### 3.1663 $$\int \frac{1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=229 $-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}$

[Out]

(-231*e^3)/(40*(b*d - a*e)^4*(d + e*x)^(5/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) + (11*e)/(12*(b*
d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) - (33*e^2)/(8*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) - (77*b*e^3)/(8
*(b*d - a*e)^5*(d + e*x)^(3/2)) - (231*b^2*e^3)/(8*(b*d - a*e)^6*Sqrt[d + e*x]) + (231*b^(5/2)*e^3*ArcTanh[(Sq
rt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(13/2))

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Rubi [A]  time = 0.184573, antiderivative size = 229, 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} $-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-231*e^3)/(40*(b*d - a*e)^4*(d + e*x)^(5/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) + (11*e)/(12*(b*
d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) - (33*e^2)/(8*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) - (77*b*e^3)/(8
*(b*d - a*e)^5*(d + e*x)^(3/2)) - (231*b^2*e^3)/(8*(b*d - a*e)^6*Sqrt[d + e*x]) + (231*b^(5/2)*e^3*ArcTanh[(Sq
rt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)^{7/2}} \, dx\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{(11 e) \int \frac{1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{6 (b d-a e)}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{\left (33 e^2\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{8 (b d-a e)^2}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{\left (231 e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{7/2}} \, dx}{16 (b d-a e)^3}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{\left (231 b e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^4}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{\left (231 b^2 e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^5}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{231 b^2 e^3}{8 (b d-a e)^6 \sqrt{d+e x}}-\frac{\left (231 b^3 e^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^6}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{231 b^2 e^3}{8 (b d-a e)^6 \sqrt{d+e x}}-\frac{\left (231 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{8 (b d-a e)^6}\\ &=-\frac{231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac{77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac{231 b^2 e^3}{8 (b d-a e)^6 \sqrt{d+e x}}+\frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}\\ \end{align*}

Mathematica [C]  time = 0.021571, size = 52, normalized size = 0.23 $-\frac{2 e^3 \, _2F_1\left (-\frac{5}{2},4;-\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{5 (d+e x)^{5/2} (a e-b d)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-2*e^3*Hypergeometric2F1[-5/2, 4, -3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^4*(d + e*x)^(5/2)
)

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Maple [A]  time = 0.212, size = 344, normalized size = 1.5 \begin{align*} -{\frac{2\,{e}^{3}}{5\, \left ( ae-bd \right ) ^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-20\,{\frac{{e}^{3}{b}^{2}}{ \left ( ae-bd \right ) ^{6}\sqrt{ex+d}}}+{\frac{8\,{e}^{3}b}{3\, \left ( ae-bd \right ) ^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{71\,{e}^{3}{b}^{5}}{8\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{59\,{b}^{4}{e}^{4}a}{3\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{59\,{e}^{3}{b}^{5}d}{3\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{89\,{e}^{5}{b}^{3}{a}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{89\,{b}^{4}{e}^{4}ad}{4\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{89\,{e}^{3}{b}^{5}{d}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{231\,{b}^{3}{e}^{3}}{8\, \left ( ae-bd \right ) ^{6}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-2/5*e^3/(a*e-b*d)^4/(e*x+d)^(5/2)-20*e^3/(a*e-b*d)^6*b^2/(e*x+d)^(1/2)+8/3*e^3/(a*e-b*d)^5*b/(e*x+d)^(3/2)-71
/8*e^3*b^5/(a*e-b*d)^6/(b*e*x+a*e)^3*(e*x+d)^(5/2)-59/3*e^4*b^4/(a*e-b*d)^6/(b*e*x+a*e)^3*(e*x+d)^(3/2)*a+59/3
*e^3*b^5/(a*e-b*d)^6/(b*e*x+a*e)^3*(e*x+d)^(3/2)*d-89/8*e^5*b^3/(a*e-b*d)^6/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a^2+89
/4*e^4*b^4/(a*e-b*d)^6/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a*d-89/8*e^3*b^5/(a*e-b*d)^6/(b*e*x+a*e)^3*(e*x+d)^(1/2)*d^
2-231/8*e^3*b^3/(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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.77801, size = 5176, normalized size = 22.6 \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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

