3.1665 \(\int \frac{(d+e x)^{13/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=224 \[ -\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}+\frac{3003 e^5 \sqrt{d+e x} (b d-a e)}{128 b^7}-\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6} \]

[Out]

(3003*e^5*(b*d - a*e)*Sqrt[d + e*x])/(128*b^7) + (1001*e^5*(d + e*x)^(3/2))/(128*b^6) - (3003*e^4*(d + e*x)^(5
/2))/(640*b^5*(a + b*x)) - (429*e^3*(d + e*x)^(7/2))/(320*b^4*(a + b*x)^2) - (143*e^2*(d + e*x)^(9/2))/(240*b^
3*(a + b*x)^3) - (13*e*(d + e*x)^(11/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(13/2)/(5*b*(a + b*x)^5) - (3003*e^5
*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))

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Rubi [A]  time = 0.139222, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \[ -\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}+\frac{3003 e^5 \sqrt{d+e x} (b d-a e)}{128 b^7}-\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(3003*e^5*(b*d - a*e)*Sqrt[d + e*x])/(128*b^7) + (1001*e^5*(d + e*x)^(3/2))/(128*b^6) - (3003*e^4*(d + e*x)^(5
/2))/(640*b^5*(a + b*x)) - (429*e^3*(d + e*x)^(7/2))/(320*b^4*(a + b*x)^2) - (143*e^2*(d + e*x)^(9/2))/(240*b^
3*(a + b*x)^3) - (13*e*(d + e*x)^(11/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(13/2)/(5*b*(a + b*x)^5) - (3003*e^5
*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))

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 47

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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{(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^{13/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{(13 e) \int \frac{(d+e x)^{11/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (143 e^2\right ) \int \frac{(d+e x)^{9/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (429 e^3\right ) \int \frac{(d+e x)^{7/2}}{(a+b x)^3} \, dx}{160 b^3}\\ &=-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (3003 e^4\right ) \int \frac{(d+e x)^{5/2}}{(a+b x)^2} \, dx}{640 b^4}\\ &=-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (3003 e^5\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{256 b^5}\\ &=\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (3003 e^5 (b d-a e)\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{256 b^6}\\ &=\frac{3003 e^5 (b d-a e) \sqrt{d+e x}}{128 b^7}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (3003 e^5 (b d-a e)^2\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^7}\\ &=\frac{3003 e^5 (b d-a e) \sqrt{d+e x}}{128 b^7}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{\left (3003 e^4 (b d-a 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 )}{128 b^7}\\ &=\frac{3003 e^5 (b d-a e) \sqrt{d+e x}}{128 b^7}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}-\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0221142, size = 52, normalized size = 0.23 \[ \frac{2 e^5 (d+e x)^{15/2} \, _2F_1\left (6,\frac{15}{2};\frac{17}{2};-\frac{b (d+e x)}{a e-b d}\right )}{15 (a e-b d)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(15/2)*Hypergeometric2F1[6, 15/2, 17/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(15*(-(b*d) + a*e)^
6)

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Maple [B]  time = 0.216, size = 908, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2/3*e^5*(e*x+d)^(3/2)/b^6+12*e^5/b^6*(e*x+d)^(1/2)*d-12*e^6/b^7*a*(e*x+d)^(1/2)-48145/96*e^8/b^4/(b*e*x+a*e)^5
*(e*x+d)^(3/2)*a^3*d^2+48145/96*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d^3-12131/64*e^6/b^2/(b*e*x+a*e)^5*(e*
x+d)^(7/2)*a*d^2+2373/64*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a*d-3003/64*e^6/b^6/((a*e-b*d)*b)^(1/2)*arctan(b*
(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a*d-22005/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^2*a^4+7335/32*e^8/b^4/(
b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^3-22005/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^2*d^4+4401/64*e^6/b^2/(b*e*
x+a*e)^5*(e*x+d)^(1/2)*a*d^5+12131/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2*d+5012/15*e^8/b^4/(b*e*x+a*e)^5*
(e*x+d)^(5/2)*a^3*d-2506/5*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2*d^2+5012/15*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(
5/2)*a*d^3-48145/192*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^4+4401/64*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^
5*d+48145/192*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^4*d+3003/128*e^7/b^7/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^
(1/2)/((a*e-b*d)*b)^(1/2))*a^2-1467/128*e^11/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^6-2373/128*e^7/b^3/(b*e*x+a*e)^
5*(e*x+d)^(9/2)*a^2-1253/15*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^4-9629/192*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(3
/2)*a^5-12131/192*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^3+3003/128*e^5/b^5/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d
)^(1/2)/((a*e-b*d)*b)^(1/2))*d^2-2373/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*d^2-1253/15*e^5/b/(b*e*x+a*e)^5*(e
*x+d)^(5/2)*d^4+9629/192*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^5-1467/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^6+
12131/192*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^3

