3.17.71 \(\int \frac {\sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1671]

Optimal. Leaf size=218 \[ -\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}-\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}} \]

[Out]

-7/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(9/2)-1/5*(e*x+d)^(1/2)/b/(b*x+a
)^5-1/40*e*(e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)^4+7/240*e^2*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/(b*x+a)^3-7/192*e^3*(e*
x+d)^(1/2)/b/(-a*e+b*d)^3/(b*x+a)^2+7/128*e^4*(e*x+d)^(1/2)/b/(-a*e+b*d)^4/(b*x+a)

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Rubi [A]
time = 0.08, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 44, 65, 214} \begin {gather*} -\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac {7 e^4 \sqrt {d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac {7 e^3 \sqrt {d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac {7 e^2 \sqrt {d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac {e \sqrt {d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac {\sqrt {d+e x}}{5 b (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*Sqrt[d + e*x]/(b*(a + b*x)^5) - (e*Sqrt[d + e*x])/(40*b*(b*d - a*e)*(a + b*x)^4) + (7*e^2*Sqrt[d + e*x])/
(240*b*(b*d - a*e)^2*(a + b*x)^3) - (7*e^3*Sqrt[d + e*x])/(192*b*(b*d - a*e)^3*(a + b*x)^2) + (7*e^4*Sqrt[d +
e*x])/(128*b*(b*d - a*e)^4*(a + b*x)) - (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(3/2)*
(b*d - a*e)^(9/2))

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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {\sqrt {d+e x}}{(a+b x)^6} \, dx\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}+\frac {e \int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx}{10 b}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}-\frac {\left (7 e^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}+\frac {\left (7 e^3\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}-\frac {\left (7 e^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac {\left (7 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b (b d-a e)^4}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}+\frac {\left (7 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 (b d-a e)^4}\\ &=-\frac {\sqrt {d+e x}}{5 b (a+b x)^5}-\frac {e \sqrt {d+e x}}{40 b (b d-a e) (a+b x)^4}+\frac {7 e^2 \sqrt {d+e x}}{240 b (b d-a e)^2 (a+b x)^3}-\frac {7 e^3 \sqrt {d+e x}}{192 b (b d-a e)^3 (a+b x)^2}+\frac {7 e^4 \sqrt {d+e x}}{128 b (b d-a e)^4 (a+b x)}-\frac {7 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.29, size = 224, normalized size = 1.03 \begin {gather*} \frac {\sqrt {d+e x} \left (-105 a^4 e^4+10 a^3 b e^3 (121 d+79 e x)+2 a^2 b^2 e^2 \left (-1052 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (744 d^3+128 d^2 e x-161 d e^2 x^2+245 e^3 x^3\right )+b^4 \left (-384 d^4-48 d^3 e x+56 d^2 e^2 x^2-70 d e^3 x^3+105 e^4 x^4\right )\right )}{1920 b (b d-a e)^4 (a+b x)^5}+\frac {7 e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{3/2} (-b d+a e)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-105*a^4*e^4 + 10*a^3*b*e^3*(121*d + 79*e*x) + 2*a^2*b^2*e^2*(-1052*d^2 - 289*d*e*x + 448*e^2*
x^2) + 2*a*b^3*e*(744*d^3 + 128*d^2*e*x - 161*d*e^2*x^2 + 245*e^3*x^3) + b^4*(-384*d^4 - 48*d^3*e*x + 56*d^2*e
^2*x^2 - 70*d*e^3*x^3 + 105*e^4*x^4)))/(1920*b*(b*d - a*e)^4*(a + b*x)^5) + (7*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[-(b*d) + a*e]])/(128*b^(3/2)*(-(b*d) + a*e)^(9/2))

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Maple [A]
time = 0.68, size = 293, normalized size = 1.34

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {\frac {7 b^{3} \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {49 b^{2} \left (e x +d \right )^{\frac {7}{2}}}{384 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 b \left (e x +d \right )^{\frac {5}{2}}}{30 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {79 \left (e x +d \right )^{\frac {3}{2}}}{384 \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}}{256 b}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(293\)
default	\(2 e^{5} \left (\frac {\frac {7 b^{3} \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {49 b^{2} \left (e x +d \right )^{\frac {7}{2}}}{384 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 b \left (e x +d \right )^{\frac {5}{2}}}{30 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {79 \left (e x +d \right )^{\frac {3}{2}}}{384 \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}}{256 b}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^5*((7/256*b^3/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(e*x+d)^(9/2)+49/384*b^2/(a^
3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)+7/30*b/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)+79/3
84/(a*e-b*d)*(e*x+d)^(3/2)-7/256/b*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b*d)^5+7/256/b/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b
^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/(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((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. 792 vs. \(2 (195) = 390\).
