 13270*(x*e + d)^(3/2)*b^5*d^3*e^5 + 4215*sqrt(x*e + d)*b^5*d^4*e^5 + 9770*(x*e + d)^(7/2)*a*b^4*e^6 - 33
536*(x*e + d)^(5/2)*a*b^4*d*e^6 + 39810*(x*e + d)^(3/2)*a*b^4*d^2*e^6 - 16860*sqrt(x*e + d)*a*b^4*d^3*e^6 + 16
768*(x*e + d)^(5/2)*a^2*b^3*e^7 - 39810*(x*e + d)^(3/2)*a^2*b^3*d*e^7 + 25290*sqrt(x*e + d)*a^2*b^3*d^2*e^7 +
13270*(x*e + d)^(3/2)*a^3*b^2*e^8 - 16860*sqrt(x*e + d)*a^3*b^2*d*e^8 + 4215*sqrt(x*e + d)*a^4*b*e^9)/((b^6*d^
6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*((
x*e + d)*b - b*d + a*e)^5)