### 3.1664 $$\int \frac{(d+e x)^{15/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=253 $-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}+\frac{3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}+\frac{9009 e^5 \sqrt{d+e x} (b d-a e)^2}{128 b^8}-\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{17/2}}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}$

[Out]

(9009*e^5*(b*d - a*e)^2*Sqrt[d + e*x])/(128*b^8) + (3003*e^5*(b*d - a*e)*(d + e*x)^(3/2))/(128*b^7) + (9009*e^
5*(d + e*x)^(5/2))/(640*b^6) - (1287*e^4*(d + e*x)^(7/2))/(128*b^5*(a + b*x)) - (143*e^3*(d + e*x)^(9/2))/(64*
b^4*(a + b*x)^2) - (13*e^2*(d + e*x)^(11/2))/(16*b^3*(a + b*x)^3) - (3*e*(d + e*x)^(13/2))/(8*b^2*(a + b*x)^4)
- (d + e*x)^(15/2)/(5*b*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/(128*b^(17/2))

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Rubi [A]  time = 0.183265, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, 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} $-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}+\frac{3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}+\frac{9009 e^5 \sqrt{d+e x} (b d-a e)^2}{128 b^8}-\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{17/2}}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(9009*e^5*(b*d - a*e)^2*Sqrt[d + e*x])/(128*b^8) + (3003*e^5*(b*d - a*e)*(d + e*x)^(3/2))/(128*b^7) + (9009*e^
5*(d + e*x)^(5/2))/(640*b^6) - (1287*e^4*(d + e*x)^(7/2))/(128*b^5*(a + b*x)) - (143*e^3*(d + e*x)^(9/2))/(64*
b^4*(a + b*x)^2) - (13*e^2*(d + e*x)^(11/2))/(16*b^3*(a + b*x)^3) - (3*e*(d + e*x)^(13/2))/(8*b^2*(a + b*x)^4)
- (d + e*x)^(15/2)/(5*b*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/(128*b^(17/2))

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)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^{15/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{(3 e) \int \frac{(d+e x)^{13/2}}{(a+b x)^5} \, dx}{2 b}\\ &=-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (39 e^2\right ) \int \frac{(d+e x)^{11/2}}{(a+b x)^4} \, dx}{16 b^2}\\ &=-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (143 e^3\right ) \int \frac{(d+e x)^{9/2}}{(a+b x)^3} \, dx}{32 b^3}\\ &=-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (1287 e^4\right ) \int \frac{(d+e x)^{7/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (9009 e^5\right ) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{256 b^5}\\ &=\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (9009 e^5 (b d-a e)\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{256 b^6}\\ &=\frac{3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (9009 e^5 (b d-a e)^2\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{256 b^7}\\ &=\frac{9009 e^5 (b d-a e)^2 \sqrt{d+e x}}{128 b^8}+\frac{3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (9009 e^5 (b d-a e)^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^8}\\ &=\frac{9009 e^5 (b d-a e)^2 \sqrt{d+e x}}{128 b^8}+\frac{3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac{\left (9009 e^4 (b d-a e)^3\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^8}\\ &=\frac{9009 e^5 (b d-a e)^2 \sqrt{d+e x}}{128 b^8}+\frac{3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac{9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac{1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac{143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac{13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac{3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{15/2}}{5 b (a+b x)^5}-\frac{9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{17/2}}\\ \end{align*}

Mathematica [C]  time = 0.0270306, size = 52, normalized size = 0.21 $\frac{2 e^5 (d+e x)^{17/2} \, _2F_1\left (6,\frac{17}{2};\frac{19}{2};-\frac{b (d+e x)}{a e-b d}\right )}{17 (a e-b d)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(17/2)*Hypergeometric2F1[6, 17/2, 19/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(17*(-(b*d) + a*e)^
6)