time = 3.21, size = 1598, normalized size = 7.33 \begin {gather*} \left [\frac {105 \, {\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 (384 \, b^{6} d^{5} + {\left (105 \, a b^{5} x^{4} + 490 \, a^{2} b^{4} x^{3} + 896 \, a^{3} b^{3} x^{2} + 790 \, a^{4} b^{2} x - 105 \, a^{5} b\right )} e^{5} - {\left (105 \, b^{6} d x^{4} + 560 \, a b^{5} d x^{3} + 1218 \, a^{2} b^{4} d x^{2} + 1368 \, a^{3} b^{3} d x - 1315 \, a^{4} b^{2} d\right )} e^{4} + 2 \, {\left (35 \, b^{6} d^{2} x^{3} + 189 \, a b^{5} d^{2} x^{2} + 417 \, a^{2} b^{4} d^{2} x - 1657 \, a^{3} b^{3} d^{2}\right )} e^{3} - 8 \, {\left (7 \, b^{6} d^{3} x^{2} + 38 \, a b^{5} d^{3} x - 449 \, a^{2} b^{4} d^{3}\right )} e^{2} + 48 \, {\left (b^{6} d^{4} x - 39 \, a b^{5} d^{4}\right )} e\right )} \sqrt {x e + d}}{3840 \, {\left (b^{12} d^{5} x^{5} + 5 \, a b^{11} d^{5} x^{4} + 10 \, a^{2} b^{10} d^{5} x^{3} + 10 \, a^{3} b^{9} d^{5} x^{2} + 5 \, a^{4} b^{8} d^{5} x + a^{5} b^{7} d^{5} - {\left (a^{5} b^{7} x^{5} + 5 \, a^{6} b^{6} x^{4} + 10 \, a^{7} b^{5} x^{3} + 10 \, a^{8} b^{4} x^{2} + 5 \, a^{9} b^{3} x + a^{10} b^{2}\right )} e^{5} + 5 \, {\left (a^{4} b^{8} d x^{5} + 5 \, a^{5} b^{7} d x^{4} + 10 \, a^{6} b^{6} d x^{3} + 10 \, a^{7} b^{5} d x^{2} + 5 \, a^{8} b^{4} d x + a^{9} b^{3} d\right )} e^{4} - 10 \, {\left (a^{3} b^{9} d^{2} x^{5} + 5 \, a^{4} b^{8} d^{2} x^{4} + 10 \, a^{5} b^{7} d^{2} x^{3} + 10 \, a^{6} b^{6} d^{2} x^{2} + 5 \, a^{7} b^{5} d^{2} x + a^{8} b^{4} d^{2}\right )} e^{3} + 10 \, {\left (a^{2} b^{10} d^{3} x^{5} + 5 \, a^{3} b^{9} d^{3} x^{4} + 10 \, a^{4} b^{8} d^{3} x^{3} + 10 \, a^{5} b^{7} d^{3} x^{2} + 5 \, a^{6} b^{6} d^{3} x + a^{7} b^{5} d^{3}\right )} e^{2} - 5 \, {\left (a b^{11} d^{4} x^{5} + 5 \, a^{2} b^{10} d^{4} x^{4} + 10 \, a^{3} b^{9} d^{4} x^{3} + 10 \, a^{4} b^{8} d^{4} x^{2} + 5 \, a^{5} b^{7} d^{4} x + a^{6} b^{6} d^{4}\right )} e\right )}}, \frac {105 \, {\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 (384 \, b^{6} d^{5} + {\left (105 \, a b^{5} x^{4} + 490 \, a^{2} b^{4} x^{3} + 