3.17.72 \(\int \frac {1}{\sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [1672]

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

[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 = 0.08, 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, 44, 65, 214} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*Sqrt[d + e*x]/((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*Sq
rt[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*
Sqrt[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*
Sqrt[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 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 )^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 ) \text {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 [A]
time = 0.60, size = 220, normalized size = 1.03 \begin {gather*} \frac {1}{640} \left (\frac {\sqrt {d+e x} \left (965 a^4 e^4+10 a^3 b e^3 (-149 d+237 e x)+6 a^2 b^2 e^2 \left (228 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (-328 d^3+384 d^2 e x-483 d e^2 x^2+735 e^3 x^3\right )+b^4 \left (128 d^4-144 d^3 e x+168 d^2 e^2 x^2-210 d e^3 x^3+315 e^4 x^4\right )\right )}{(-b d+a e)^5 (a+b x)^5}+\frac {315 e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{11/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[d + e*x]*(965*a^4*e^4 + 10*a^3*b*e^3*(-149*d + 237*e*x) + 6*a^2*b^2*e^2*(228*d^2 - 289*d*e*x + 448*e^2*
x^2) + 2*a*b^3*e*(-328*d^3 + 384*d^2*e*x - 483*d*e^2*x^2 + 735*e^3*x^3) + b^4*(128*d^4 - 144*d^3*e*x + 168*d^2
*e^2*x^2 - 210*d*e^3*x^3 + 315*e^4*x^4)))/((-(b*d) + a*e)^5*(a + b*x)^5) + (315*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e
*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(11/2)))/640

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Maple [A]
time = 0.66, size = 285, normalized size = 1.34

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{10 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {\frac {9 \sqrt {e x +d}}{80 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{4}}+\frac {9 \left (\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}\right )}{10 \left (a e -b d \right )}}{a e -b d}\right )\)	\(285\)
default	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{10 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {\frac {9 \sqrt {e x +d}}{80 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{4}}+\frac {9 \left (\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}\right )}{10 \left (a e -b d \right )}}{a e -b d}\right )\)	\(285\)

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,method=_RETURNVERBOSE)

[Out]

