### 3.1666 $$\int \frac{(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=197 $-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6}$

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)) - (231*e^3*(d + e*x)^(5/2))/
(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*
x)^4) - (d + e*x)^(11/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*b^(13/2))

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Rubi [A]  time = 0.10289, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, 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} $-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)) - (231*e^3*(d + e*x)^(5/2))/
(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*
x)^4) - (d + e*x)^(11/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*b^(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 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)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^{11/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{(11 e) \int \frac{(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (99 e^2\right ) \int \frac{(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (231 e^3\right ) \int \frac{(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3}\\ &=-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (231 e^4\right ) \int \frac{(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (693 e^5\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{256 b^5}\\ &=\frac{693 e^5 \sqrt{d+e x}}{128 b^6}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (693 e^5 (b d-a e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^6}\\ &=\frac{693 e^5 \sqrt{d+e x}}{128 b^6}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{\left (693 e^4 (b d-a e)\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^6}\\ &=\frac{693 e^5 \sqrt{d+e x}}{128 b^6}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}\\ \end{align*}

Mathematica [C]  time = 0.022983, size = 52, normalized size = 0.26 $\frac{2 e^5 (d+e x)^{13/2} \, _2F_1\left (6,\frac{13}{2};\frac{15}{2};-\frac{b (d+e x)}{a e-b d}\right )}{13 (a e-b d)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(13/2)*Hypergeometric2F1[6, 13/2, 15/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(13*(-(b*d) + a*e)^
6)

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Maple [B]  time = 0.212, size = 673, normalized size = 3.4 \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)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*e^5*(e*x+d)^(1/2)/b^6+843/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a-843/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2
)*d+1327/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2-1327/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a*d+1327/64*e^
5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^2+131/5*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3-393/5*e^7/b^3/(b*e*x+a*e)^5*
(e*x+d)^(5/2)*a^2*d+393/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d^2-131/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^
3+977/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^4-977/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d+2931/32*e^7/
b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d^2-977/16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^3+977/64*e^5/b/(b*e*x+a
*e)^5*(e*x+d)^(3/2)*d^4+437/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^5-2185/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d
)^(1/2)*d*a^4+2185/64*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^2-2185/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*
a^2*d^3+2185/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^4-437/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^5-693/1
28*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+693/128*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

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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)^(11/2)/(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 = 2.22922, size = 1976, normalized size = 10.03 \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)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(3465*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e
^5)*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*(1280
*b^5*e^5*x^5 - 128*b^5*d^5 - 176*a*b^4*d^4*e - 264*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 +
3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e
^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3 + 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*
d^4*e + 616*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 + 2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^1
1*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6), -1/640*(3465*(b^5*e^5*x^5 + 5
*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(-(b*d - a*e)/b)*arcta
n(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (1280*b^5*e^5*x^5 - 128*b^5*d^5 - 176*a*b^4*d^4*e - 264
*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 + 3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*
x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3
+ 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*d^4*e + 616*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 +
2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8
*x^2 + 5*a^4*b^7*x + a^5*b^6)]

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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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.34365, size = 620, normalized size = 3.15 \begin{align*} \frac{693 \,{\left (b d e^{5} - a e^{6}\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^{6}} + \frac{2 \, \sqrt{x e + d} e^{5}}{b^{6}} - \frac{4215 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} d e^{5} - 13270 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d^{2} e^{5} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{5} - 9770 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{5} + 2185 \, \sqrt{x e + d} b^{5} d^{5} e^{5} - 4215 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{4} e^{6} + 26540 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} d e^{6} - 50304 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{6} + 39080 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{6} - 10925 \, \sqrt{x e + d} a b^{4} d^{4} e^{6} - 13270 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{3} e^{7} + 50304 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{7} - 58620 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{7} + 21850 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{7} - 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{8} + 39080 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{8} - 21850 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{8} - 9770 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{9} + 10925 \, \sqrt{x e + d} a^{4} b d e^{9} - 2185 \, \sqrt{x e + d} a^{5} e^{10}}{640 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

693/128*(b*d*e^5 - a*e^6)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) + 2*sqrt(x*e
+ d)*e^5/b^6 - 1/640*(4215*(x*e + d)^(9/2)*b^5*d*e^5 - 13270*(x*e + d)^(7/2)*b^5*d^2*e^5 + 16768*(x*e + d)^(5
/2)*b^5*d^3*e^5 - 9770*(x*e + d)^(3/2)*b^5*d^4*e^5 + 2185*sqrt(x*e + d)*b^5*d^5*e^5 - 4215*(x*e + d)^(9/2)*a*b
^4*e^6 + 26540*(x*e + d)^(7/2)*a*b^4*d*e^6 - 50304*(x*e + d)^(5/2)*a*b^4*d^2*e^6 + 39080*(x*e + d)^(3/2)*a*b^4
*d^3*e^6 - 10925*sqrt(x*e + d)*a*b^4*d^4*e^6 - 13270*(x*e + d)^(7/2)*a^2*b^3*e^7 + 50304*(x*e + d)^(5/2)*a^2*b
^3*d*e^7 - 58620*(x*e + d)^(3/2)*a^2*b^3*d^2*e^7 + 21850*sqrt(x*e + d)*a^2*b^3*d^3*e^7 - 16768*(x*e + d)^(5/2)
*a^3*b^2*e^8 + 39080*(x*e + d)^(3/2)*a^3*b^2*d*e^8 - 21850*sqrt(x*e + d)*a^3*b^2*d^2*e^8 - 9770*(x*e + d)^(3/2
)*a^4*b*e^9 + 10925*sqrt(x*e + d)*a^4*b*d*e^9 - 2185*sqrt(x*e + d)*a^5*e^10)/(((x*e + d)*b - b*d + a*e)^5*b^6)