896 \, a^{3} b^{3} x^{2} + 790 \, a^{4} b^{2} x - 105 \, a^{5} b\right )} e^{5} - {\left (105 \, b^{6} d x^{4} + 560 \, a b^{5} d x^{3} + 1218 \, a^{2} b^{4} d x^{2} + 1368 \, a^{3} b^{3} d x - 1315 \, a^{4} b^{2} d\right )} e^{4} + 2 \, {\left (35 \, b^{6} d^{2} x^{3} + 189 \, a b^{5} d^{2} x^{2} + 417 \, a^{2} b^{4} d^{2} x - 1657 \, a^{3} b^{3} d^{2}\right )} e^{3} - 8 \, {\left (7 \, b^{6} d^{3} x^{2} + 38 \, a b^{5} d^{3} x - 449 \, a^{2} b^{4} d^{3}\right )} e^{2} + 48 \, {\left (b^{6} d^{4} x - 39 \, a b^{5} d^{4}\right )} e\right )} \sqrt {x e + d}}{1920 \, {\left (b^{12} d^{5} x^{5} + 5 \, a b^{11} d^{5} x^{4} + 10 \, a^{2} b^{10} d^{5} x^{3} + 10 \, a^{3} b^{9} d^{5} x^{2} + 5 \, a^{4} b^{8} d^{5} x + a^{5} b^{7} d^{5} - {\left (a^{5} b^{7} x^{5} + 5 \, a^{6} b^{6} x^{4} + 10 \, a^{7} b^{5} x^{3} + 10 \, a^{8} b^{4} x^{2} + 5 \, a^{9} b^{3} x + a^{10} b^{2}\right )} e^{5} + 5 \, {\left (a^{4} b^{8} d x^{5} + 5 \, a^{5} b^{7} d x^{4} + 10 \, a^{6} b^{6} d x^{3} + 10 \, a^{7} b^{5} d x^{2} + 5 \, a^{8} b^{4} d x + a^{9} b^{3} d\right )} e^{4} - 10 \, {\left (a^{3} b^{9} d^{2} x^{5} + 5 \, a^{4} b^{8} d^{2} x^{4} + 10 \, a^{5} b^{7} d^{2} x^{3} + 10 \, a^{6} b^{6} d^{2} x^{2} + 5 \, a^{7} b^{5} d^{2} x + a^{8} b^{4} d^{2}\right )} e^{3} + 10 \, {\left (a^{2} b^{10} d^{3} x^{5} + 5 \, a^{3} b^{9} d^{3} x^{4} + 10 \, a^{4} b^{8} d^{3} x^{3} + 10 \, a^{5} b^{7} d^{3} x^{2} + 5 \, a^{6} b^{6} d^{3} x + a^{7} b^{5} d^{3}\right )} e^{2} - 5 \, {\left (a b^{11} d^{4} x^{5} + 5 \, a^{2} b^{10} d^{4} x^{4} + 10 \, a^{3} b^{9} d^{4} x^{3} + 10 \, a^{4} b^{8} d^{4} x^{2} + 5 \, a^{5} b^{7} d^{4} x + a^{6} b^{6} d^{4}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/3840*(105*(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*(384*b^6*d^5 + (105*a*b^5*x^
4 + 490*a^2*b^4*x^3 + 896*a^3*b^3*x^2 + 790*a^4*b^2*x - 105*a^5*b)*e^5 - (105*b^6*d*x^4 + 560*a*b^5*d*x^3 + 12
18*a^2*b^4*d*x^2 + 1368*a^3*b^3*d*x - 1315*a^4*b^2*d)*e^4 + 2*(35*b^6*d^2*x^3 + 189*a*b^5*d^2*x^2 + 417*a^2*b^
