3.17.60 \(\int \frac {1}{\sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^2} \, dx\) [1660]

Optimal. Leaf size=147 \[ -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}+\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}} \]

[Out]

5/8*e^3*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(7/2)/b^(1/2)-1/3*(e*x+d)^(1/2)/(-a*e+b*d)/
(b*x+a)^3+5/12*e*(e*x+d)^(1/2)/(-a*e+b*d)^2/(b*x+a)^2-5/8*e^2*(e*x+d)^(1/2)/(-a*e+b*d)^3/(b*x+a)

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Rubi [A]
time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 44, 65, 214} \begin {gather*} \frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}}-\frac {5 e^2 \sqrt {d+e x}}{8 (a+b x) (b d-a e)^3}+\frac {5 e \sqrt {d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Sqrt[d + e*x]/((b*d - a*e)*(a + b*x)^3) + (5*e*Sqrt[d + e*x])/(12*(b*d - a*e)^2*(a + b*x)^2) - (5*e^2*Sqr
t[d + e*x])/(8*(b*d - a*e)^3*(a + b*x)) + (5*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*
(b*d - a*e)^(7/2))

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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], 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 )^2} \, dx &=\int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}-\frac {(5 e) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{6 (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}+\frac {\left (5 e^2\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac {\left (5 e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac {\left (5 e^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}+\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 128, normalized size = 0.87 \begin {gather*} \frac {\sqrt {d+e x} \left (33 a^2 e^2+2 a b e (-13 d+20 e x)+b^2 \left (8 d^2-10 d e x+15 e^2 x^2\right )\right )}{24 (-b d+a e)^3 (a+b x)^3}+\frac {5 e^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 \sqrt {b} (-b d+a e)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(33*a^2*e^2 + 2*a*b*e*(-13*d + 20*e*x) + b^2*(8*d^2 - 10*d*e*x + 15*e^2*x^2)))/(24*(-(b*d) + a*
e)^3*(a + b*x)^3) + (5*e^3*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*Sqrt[b]*(-(b*d) + a*e)^(7/2)
)

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Maple [A]
time = 0.75, size = 187, normalized size = 1.27

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{6 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}}{a e -b d}\right )\)	\(187\)
default	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{6 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}}{a e -b d}\right )\)	\(187\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^3*(1/6*(e*x+d)^(1/2)/(a*e-b*d)/((e*x+d)*b+a*e-b*d)^3+5/6/(a*e-b*d)*(1/4*(e*x+d)^(1/2)/(a*e-b*d)/((e*x+d)*b
+a*e-b*d)^2+3/4/(a*e-b*d)*(1/2*(e*x+d)^(1/2)/(a*e-b*d)/((e*x+d)*b+a*e-b*d)+1/2/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*a
rctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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)^2,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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (131) = 262\).
time = 3.48, size = 857, normalized size = 5.83 \begin {gather*} \left [-\frac {15 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {b^{2} d - a b e} e^{3} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) + 2 \, {\left (8 \, b^{4} d^{3} - {\left (15 \, a b^{3} x^{2} + 40 \, a^{2} b^{2} x + 33 \, a^{3} b\right )} e^{3} + {\left (15 \, b^{4} d x^{2} + 50 \, a b^{3} d x + 59 \, a^{2} b^{2} d\right )} e^{2} - 2 \, {\left (5 \, b^{4} d^{2} x + 17 \, a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{48 \, {\left (b^{8} d^{4} x^{3} + 3 \, a b^{7} d^{4} x^{2} + 3 \, a^{2} b^{6} d^{4} x + a^{3} b^{5} d^{4} + {\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )} e^{4} - 4 \, {\left (a^{3} b^{5} d x^{3} + 3 \, a^{4} b^{4} d x^{2} + 3 \, a^{5} b^{3} d x + a^{6} b^{2} d\right )} e^{3} + 6 \, {\left (a^{2} b^{6} d^{2} x^{3} + 3 \, a^{3} b^{5} d^{2} x^{2} + 3 \, a^{4} b^{4} d^{2} x + a^{5} b^{3} d^{2}\right )} e^{2} - 4 \, {\left (a b^{7} d^{3} x^{3} + 3 \, a^{2} b^{6} d^{3} x^{2} + 3 \, a^{3} b^{5} d^{3} x + a^{4} b^{4} d^{3}\right )} e\right )}}, -\frac {15 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{3} + {\left (8 \, b^{4} d^{3} - {\left (15 \, a b^{3} x^{2} + 40 \, a^{2} b^{2} x + 33 \, a^{3} b\right )} e^{3} + {\left (15 \, b^{4} d x^{2} + 50 \, a b^{3} d x + 59 \, a^{2} b^{2} d\right )} e^{2} - 2 \, {\left (5 \, b^{4} d^{2} x + 17 \, a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{8} d^{4} x^{3} + 3 \, a b^{7} d^{4} x^{2} + 3 \, a^{2} b^{6} d^{4} x + a^{3} b^{5} d^{4} + {\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )} e^{4} - 4 \, {\left (a^{3} b^{5} d x^{3} + 3 \, a^{4} b^{4} d x^{2} + 3 \, a^{5} b^{3} d x + a^{6} b^{2} d\right )} e^{3} + 6 \, {\left (a^{2} b^{6} d^{2} x^{3} + 3 \, a^{3} b^{5} d^{2} x^{2} + 3 \, a^{4} b^{4} d^{2} x + a^{5} b^{3} d^{2}\right )} e^{2} - 4 \, {\left (a b^{7} d^{3} x^{3} + 3 \, a^{2} b^{6} d^{3} x^{2} + 3 \, a^{3} b^{5} d^{3} x + a^{4} b^{4} d^{3}\right )} e\right )}}\right ] \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