[1/240*(3465*(b^5*e^6*x^6 + a^3*b^2*d^3*e^3 + 3*(b^5*d*e^5 + a*b^4*e^6)*x^5 + 3*(b^5*d^2*e^4 + 3*a*b^4*d*e^5 +
a^2*b^3*e^6)*x^4 + (b^5*d^3*e^3 + 9*a*b^4*d^2*e^4 + 9*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 3*(a*b^4*d^3*e^3 + 3
*a^2*b^3*d^2*e^4 + a^3*b^2*d*e^5)*x^2 + 3*(a^2*b^3*d^3*e^3 + a^3*b^2*d^2*e^4)*x)*sqrt(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*(3465*b^5*e^5*x^5 + 40*b^5*d
^5 - 310*a*b^4*d^4*e + 1335*a^2*b^3*d^3*e^2 + 2768*a^3*b^2*d^2*e^3 - 416*a^4*b*d*e^4 + 48*a^5*e^5 + 1155*(7*b^
5*d*e^4 + 8*a*b^4*e^5)*x^4 + 231*(23*b^5*d^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2*b^3*e^5)*x^3 + 99*(5*b^5*d^3*e^2 +
146*a*b^4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 11*(10*b^5*d^4*e - 130*a*b^4*d^3*e^2 - 1119*a^2*
b^3*d^2*e^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^3*b^6*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*
d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*d^3*e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5
*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b^5*d^2*e^7 - 6*a^5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(
b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*a^4*b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7
- 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a*b^8*d^7*e^2 - 2*a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4
- 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2*d*e^8 + a^8*b*e^9)*x^4 + (b^9*d^9 +
3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3 - 36*a^4*b^5*d^5*e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^
3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7
*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^
9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 +
9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x), 1/120*(3465*(b^5*e^6*x^6 + a^3*b^2*d^3*e^3 + 3*(b^5*d*e
^5 + a*b^4*e^6)*x^5 + 3*(b^5*d^2*e^4 + 3*a*b^4*d*e^5 + a^2*b^3*e^6)*x^4 + (b^5*d^3*e^3 + 9*a*b^4*d^2*e^4 + 9*a
^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 3*(a*b^4*d^3*e^3 + 3*a^2*b^3*d^2*e^4 + a^3*b^2*d*e^5)*x^2 + 3*(a^2*b^3*d^3*e
^3 + a^3*b^2*d^2*e^4)*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)) - (3465*b^5*e^5*x^5 + 40*b^5*d^5 - 310*a*b^4*d^4*e + 1335*a^2*b^3*d^3*e^2 + 2768*a^3*b^2*d^2*e^3 - 416*a
^4*b*d*e^4 + 48*a^5*e^5 + 1155*(7*b^5*d*e^4 + 8*a*b^4*e^5)*x^4 + 231*(23*b^5*d^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2
*b^3*e^5)*x^3 + 99*(5*b^5*d^3*e^2 + 146*a*b^4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 11*(10*b^5*d
^4*e - 130*a*b^4*d^3*e^2 - 1119*a^2*b^3*d^2*e^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^3*b^6
*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*
d^3*e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b^5*d^2*e^7 - 6*a^
5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*
a^4*b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a*b^8*d^7*e^2 - 2*
a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2
*d*e^8 + a^8*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3 - 36*a^4*b^5*d^5*
e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d
^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^
7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^
5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.20805, size = 635, normalized size = 2.77 \begin{align*} -\frac{231 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\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 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{2} e^{3} + 20 \,{\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3} - 20 \,{\left (x e + d\right )} a b e^{4} - 6 \, a b d e^{4} + 3 \, a^{2} e^{5}\right )}}{15 \,{\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 (x e + d\right )}^{\frac{5}{2}}} - \frac{213 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{3} - 472 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{3} + 267 \, \sqrt{x e + d} b^{5} d^{2} e^{3} + 472 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{4} - 534 \, \sqrt{x e + d} a b^{4} d e^{4} + 267 \, \sqrt{x e + d} a^{2} b^{3} e^{5}}{24 \,{\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 )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

-231/8*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((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)) - 2/15*(150*(x*e + d)^
2*b^2*e^3 + 20*(x*e + d)*b^2*d*e^3 + 3*b^2*d^2*e^3 - 20*(x*e + d)*a*b*e^4 - 6*a*b*d*e^4 + 3*a^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)*(x*
e + d)^(5/2)) - 1/24*(213*(x*e + d)^(5/2)*b^5*e^3 - 472*(x*e + d)^(3/2)*b^5*d*e^3 + 267*sqrt(x*e + d)*b^5*d^2*
e^3 + 472*(x*e + d)^(3/2)*a*b^4*e^4 - 534*sqrt(x*e + d)*a*b^4*d*e^4 + 267*sqrt(x*e + d)*a^2*b^3*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)*((x
*e + d)*b - b*d + a*e)^3)