4*d^2*x - 1657*a^3*b^3*d^2)*e^3 - 8*(7*b^6*d^3*x^2 + 38*a*b^5*d^3*x - 449*a^2*b^4*d^3)*e^2 + 48*(b^6*d^4*x - 3
9*a*b^5*d^4)*e)*sqrt(x*e + d))/(b^12*d^5*x^5 + 5*a*b^11*d^5*x^4 + 10*a^2*b^10*d^5*x^3 + 10*a^3*b^9*d^5*x^2 + 5
*a^4*b^8*d^5*x + a^5*b^7*d^5 - (a^5*b^7*x^5 + 5*a^6*b^6*x^4 + 10*a^7*b^5*x^3 + 10*a^8*b^4*x^2 + 5*a^9*b^3*x +
a^10*b^2)*e^5 + 5*(a^4*b^8*d*x^5 + 5*a^5*b^7*d*x^4 + 10*a^6*b^6*d*x^3 + 10*a^7*b^5*d*x^2 + 5*a^8*b^4*d*x + a^9
*b^3*d)*e^4 - 10*(a^3*b^9*d^2*x^5 + 5*a^4*b^8*d^2*x^4 + 10*a^5*b^7*d^2*x^3 + 10*a^6*b^6*d^2*x^2 + 5*a^7*b^5*d^
2*x + a^8*b^4*d^2)*e^3 + 10*(a^2*b^10*d^3*x^5 + 5*a^3*b^9*d^3*x^4 + 10*a^4*b^8*d^3*x^3 + 10*a^5*b^7*d^3*x^2 +
5*a^6*b^6*d^3*x + a^7*b^5*d^3)*e^2 - 5*(a*b^11*d^4*x^5 + 5*a^2*b^10*d^4*x^4 + 10*a^3*b^9*d^4*x^3 + 10*a^4*b^8*
d^4*x^2 + 5*a^5*b^7*d^4*x + a^6*b^6*d^4)*e), 1/1920*(105*(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 - (38
4*b^6*d^5 + (105*a*b^5*x^4 + 490*a^2*b^4*x^3 + 896*a^3*b^3*x^2 + 790*a^4*b^2*x - 105*a^5*b)*e^5 - (105*b^6*d*x
^4 + 560*a*b^5*d*x^3 + 1218*a^2*b^4*d*x^2 + 1368*a^3*b^3*d*x - 1315*a^4*b^2*d)*e^4 + 2*(35*b^6*d^2*x^3 + 189*a
*b^5*d^2*x^2 + 417*a^2*b^4*d^2*x - 1657*a^3*b^3*d^2)*e^3 - 8*(7*b^6*d^3*x^2 + 38*a*b^5*d^3*x - 449*a^2*b^4*d^3
)*e^2 + 48*(b^6*d^4*x - 39*a*b^5*d^4)*e)*sqrt(x*e + d))/(b^12*d^5*x^5 + 5*a*b^11*d^5*x^4 + 10*a^2*b^10*d^5*x^3
+ 10*a^3*b^9*d^5*x^2 + 5*a^4*b^8*d^5*x + a^5*b^7*d^5 - (a^5*b^7*x^5 + 5*a^6*b^6*x^4 + 10*a^7*b^5*x^3 + 10*a^8
*b^4*x^2 + 5*a^9*b^3*x + a^10*b^2)*e^5 + 5*(a^4*b^8*d*x^5 + 5*a^5*b^7*d*x^4 + 10*a^6*b^6*d*x^3 + 10*a^7*b^5*d*
x^2 + 5*a^8*b^4*d*x + a^9*b^3*d)*e^4 - 10*(a^3*b^9*d^2*x^5 + 5*a^4*b^8*d^2*x^4 + 10*a^5*b^7*d^2*x^3 + 10*a^6*b
^6*d^2*x^2 + 5*a^7*b^5*d^2*x + a^8*b^4*d^2)*e^3 + 10*(a^2*b^10*d^3*x^5 + 5*a^3*b^9*d^3*x^4 + 10*a^4*b^8*d^3*x^
3 + 10*a^5*b^7*d^3*x^2 + 5*a^6*b^6*d^3*x + a^7*b^5*d^3)*e^2 - 5*(a*b^11*d^4*x^5 + 5*a^2*b^10*d^4*x^4 + 10*a^3*
b^9*d^4*x^3 + 10*a^4*b^8*d^4*x^2 + 5*a^5*b^7*d^4*x + a^6*b^6*d^4)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 25549 vs. \(2 (187) = 374\).