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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((e*x+d)^(13/2)/(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 = 2.41751, size = 2743, normalized size = 12.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*
(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 5*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sq
rt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(1280*b^6*e
^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*b^2*d^2*e^4 +
60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4 - 38558*a*b^5
*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^5 + 33891*a^3*
b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^3*d*e^5 + 1921
92*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*b^3*d^2*e^4 -
141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3
*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7), -1/1920*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*
b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 5
*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))
- (1280*b^6*e^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*
b^2*d^2*e^4 + 60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4
- 38558*a*b^5*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^
5 + 33891*a^3*b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^
3*d*e^5 + 192192*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*
b^3*d^2*e^4 - 141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^1
0*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)]

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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((e*x+d)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.32275, size = 828, normalized size = 3.7 \begin{align*} \frac{3003 \,{\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{128 \, \sqrt{-b^{2} d + a b e} b^{7}} - \frac{35595 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} d^{2} e^{5} - 121310 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d^{3} e^{5} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{4} e^{5} - 96290 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{5} e^{5} + 22005 \, \sqrt{x e + d} b^{6} d^{6} e^{5} - 71190 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{5} d e^{6} + 363930 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} d^{2} e^{6} - 641536 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d^{3} e^{6} + 481450 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{4} e^{6} - 132030 \, \sqrt{x e + d} a b^{5} d^{5} e^{6} + 35595 \,{\left (x e + d\right )}^{\frac{9}{2}} a^{2} b^{4} e^{7} - 363930 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{4} d e^{7} + 962304 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} d^{2} e^{7} - 962900 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d^{3} e^{7} + 330075 \, \sqrt{x e + d} a^{2} b^{4} d^{4} e^{7} + 121310 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{3} b^{3} e^{8} - 641536 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{3} d e^{8} + 962900 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} d^{2} e^{8} - 440100 \, \sqrt{x e + d} a^{3} b^{3} d^{3} e^{8} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{4} b^{2} e^{9} - 481450 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b^{2} d e^{9} + 330075 \, \sqrt{x e + d} a^{4} b^{2} d^{2} e^{9} + 96290 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{5} b e^{10} - 132030 \, \sqrt{x e + d} a^{5} b d e^{10} + 22005 \, \sqrt{x e + d} a^{6} e^{11}}{1920 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{7}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{12} e^{5} + 18 \, \sqrt{x e + d} b^{12} d e^{5} - 18 \, \sqrt{x e + d} a b^{11} e^{6}\right )}}{3 \, b^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

3003/128*(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b
*e)*b^7) - 1/1920*(35595*(x*e + d)^(9/2)*b^6*d^2*e^5 - 121310*(x*e + d)^(7/2)*b^6*d^3*e^5 + 160384*(x*e + d)^(
5/2)*b^6*d^4*e^5 - 96290*(x*e + d)^(3/2)*b^6*d^5*e^5 + 22005*sqrt(x*e + d)*b^6*d^6*e^5 - 71190*(x*e + d)^(9/2)
*a*b^5*d*e^6 + 363930*(x*e + d)^(7/2)*a*b^5*d^2*e^6 - 641536*(x*e + d)^(5/2)*a*b^5*d^3*e^6 + 481450*(x*e + d)^
(3/2)*a*b^5*d^4*e^6 - 132030*sqrt(x*e + d)*a*b^5*d^5*e^6 + 35595*(x*e + d)^(9/2)*a^2*b^4*e^7 - 363930*(x*e + d
)^(7/2)*a^2*b^4*d*e^7 + 962304*(x*e + d)^(5/2)*a^2*b^4*d^2*e^7 - 962900*(x*e + d)^(3/2)*a^2*b^4*d^3*e^7 + 3300
75*sqrt(x*e + d)*a^2*b^4*d^4*e^7 + 121310*(x*e + d)^(7/2)*a^3*b^3*e^8 - 641536*(x*e + d)^(5/2)*a^3*b^3*d*e^8 +
962900*(x*e + d)^(3/2)*a^3*b^3*d^2*e^8 - 440100*sqrt(x*e + d)*a^3*b^3*d^3*e^8 + 160384*(x*e + d)^(5/2)*a^4*b^
2*e^9 - 481450*(x*e + d)^(3/2)*a^4*b^2*d*e^9 + 330075*sqrt(x*e + d)*a^4*b^2*d^2*e^9 + 96290*(x*e + d)^(3/2)*a^
5*b*e^10 - 132030*sqrt(x*e + d)*a^5*b*d*e^10 + 22005*sqrt(x*e + d)*a^6*e^11)/(((x*e + d)*b - b*d + a*e)^5*b^7)
+ 2/3*((x*e + d)^(3/2)*b^12*e^5 + 18*sqrt(x*e + d)*b^12*d*e^5 - 18*sqrt(x*e + d)*a*b^11*e^6)/b^18