3.1672 $$\int \frac{1}{\sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=213 $-\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)}$

[Out]

-Sqrt[d + e*x]/(5*(b*d - a*e)*(a + b*x)^5) + (9*e*Sqrt[d + e*x])/(40*(b*d - a*e)^2*(a + b*x)^4) - (21*e^2*Sqrt
[d + e*x])/(80*(b*d - a*e)^3*(a + b*x)^3) + (21*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)^2) - (63*e^4*Sq
rt[d + e*x])/(128*(b*d - a*e)^5*(a + b*x)) + (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*Sq
rt[b]*(b*d - a*e)^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.116132, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {27, 51, 63, 208} $-\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[d + e*x]/(5*(b*d - a*e)*(a + b*x)^5) + (9*e*Sqrt[d + e*x])/(40*(b*d - a*e)^2*(a + b*x)^4) - (21*e^2*Sqrt
[d + e*x])/(80*(b*d - a*e)^3*(a + b*x)^3) + (21*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)^2) - (63*e^4*Sq
rt[d + e*x])/(128*(b*d - a*e)^5*(a + b*x)) + (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*Sq
rt[b]*(b*d - a*e)^(11/2))

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}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^6 \sqrt{d+e x}} \, dx\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}-\frac{(9 e) \int \frac{1}{(a+b x)^5 \sqrt{d+e x}} \, dx}{10 (b d-a e)}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}+\frac{\left (63 e^2\right ) \int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx}{80 (b d-a e)^2}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}-\frac{\left (21 e^3\right ) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{32 (b d-a e)^3}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}+\frac{\left (63 e^4\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 (b d-a e)^4}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac{63 e^4 \sqrt{d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac{\left (63 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^5}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac{63 e^4 \sqrt{d+e x}}{128 (b d-a e)^5 (a+b x)}-\frac{\left (63 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)^5}\\ &=-\frac{\sqrt{d+e x}}{5 (b d-a e) (a+b x)^5}+\frac{9 e \sqrt{d+e x}}{40 (b d-a e)^2 (a+b x)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (b d-a e)^3 (a+b x)^3}+\frac{21 e^3 \sqrt{d+e x}}{64 (b d-a e)^4 (a+b x)^2}-\frac{63 e^4 \sqrt{d+e x}}{128 (b d-a e)^5 (a+b x)}+\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0132862, size = 50, normalized size = 0.23 $\frac{2 e^5 \sqrt{d+e x} \, _2F_1\left (\frac{1}{2},6;\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{(a e-b d)^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*e^5*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 6, 3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(-(b*d) + a*e)^6