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Maple [B]  time = 0.244, size = 1164, normalized size = 4.6 \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)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2/5*e^5*(e*x+d)^(5/2)/b^6-84*e^6/b^7*a*d*(e*x+d)^(1/2)+9443/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^4+5327/12
8*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a^3+1001/5*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^5-5327/128*e^5/b/(b*e*
x+a*e)^5*(e*x+d)^(9/2)*d^3-1001/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^5+7837/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3
/2)*d^6-3633/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^7+9443/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^4+9009/128*
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^3+3633/128*e^12/b^8/(b*e*x+a*e)^5*(e
*x+d)^(1/2)*a^7-9009/128*e^8/b^8/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3+7837/64*e
^11/b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^6+117555/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d^4-23511/32*e^6/b^2
/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^5-27027/128*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^2-25431/128*e^11/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^6*d+76293/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/
2)*a^5*d^2+127155/128*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^4-76293/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*a^2*d^5+4*e^5/b^6*(e*x+d)^(3/2)*d+42*e^5/b^6*d^2*(e*x+d)^(1/2)-4*e^6/b^7*(e*x+d)^(3/2)*a+42*e^7/b^8*a^2*(e*x
+d)^(1/2)+25431/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^6-15981/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a^
2*d+15981/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a*d^2-1001*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^4*d+2002*e^
8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3*d^2-2002*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2*d^3+1001*e^6/b^2/(b*e*x
+a*e)^5*(e*x+d)^(5/2)*a*d^4-23511/32*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^5*d+117555/64*e^9/b^5/(b*e*x+a*e)^
5*(e*x+d)^(3/2)*a^4*d^2+27027/128*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))*d*a^
2-9443/16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a*d^3+28329/32*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2*d^2-39185
/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d^3-127155/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4*d^3-9443/16
*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^3*d

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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)^(15/2)/(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 = 2.28154, size = 3553, normalized size = 14.04 \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)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(45045*(a^5*b^2*d^2*e^5 - 2*a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b^6*d*e^6 + a^2*b^5*e^7)*x^5 +
5*(a*b^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x
^3 + 10*(a^3*b^4*d^2*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + 5*(a^4*b^3*d^2*e^5 - 2*a^5*b^2*d*e^6 + a^6*b*e
^7)*x)*sqrt((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*(2
56*b^7*e^7*x^7 - 128*b^7*d^7 - 240*a*b^6*d^6*e - 520*a^2*b^5*d^5*e^2 - 1430*a^3*b^4*d^4*e^3 - 6435*a^4*b^3*d^3
*e^4 + 69069*a^5*b^2*d^2*e^5 - 105105*a^6*b*d*e^6 + 45045*a^7*e^7 + 256*(12*b^7*d*e^6 - 5*a*b^6*e^7)*x^6 + 256
*(116*b^7*d^2*e^5 - 160*a*b^6*d*e^6 + 65*a^2*b^5*e^7)*x^5 - 5*(5327*b^7*d^3*e^4 - 45677*a*b^6*d^2*e^5 + 66157*
a^2*b^5*d*e^6 - 27599*a^3*b^4*e^7)*x^4 - 10*(1211*b^7*d^4*e^3 + 5810*a*b^6*d^3*e^4 - 54392*a^2*b^5*d^2*e^5 + 8
0366*a^3*b^4*d*e^6 - 33891*a^4*b^3*e^7)*x^3 - 2*(2324*b^7*d^5*e^2 + 6545*a*b^6*d^4*e^3 + 30485*a^2*b^5*d^3*e^4
- 302445*a^3*b^4*d^2*e^5 + 452595*a^4*b^3*d*e^6 - 192192*a^5*b^2*e^7)*x^2 - 2*(568*b^7*d^6*e + 1240*a*b^6*d^5
*e^2 + 3445*a^2*b^5*d^4*e^3 + 15730*a^3*b^4*d^3*e^4 - 163020*a^4*b^3*d^2*e^5 + 246246*a^5*b^2*d*e^6 - 105105*a
^6*b*e^7)*x)*sqrt(e*x + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b
^8), -1/640*(45045*(a^5*b^2*d^2*e^5 - 2*a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b^6*d*e^6 + a^2*b^5*e^7)*x^
5 + 5*(a*b^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*b^4*d*e^6 + a^4*b^3*e^
7)*x^3 + 10*(a^3*b^4*d^2*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + 5*(a^4*b^3*d^2*e^5 - 2*a^5*b^2*d*e^6 + a^6
*b*e^7)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (256*b^7*e^7*x^7 -
128*b^7*d^7 - 240*a*b^6*d^6*e - 520*a^2*b^5*d^5*e^2 - 1430*a^3*b^4*d^4*e^3 - 6435*a^4*b^3*d^3*e^4 + 69069*a^5
*b^2*d^2*e^5 - 105105*a^6*b*d*e^6 + 45045*a^7*e^7 + 256*(12*b^7*d*e^6 - 5*a*b^6*e^7)*x^6 + 256*(116*b^7*d^2*e^
5 - 160*a*b^6*d*e^6 + 65*a^2*b^5*e^7)*x^5 - 5*(5327*b^7*d^3*e^4 - 45677*a*b^6*d^2*e^5 + 66157*a^2*b^5*d*e^6 -
27599*a^3*b^4*e^7)*x^4 - 10*(1211*b^7*d^4*e^3 + 5810*a*b^6*d^3*e^4 - 54392*a^2*b^5*d^2*e^5 + 80366*a^3*b^4*d*e
^6 - 33891*a^4*b^3*e^7)*x^3 - 2*(2324*b^7*d^5*e^2 + 6545*a*b^6*d^4*e^3 + 30485*a^2*b^5*d^3*e^4 - 302445*a^3*b^
4*d^2*e^5 + 452595*a^4*b^3*d*e^6 - 192192*a^5*b^2*e^7)*x^2 - 2*(568*b^7*d^6*e + 1240*a*b^6*d^5*e^2 + 3445*a^2*
b^5*d^4*e^3 + 15730*a^3*b^4*d^3*e^4 - 163020*a^4*b^3*d^2*e^5 + 246246*a^5*b^2*d*e^6 - 105105*a^6*b*e^7)*x)*sqr
t(e*x + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8)]