2*e^5*(1/10*(e*x+d)^(1/2)/(a*e-b*d)/((e*x+d)*b+a*e-b*d)^5+9/10/(a*e-b*d)*(1/8*(e*x+d)^(1/2)/(a*e-b*d)/((e*x+d)
*b+a*e-b*d)^4+7/8/(a*e-b*d)*(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)*arctan(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)^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(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. 871 vs. \(2 (191) = 382\).
time = 2.85, size = 1757, normalized size = 8.25 \begin {gather*} \left [-\frac {315 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt {b^{2} d - a b e} e^{5} \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 (128 \, b^{6} d^{5} - {\left (315 \, a b^{5} x^{4} + 1470 \, a^{2} b^{4} x^{3} + 2688 \, a^{3} b^{3} x^{2} + 2370 \, a^{4} b^{2} x + 965 \, a^{5} b\right )} e^{5} + {\left (315 \, b^{6} d x^{4} + 1680 \, a b^{5} d x^{3} + 3654 \, a^{2} b^{4} d x^{2} + 4104 \, a^{3} b^{3} d x + 2455 \, a^{4} b^{2} d\right )} e^{4} - 2 \, {\left (105 \, b^{6} d^{2} x^{3} + 567 \, a b^{5} d^{2} x^{2} + 1251 \, a^{2} b^{4} d^{2} x + 1429 \, a^{3} b^{3} d^{2}\right )} e^{3} + 8 \, {\left (21 \, b^{6} d^{3} x^{2} + 114 \, a b^{5} d^{3} x + 253 \, a^{2} b^{4} d^{3}\right )} e^{2} - 16 \, {\left (9 \, b^{6} d^{4} x + 49 \, a b^{5} d^{4}\right )} e\right )} \sqrt {x e + d}}{1280 \, {\left (b^{12} d^{6} x^{5} + 5 \, a b^{11} d^{6} x^{4} + 10 \, a^{2} b^{10} d^{6} x^{3} + 10 \, a^{3} b^{9} d^{6} x^{2} + 5 \, a^{4} b^{8} d^{6} x + a^{5} b^{7} d^{6} + {\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )} e^{6} - 6 \, {\left (a^{5} b^{7} d x^{5} + 5 \, a^{6} b^{6} d x^{4} + 10 \, a^{7} b^{5} d x^{3} + 10 \, a^{8} b^{4} d x^{2} + 5 \, a^{9} b^{3} d x + a^{10} b^{2} d\right )} e^{5} + 15 \, {\left (a^{4} b^{8} d^{2} x^{5} + 5 \, a^{5} b^{7} d^{2} x^{4} + 10 \, a^{6} b^{6} d^{2} x^{3} + 10 \, a^{7} b^{5} d^{2} x^{2} + 5 \, a^{8} b^{4} d^{2} x + a^{9} b^{3} d^{2}\right )} e^{4} - 20 \, {\left (a^{3} b^{9} d^{3} x^{5} + 5 \, a^{4} b^{8} d^{3} x^{4} + 10 \, a^{5} b^{7} d^{3} x^{3} + 10 \, a^{6} b^{6} d^{3} x^{2} + 5 \, a^{7} b^{5} d^{3} x + a^{8} b^{4} d^{3}\right )} e^{3} + 15 \, {\left (a^{2} b^{10} d^{4} x^{5} + 5 \, a^{3} b^{9} d^{4} x^{4} + 10 \, a^{4} b^{8} d^{4} x^{3} + 10 \, a^{5} b^{7} d^{4} x^{2} + 5 \, a^{6} b^{6} d^{4} x + a^{7} b^{5} d^{4}\right )} e^{2} - 6 \, {\left (a b^{11} d^{5} x^{5} + 5 \, a^{2} b^{10} d^{5} x^{4} + 10 \, a^{3} b^{9} d^{5} x^{3} + 10 \, a^{4} b^{8} d^{5} x^{2} + 5 \, a^{5} b^{7} d^{5} x + a^{6} b^{6} d^{5}\right )} e\right )}}, -\frac {315 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\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^{5} + {\left (128 \, b^{6} d^{5} - {\left (315 \, a b^{5} x^{4} + 1470 \, a^{2} b^{4} x^{3} + 2688 \, a^{3} b^{3} x^{2} + 2370 \, a^{4} b^{2} x + 965 \, a^{5} b\right )} e^{5} + {\left (315 \, b^{6} d x^{4} + 1680 \, a b^{5} d x^{3} + 3654 \, a^{2} b^{4} d x^{2} + 4104 \, a^{3} b^{3} d x + 2455 \, a^{4} b^{2} d\right )} e^{4} - 2 \, {\left (105 \, b^{6} d^{2} x^{3} + 567 \, a b^{5} d^{2} x^{2} + 1251 \, a^{2} b^{4} d^{2} x + 1429 \, a^{3} b^{3} d^{2}\right )} e^{3} + 8 \, {\left (21 \, b^{6} d^{3} x^{2} + 114 \, a b^{5} d^{3} x + 253 \, a^{2} b^{4} d^{3}\right )} e^{2} - 16 \, {\left (9 \, b^{6} d^{4} x + 49 \, a b^{5} d^{4}\right )} e\right )} \sqrt {x e + d}}{640 \, {\left (b^{12} d^{6} x^{5} + 5 \, a b^{11} d^{6} x^{4} + 10 \, a^{2} b^{10} d^{6} x^{3} + 10 \, a^{3} b^{9} d^{6} x^{2} + 5 \, a^{4} b^{8} d^{6} x + a^{5} b^{7} d^{6} + {\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )} e^{6} - 6 \, {\left (a^{5} b^{7} d x^{5} + 5 \, a^{6} b^{6} d x^{4} + 10 \, a^{7} b^{5} d x^{3} + 10 \, a^{8} b^{4} d x^{2} + 5 \, a^{9} b^{3} d x + a^{10} b^{2} d\right )} e^{5} + 15 \, {\left (a^{4} b^{8} d^{2} x^{5} + 5 \, a^{5} b^{7} d^{2} x^{4} + 10 \, a^{6} b^{6} d^{2} x^{3} + 10 \, a^{7} b^{5} d^{2} x^{2} + 5 \, a^{8} b^{4} d^{2} x + a^{9} b^{3} d^{2}\right )} e^{4} - 20 \, {\left (a^{3} b^{9} d^{3} x^{5} + 5 \, a^{4} b^{8} d^{3} x^{4} + 10 \, a^{5} b^{7} d^{3} x^{3} + 10 \, a^{6} b^{6} d^{3} x^{2} + 5 \, a^{7} b^{5} d^{3} x + a^{8} b^{4} d^{3}\right )} e^{3} + 15 \, {\left (a^{2} b^{10} d^{4} x^{5} + 5 \, a^{3} b^{9} d^{4} x^{4} + 10 \, a^{4} b^{8} d^{4} x^{3} + 10 \, a^{5} b^{7} d^{4} x^{2} + 5 \, a^{6} b^{6} d^{4} x + a^{7} b^{5} d^{4}\right )} e^{2} - 6 \, {\left (a b^{11} d^{5} x^{5} + 5 \, a^{2} b^{10} d^{5} x^{4} + 10 \, a^{3} b^{9} d^{5} x^{3} + 10 \, a^{4} b^{8} d^{5} x^{2} + 5 \, a^{5} b^{7} d^{5} x + a^{6} b^{6} d^{5}\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)^3,x, algorithm="fricas")