[-1/48*(15*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b^2*d - a*b*e)*e^3*log((2*b*d + (b*x - a)*e - 2*sqrt
(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) + 2*(8*b^4*d^3 - (15*a*b^3*x^2 + 40*a^2*b^2*x + 33*a^3*b)*e^3 + (15*
b^4*d*x^2 + 50*a*b^3*d*x + 59*a^2*b^2*d)*e^2 - 2*(5*b^4*d^2*x + 17*a*b^3*d^2)*e)*sqrt(x*e + d))/(b^8*d^4*x^3 +
3*a*b^7*d^4*x^2 + 3*a^2*b^6*d^4*x + a^3*b^5*d^4 + (a^4*b^4*x^3 + 3*a^5*b^3*x^2 + 3*a^6*b^2*x + a^7*b)*e^4 - 4
*(a^3*b^5*d*x^3 + 3*a^4*b^4*d*x^2 + 3*a^5*b^3*d*x + a^6*b^2*d)*e^3 + 6*(a^2*b^6*d^2*x^3 + 3*a^3*b^5*d^2*x^2 +
3*a^4*b^4*d^2*x + a^5*b^3*d^2)*e^2 - 4*(a*b^7*d^3*x^3 + 3*a^2*b^6*d^3*x^2 + 3*a^3*b^5*d^3*x + a^4*b^4*d^3)*e),
-1/24*(15*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(x*e
+ d)/(b*x*e + b*d))*e^3 + (8*b^4*d^3 - (15*a*b^3*x^2 + 40*a^2*b^2*x + 33*a^3*b)*e^3 + (15*b^4*d*x^2 + 50*a*b^
3*d*x + 59*a^2*b^2*d)*e^2 - 2*(5*b^4*d^2*x + 17*a*b^3*d^2)*e)*sqrt(x*e + d))/(b^8*d^4*x^3 + 3*a*b^7*d^4*x^2 +
3*a^2*b^6*d^4*x + a^3*b^5*d^4 + (a^4*b^4*x^3 + 3*a^5*b^3*x^2 + 3*a^6*b^2*x + a^7*b)*e^4 - 4*(a^3*b^5*d*x^3 + 3
*a^4*b^4*d*x^2 + 3*a^5*b^3*d*x + a^6*b^2*d)*e^3 + 6*(a^2*b^6*d^2*x^3 + 3*a^3*b^5*d^2*x^2 + 3*a^4*b^4*d^2*x + a
^5*b^3*d^2)*e^2 - 4*(a*b^7*d^3*x^3 + 3*a^2*b^6*d^3*x^2 + 3*a^3*b^5*d^3*x + a^4*b^4*d^3)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{4} \sqrt {d + e x}}\, dx \end {gather*}

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)**2,x)

[Out]

Integral(1/((a + b*x)**4*sqrt(d + e*x)), x)

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Giac [A]
time = 0.97, size = 233, normalized size = 1.59 \begin {gather*} -\frac {5 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} - \frac {15 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} e^{3} - 40 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} d e^{3} + 33 \, \sqrt {x e + d} b^{2} d^{2} e^{3} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} a b e^{4} - 66 \, \sqrt {x e + d} a b d e^{4} + 33 \, \sqrt {x e + d} a^{2} e^{5}}{24 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

-5/8*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqr
t(-b^2*d + a*b*e)) - 1/24*(15*(x*e + d)^(5/2)*b^2*e^3 - 40*(x*e + d)^(3/2)*b^2*d*e^3 + 33*sqrt(x*e + d)*b^2*d^
2*e^3 + 40*(x*e + d)^(3/2)*a*b*e^4 - 66*sqrt(x*e + d)*a*b*d*e^4 + 33*sqrt(x*e + d)*a^2*e^5)/((b^3*d^3 - 3*a*b^
2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((x*e + d)*b - b*d + a*e)^3)

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Mupad [B]
time = 0.65, size = 218, normalized size = 1.48 \begin {gather*} \frac {\frac {11\,e^3\,\sqrt {d+e\,x}}{8\,\left (a\,e-b\,d\right )}+\frac {5\,b^2\,e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,{\left (a\,e-b\,d\right )}^3}+\frac {5\,b\,e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2}+\frac {5\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}

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)^2),x)

[Out]

((11*e^3*(d + e*x)^(1/2))/(8*(a*e - b*d)) + (5*b^2*e^3*(d + e*x)^(5/2))/(8*(a*e - b*d)^3) + (5*b*e^3*(d + e*x)
^(3/2))/(3*(a*e - b*d)^2))/((d + e*x)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a*b^2*d*e) + b^3*(d + e*x)^3 - (3*b^3*d - 3
*a*b^2*e)*(d + e*x)^2 + a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + (5*e^3*atan((b^(1/2)*(d + e*x)^(1
/2))/(a*e - b*d)^(1/2)))/(8*b^(1/2)*(a*e - b*d)^(7/2))

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