________________________________________________________________________________________

Maple [A]  time = 0.198, size = 211, normalized size = 1. \begin{align*}{\frac{{e}^{5}}{ \left ( 5\,ae-5\,bd \right ) \left ( bxe+ae \right ) ^{5}}\sqrt{ex+d}}+{\frac{9\,{e}^{5}}{40\, \left ( ae-bd \right ) ^{2} \left ( bxe+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{21\,{e}^{5}}{80\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{21\,{e}^{5}}{64\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\, \left ( ae-bd \right ) ^{5}}\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)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/5*e^5*(e*x+d)^(1/2)/(a*e-b*d)/(b*e*x+a*e)^5+9/40*e^5/(a*e-b*d)^2*(e*x+d)^(1/2)/(b*e*x+a*e)^4+21/80*e^5/(a*e-
b*d)^3*(e*x+d)^(1/2)/(b*e*x+a*e)^3+21/64*e^5/(a*e-b*d)^4*(e*x+d)^(1/2)/(b*e*x+a*e)^2+63/128*e^5/(a*e-b*d)^5*(e
*x+d)^(1/2)/(b*e*x+a*e)+63/128*e^5/(a*e-b*d)^5/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.91918, size = 3794, normalized size = 17.81 \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)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/1280*(315*(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^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) + 2*(128*b^
6*d^5 - 784*a*b^5*d^4*e + 2024*a^2*b^4*d^3*e^2 - 2858*a^3*b^3*d^2*e^3 + 2455*a^4*b^2*d*e^4 - 965*a^5*b*e^5 + 3
15*(b^6*d*e^4 - a*b^5*e^5)*x^4 - 210*(b^6*d^2*e^3 - 8*a*b^5*d*e^4 + 7*a^2*b^4*e^5)*x^3 + 42*(4*b^6*d^3*e^2 - 2
7*a*b^5*d^2*e^3 + 87*a^2*b^4*d*e^4 - 64*a^3*b^3*e^5)*x^2 - 6*(24*b^6*d^4*e - 152*a*b^5*d^3*e^2 + 417*a^2*b^4*d
^2*e^3 - 684*a^3*b^3*d*e^4 + 395*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^6 - 6*a^6*b^6*d^5*e + 15*a^7*b^5*d^
4*e^2 - 20*a^8*b^4*d^3*e^3 + 15*a^9*b^3*d^2*e^4 - 6*a^10*b^2*d*e^5 + a^11*b*e^6 + (b^12*d^6 - 6*a*b^11*d^5*e +
15*a^2*b^10*d^4*e^2 - 20*a^3*b^9*d^3*e^3 + 15*a^4*b^8*d^2*e^4 - 6*a^5*b^7*d*e^5 + a^6*b^6*e^6)*x^5 + 5*(a*b^1
1*d^6 - 6*a^2*b^10*d^5*e + 15*a^3*b^9*d^4*e^2 - 20*a^4*b^8*d^3*e^3 + 15*a^5*b^7*d^2*e^4 - 6*a^6*b^6*d*e^5 + a^
7*b^5*e^6)*x^4 + 10*(a^2*b^10*d^6 - 6*a^3*b^9*d^5*e + 15*a^4*b^8*d^4*e^2 - 20*a^5*b^7*d^3*e^3 + 15*a^6*b^6*d^2
*e^4 - 6*a^7*b^5*d*e^5 + a^8*b^4*e^6)*x^3 + 10*(a^3*b^9*d^6 - 6*a^4*b^8*d^5*e + 15*a^5*b^7*d^4*e^2 - 20*a^6*b^
6*d^3*e^3 + 15*a^7*b^5*d^2*e^4 - 6*a^8*b^4*d*e^5 + a^9*b^3*e^6)*x^2 + 5*(a^4*b^8*d^6 - 6*a^5*b^7*d^5*e + 15*a^
6*b^6*d^4*e^2 - 20*a^7*b^5*d^3*e^3 + 15*a^8*b^4*d^2*e^4 - 6*a^9*b^3*d*e^5 + a^10*b^2*e^6)*x), -1/640*(315*(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^2*d +
a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (128*b^6*d^5 - 784*a*b^5*d^4*e + 2024*a^2*b^
4*d^3*e^2 - 2858*a^3*b^3*d^2*e^3 + 2455*a^4*b^2*d*e^4 - 965*a^5*b*e^5 + 315*(b^6*d*e^4 - a*b^5*e^5)*x^4 - 210*
(b^6*d^2*e^3 - 8*a*b^5*d*e^4 + 7*a^2*b^4*e^5)*x^3 + 42*(4*b^6*d^3*e^2 - 27*a*b^5*d^2*e^3 + 87*a^2*b^4*d*e^4 -
64*a^3*b^3*e^5)*x^2 - 6*(24*b^6*d^4*e - 152*a*b^5*d^3*e^2 + 417*a^2*b^4*d^2*e^3 - 684*a^3*b^3*d*e^4 + 395*a^4*
b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^6 - 6*a^6*b^6*d^5*e + 15*a^7*b^5*d^4*e^2 - 20*a^8*b^4*d^3*e^3 + 15*a^9*b
^3*d^2*e^4 - 6*a^10*b^2*d*e^5 + a^11*b*e^6 + (b^12*d^6 - 6*a*b^11*d^5*e + 15*a^2*b^10*d^4*e^2 - 20*a^3*b^9*d^3
*e^3 + 15*a^4*b^8*d^2*e^4 - 6*a^5*b^7*d*e^5 + a^6*b^6*e^6)*x^5 + 5*(a*b^11*d^6 - 6*a^2*b^10*d^5*e + 15*a^3*b^9
*d^4*e^2 - 20*a^4*b^8*d^3*e^3 + 15*a^5*b^7*d^2*e^4 - 6*a^6*b^6*d*e^5 + a^7*b^5*e^6)*x^4 + 10*(a^2*b^10*d^6 - 6
*a^3*b^9*d^5*e + 15*a^4*b^8*d^4*e^2 - 20*a^5*b^7*d^3*e^3 + 15*a^6*b^6*d^2*e^4 - 6*a^7*b^5*d*e^5 + a^8*b^4*e^6)
*x^3 + 10*(a^3*b^9*d^6 - 6*a^4*b^8*d^5*e + 15*a^5*b^7*d^4*e^2 - 20*a^6*b^6*d^3*e^3 + 15*a^7*b^5*d^2*e^4 - 6*a^
8*b^4*d*e^5 + a^9*b^3*e^6)*x^2 + 5*(a^4*b^8*d^6 - 6*a^5*b^7*d^5*e + 15*a^6*b^6*d^4*e^2 - 20*a^7*b^5*d^3*e^3 +
15*a^8*b^4*d^2*e^4 - 6*a^9*b^3*d*e^5 + a^10*b^2*e^6)*x)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.19648, size = 613, normalized size = 2.88 \begin{align*} -\frac{63 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} - \frac{315 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 1470 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} + 965 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 1470 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 5376 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 7110 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} - 3860 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 7110 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} + 5790 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} - 3860 \, \sqrt{x e + d} a^{3} b d e^{8} + 965 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\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)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-63/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a
^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^2*d + a*b*e)) - 1/640*(315*(x*e + d)^(9/2)*b^4*e^5 - 1470*(x
*e + d)^(7/2)*b^4*d*e^5 + 2688*(x*e + d)^(5/2)*b^4*d^2*e^5 - 2370*(x*e + d)^(3/2)*b^4*d^3*e^5 + 965*sqrt(x*e +
d)*b^4*d^4*e^5 + 1470*(x*e + d)^(7/2)*a*b^3*e^6 - 5376*(x*e + d)^(5/2)*a*b^3*d*e^6 + 7110*(x*e + d)^(3/2)*a*b
^3*d^2*e^6 - 3860*sqrt(x*e + d)*a*b^3*d^3*e^6 + 2688*(x*e + d)^(5/2)*a^2*b^2*e^7 - 7110*(x*e + d)^(3/2)*a^2*b^
2*d*e^7 + 5790*sqrt(x*e + d)*a^2*b^2*d^2*e^7 + 2370*(x*e + d)^(3/2)*a^3*b*e^8 - 3860*sqrt(x*e + d)*a^3*b*d*e^8
+ 965*sqrt(x*e + d)*a^4*e^9)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*
e^4 - a^5*e^5)*((x*e + d)*b - b*d + a*e)^5)