[Out]

[-1/1280*(315*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)*sqrt(b^2*d - a*b*e)*
e^5*log((2*b*d + (b*x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) + 2*(128*b^6*d^5 - (315*a*b^5*x
^4 + 1470*a^2*b^4*x^3 + 2688*a^3*b^3*x^2 + 2370*a^4*b^2*x + 965*a^5*b)*e^5 + (315*b^6*d*x^4 + 1680*a*b^5*d*x^3
+ 3654*a^2*b^4*d*x^2 + 4104*a^3*b^3*d*x + 2455*a^4*b^2*d)*e^4 - 2*(105*b^6*d^2*x^3 + 567*a*b^5*d^2*x^2 + 1251
*a^2*b^4*d^2*x + 1429*a^3*b^3*d^2)*e^3 + 8*(21*b^6*d^3*x^2 + 114*a*b^5*d^3*x + 253*a^2*b^4*d^3)*e^2 - 16*(9*b^
6*d^4*x + 49*a*b^5*d^4)*e)*sqrt(x*e + d))/(b^12*d^6*x^5 + 5*a*b^11*d^6*x^4 + 10*a^2*b^10*d^6*x^3 + 10*a^3*b^9*
d^6*x^2 + 5*a^4*b^8*d^6*x + a^5*b^7*d^6 + (a^6*b^6*x^5 + 5*a^7*b^5*x^4 + 10*a^8*b^4*x^3 + 10*a^9*b^3*x^2 + 5*a
^10*b^2*x + a^11*b)*e^6 - 6*(a^5*b^7*d*x^5 + 5*a^6*b^6*d*x^4 + 10*a^7*b^5*d*x^3 + 10*a^8*b^4*d*x^2 + 5*a^9*b^3
*d*x + a^10*b^2*d)*e^5 + 15*(a^4*b^8*d^2*x^5 + 5*a^5*b^7*d^2*x^4 + 10*a^6*b^6*d^2*x^3 + 10*a^7*b^5*d^2*x^2 + 5
*a^8*b^4*d^2*x + a^9*b^3*d^2)*e^4 - 20*(a^3*b^9*d^3*x^5 + 5*a^4*b^8*d^3*x^4 + 10*a^5*b^7*d^3*x^3 + 10*a^6*b^6*
d^3*x^2 + 5*a^7*b^5*d^3*x + a^8*b^4*d^3)*e^3 + 15*(a^2*b^10*d^4*x^5 + 5*a^3*b^9*d^4*x^4 + 10*a^4*b^8*d^4*x^3 +
10*a^5*b^7*d^4*x^2 + 5*a^6*b^6*d^4*x + a^7*b^5*d^4)*e^2 - 6*(a*b^11*d^5*x^5 + 5*a^2*b^10*d^5*x^4 + 10*a^3*b^9
*d^5*x^3 + 10*a^4*b^8*d^5*x^2 + 5*a^5*b^7*d^5*x + a^6*b^6*d^5)*e), -1/640*(315*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2
*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)*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^5 + (128*b^6*d^5 - (315*a*b^5*x^4 + 1470*a^2*b^4*x^3 + 2688*a^3*b^3*x^2 + 2370*a^4*b^2*x + 965*
a^5*b)*e^5 + (315*b^6*d*x^4 + 1680*a*b^5*d*x^3 + 3654*a^2*b^4*d*x^2 + 4104*a^3*b^3*d*x + 2455*a^4*b^2*d)*e^4 -
2*(105*b^6*d^2*x^3 + 567*a*b^5*d^2*x^2 + 1251*a^2*b^4*d^2*x + 1429*a^3*b^3*d^2)*e^3 + 8*(21*b^6*d^3*x^2 + 114
*a*b^5*d^3*x + 253*a^2*b^4*d^3)*e^2 - 16*(9*b^6*d^4*x + 49*a*b^5*d^4)*e)*sqrt(x*e + d))/(b^12*d^6*x^5 + 5*a*b^
11*d^6*x^4 + 10*a^2*b^10*d^6*x^3 + 10*a^3*b^9*d^6*x^2 + 5*a^4*b^8*d^6*x + a^5*b^7*d^6 + (a^6*b^6*x^5 + 5*a^7*b
^5*x^4 + 10*a^8*b^4*x^3 + 10*a^9*b^3*x^2 + 5*a^10*b^2*x + a^11*b)*e^6 - 6*(a^5*b^7*d*x^5 + 5*a^6*b^6*d*x^4 + 1
0*a^7*b^5*d*x^3 + 10*a^8*b^4*d*x^2 + 5*a^9*b^3*d*x + a^10*b^2*d)*e^5 + 15*(a^4*b^8*d^2*x^5 + 5*a^5*b^7*d^2*x^4
+ 10*a^6*b^6*d^2*x^3 + 10*a^7*b^5*d^2*x^2 + 5*a^8*b^4*d^2*x + a^9*b^3*d^2)*e^4 - 20*(a^3*b^9*d^3*x^5 + 5*a^4*
b^8*d^3*x^4 + 10*a^5*b^7*d^3*x^3 + 10*a^6*b^6*d^3*x^2 + 5*a^7*b^5*d^3*x + a^8*b^4*d^3)*e^3 + 15*(a^2*b^10*d^4*
x^5 + 5*a^3*b^9*d^4*x^4 + 10*a^4*b^8*d^4*x^3 + 10*a^5*b^7*d^4*x^2 + 5*a^6*b^6*d^4*x + a^7*b^5*d^4)*e^2 - 6*(a*
b^11*d^5*x^5 + 5*a^2*b^10*d^5*x^4 + 10*a^3*b^9*d^5*x^3 + 10*a^4*b^8*d^5*x^2 + 5*a^5*b^7*d^5*x + a^6*b^6*d^5)*e
)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (191) = 382\).
time = 0.92, size = 454, normalized size = 2.13 \begin {gather*} -\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 {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)^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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Mupad [B]
time = 0.74, size = 252, normalized size = 1.18 \begin {gather*} \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}} \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)^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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