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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)**(15/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.38142, size = 1060, normalized size = 4.19 \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)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

9009/128*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e)
)/(sqrt(-b^2*d + a*b*e)*b^8) - 1/640*(26635*(x*e + d)^(9/2)*b^7*d^3*e^5 - 94430*(x*e + d)^(7/2)*b^7*d^4*e^5 +
128128*(x*e + d)^(5/2)*b^7*d^5*e^5 - 78370*(x*e + d)^(3/2)*b^7*d^6*e^5 + 18165*sqrt(x*e + d)*b^7*d^7*e^5 - 799
05*(x*e + d)^(9/2)*a*b^6*d^2*e^6 + 377720*(x*e + d)^(7/2)*a*b^6*d^3*e^6 - 640640*(x*e + d)^(5/2)*a*b^6*d^4*e^6
+ 470220*(x*e + d)^(3/2)*a*b^6*d^5*e^6 - 127155*sqrt(x*e + d)*a*b^6*d^6*e^6 + 79905*(x*e + d)^(9/2)*a^2*b^5*d
*e^7 - 566580*(x*e + d)^(7/2)*a^2*b^5*d^2*e^7 + 1281280*(x*e + d)^(5/2)*a^2*b^5*d^3*e^7 - 1175550*(x*e + d)^(3
/2)*a^2*b^5*d^4*e^7 + 381465*sqrt(x*e + d)*a^2*b^5*d^5*e^7 - 26635*(x*e + d)^(9/2)*a^3*b^4*e^8 + 377720*(x*e +
d)^(7/2)*a^3*b^4*d*e^8 - 1281280*(x*e + d)^(5/2)*a^3*b^4*d^2*e^8 + 1567400*(x*e + d)^(3/2)*a^3*b^4*d^3*e^8 -
635775*sqrt(x*e + d)*a^3*b^4*d^4*e^8 - 94430*(x*e + d)^(7/2)*a^4*b^3*e^9 + 640640*(x*e + d)^(5/2)*a^4*b^3*d*e^
9 - 1175550*(x*e + d)^(3/2)*a^4*b^3*d^2*e^9 + 635775*sqrt(x*e + d)*a^4*b^3*d^3*e^9 - 128128*(x*e + d)^(5/2)*a^
5*b^2*e^10 + 470220*(x*e + d)^(3/2)*a^5*b^2*d*e^10 - 381465*sqrt(x*e + d)*a^5*b^2*d^2*e^10 - 78370*(x*e + d)^(
3/2)*a^6*b*e^11 + 127155*sqrt(x*e + d)*a^6*b*d*e^11 - 18165*sqrt(x*e + d)*a^7*e^12)/(((x*e + d)*b - b*d + a*e)
^5*b^8) + 2/5*((x*e + d)^(5/2)*b^24*e^5 + 10*(x*e + d)^(3/2)*b^24*d*e^5 + 105*sqrt(x*e + d)*b^24*d^2*e^5 - 10*
(x*e + d)^(3/2)*a*b^23*e^6 - 210*sqrt(x*e + d)*a*b^23*d*e^6 + 105*sqrt(x*e + d)*a^2*b^22*e^7)/b^30