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 \[ \frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{11/2}}-\frac {63 e^4 \sqrt {d+e x}}{128 (a+b x) (b d-a e)^5}+\frac {21 e^3 \sqrt {d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac {21 e^2 \sqrt {d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac {9 e \sqrt {d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac {\sqrt {d+e x}}{5 (a+b x)^5 (b d-a e)} \]

[Out]

63/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(11/2)/b^(1/2)-1/5*(e*x+d)^(1/2)/(-a*e+b
*d)/(b*x+a)^5+9/40*e*(e*x+d)^(1/2)/(-a*e+b*d)^2/(b*x+a)^4-21/80*e^2*(e*x+d)^(1/2)/(-a*e+b*d)^3/(b*x+a)^3+21/64
*e^3*(e*x+d)^(1/2)/(-a*e+b*d)^4/(b*x+a)^2-63/128*e^4*(e*x+d)^(1/2)/(-a*e+b*d)^5/(b*x+a)

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Rubi [A]  time = 0.12, antiderivative size = 213, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.01, 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

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fricas [B]  time = 1.06, size = 1824, normalized size = 8.56 \[ \text {result too large to display} \]

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)]

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giac [B]  time = 0.20, size = 454, normalized size = 2.13 \[ -\frac {63 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} - \frac {315 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 1470 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 965 \, \sqrt {x e + d} b^{4} d^{4} e^{5} + 1470 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 7110 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 3860 \, \sqrt {x e + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 7110 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 5790 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{7} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 3860 \, \sqrt {x e + d} a^{3} b d e^{8} + 965 \, \sqrt {x e + d} a^{4} e^{9}}{640 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

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)

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maple [A]  time = 0.06, size = 211, normalized size = 0.99 \[ \frac {63 e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}+\frac {\sqrt {e x +d}\, e^{5}}{5 \left (a e -b d \right ) \left (b e x +a e \right )^{5}}+\frac {9 \sqrt {e x +d}\, e^{5}}{40 \left (a e -b d \right )^{2} \left (b e x +a e \right )^{4}}+\frac {21 \sqrt {e x +d}\, e^{5}}{80 \left (a e -b d \right )^{3} \left (b e x +a e \right )^{3}}+\frac {21 \sqrt {e x +d}\, e^{5}}{64 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{2}}+\frac {63 \sqrt {e x +d}\, e^{5}}{128 \left (a e -b d \right )^{5} \left (b e x +a e \right )} \]

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((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 0.74, size = 252, normalized size = 1.18 \[ \frac {\frac {965\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{9/2}\,\sqrt {d+e\,x}+2370\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{7/2}\,{\left (d+e\,x\right )}^{3/2}+2688\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{5/2}\,{\left (d+e\,x\right )}^{5/2}+1470\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{3/2}\,{\left (d+e\,x\right )}^{7/2}+315\,b^{9/2}\,\sqrt {a\,e-b\,d}\,{\left (d+e\,x\right )}^{9/2}+315\,b^5\,e^5\,x^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{640\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{11/2}}-\frac {63\,b^{9/2}\,e^5\,x^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,{\left (a\,e-b\,d\right )}^{11/2}}}{{\left (a+b\,x\right )}^5}+\frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((965*b^(1/2)*(a*e - b*d)^(9/2)*(d + e*x)^(1/2) + 2370*b^(3/2)*(a*e - b*d)^(7/2)*(d + e*x)^(3/2) + 2688*b^(5/2
)*(a*e - b*d)^(5/2)*(d + e*x)^(5/2) + 1470*b^(7/2)*(a*e - b*d)^(3/2)*(d + e*x)^(7/2) + 315*b^(9/2)*(a*e - b*d)
^(1/2)*(d + e*x)^(9/2) + 315*b^5*e^5*x^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(640*b^(1/2)*(a*e
- b*d)^(11/2)) - (63*b^(9/2)*e^5*x^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*(a*e - b*d)^(11/2
)))/(a + b*x)^5 + (63*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*b^(1/2)*(a*e - b*d)^(11/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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