time = 219.32, size = 25549, normalized size = 117.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1930*a**5*e**10*sqrt(d + e*x)/(1280*a**10*b*e**10 - 6400*a**9*b**2*d*e**9 + 6400*a**9*b**2*e**10*x - 57600*a*
*8*b**3*d*e**9*x + 12800*a**8*b**3*e**8*(d + e*x)**2 + 76800*a**7*b**4*d**3*e**7 + 230400*a**7*b**4*d**2*e**8*
x - 102400*a**7*b**4*d*e**7*(d + e*x)**2 + 12800*a**7*b**4*e**7*(d + e*x)**3 - 268800*a**6*b**5*d**4*e**6 - 53
7600*a**6*b**5*d**3*e**7*x + 358400*a**6*b**5*d**2*e**6*(d + e*x)**2 - 89600*a**6*b**5*d*e**6*(d + e*x)**3 + 6
400*a**6*b**5*e**6*(d + e*x)**4 + 483840*a**5*b**6*d**5*e**5 + 806400*a**5*b**6*d**4*e**6*x - 716800*a**5*b**6
*d**3*e**5*(d + e*x)**2 + 268800*a**5*b**6*d**2*e**5*(d + e*x)**3 - 38400*a**5*b**6*d*e**5*(d + e*x)**4 + 1280
*a**5*b**6*e**5*(d + e*x)**5 - 537600*a**4*b**7*d**6*e**4 - 806400*a**4*b**7*d**5*e**5*x + 896000*a**4*b**7*d*
*4*e**4*(d + e*x)**2 - 448000*a**4*b**7*d**3*e**4*(d + e*x)**3 + 96000*a**4*b**7*d**2*e**4*(d + e*x)**4 - 6400
*a**4*b**7*d*e**4*(d + e*x)**5 + 384000*a**3*b**8*d**7*e**3 + 537600*a**3*b**8*d**6*e**4*x - 716800*a**3*b**8*
d**5*e**3*(d + e*x)**2 + 448000*a**3*b**8*d**4*e**3*(d + e*x)**3 - 128000*a**3*b**8*d**3*e**3*(d + e*x)**4 + 1
2800*a**3*b**8*d**2*e**3*(d + e*x)**5 - 172800*a**2*b**9*d**8*e**2 - 230400*a**2*b**9*d**7*e**3*x + 358400*a**
2*b**9*d**6*e**2*(d + e*x)**2 - 268800*a**2*b**9*d**5*e**2*(d + e*x)**3 + 96000*a**2*b**9*d**4*e**2*(d + e*x)*
*4 - 12800*a**2*b**9*d**3*e**2*(d + e*x)**5 + 44800*a*b**10*d**9*e + 57600*a*b**10*d**8*e**2*x - 102400*a*b**1
0*d**7*e*(d + e*x)**2 + 89600*a*b**10*d**6*e*(d + e*x)**3 - 38400*a*b**10*d**5*e*(d + e*x)**4 + 6400*a*b**10*d
**4*e*(d + e*x)**5 - 5120*b**11*d**10 - 6400*b**11*d**9*e*x + 12800*b**11*d**8*(d + e*x)**2 - 12800*b**11*d**7
*(d + e*x)**3 + 6400*b**11*d**6*(d + e*x)**4 - 1280*b**11*d**5*(d + e*x)**5) + 9650*a**4*d*e**9*sqrt(d + e*x)/
(1280*a**10*e**10 - 6400*a**9*b*d*e**9 + 6400*a**9*b*e**10*x - 57600*a**8*b**2*d*e**9*x + 12800*a**8*b**2*e**8
*(d + e*x)**2 + 76800*a**7*b**3*d**3*e**7 + 230400*a**7*b**3*d**2*e**8*x - 102400*a**7*b**3*d*e**7*(d + e*x)**
2 + 12800*a**7*b**3*e**7*(d + e*x)**3 - 268800*a**6*b**4*d**4*e**6 - 537600*a**6*b**4*d**3*e**7*x + 358400*a**
6*b**4*d**2*e**6*(d + e*x)**2 - 89600*a**6*b**4*d*e**6*(d + e*x)**3 + 6400*a**6*b**4*e**6*(d + e*x)**4 + 48384
0*a**5*b**5*d**5*e**5 + 806400*a**5*b**5*d**4*e**6*x - 716800*a**5*b**5*d**3*e**5*(d + e*x)**2 + 268800*a**5*b
**5*d**2*e**5*(d + e*x)**3 - 38400*a**5*b**5*d*e**5*(d + e*x)**4 + 1280*a**5*b**5*e**5*(d + e*x)**5 - 537600*a
**4*b**6*d**6*e**4 - 806400*a**4*b**6*d**5*e**5*x + 896000*a**4*b**6*d**4*e**4*(d + e*x)**2 - 448000*a**4*b**6
*d**3*e**4*(d + e*x)**3 + 96000*a**4*b**6*d**2*e**4*(d + e*x)**4 - 6400*a**4*b**6*d*e**4*(d + e*x)**5 + 384000
*a**3*b**7*d**7*e**3 + 537600*a**3*b**7*d**6*e**4*x - 716800*a**3*b**7*d**5*e**3*(d + e*x)**2 + 448000*a**3*b*
*7*d**4*e**3*(d + e*x)**3 - 128000*a**3*b**7*d**3*e**3*(d + e*x)**4 + 12800*a**3*b**7*d**2*e**3*(d + e*x)**5 -
172800*a**2*b**8*d**8*e**2 - 230400*a**2*b**8*d**7*e**3*x + 358400*a**2*b**8*d**6*e**2*(d + e*x)**2 - 268800*
a**2*b**8*d**5*e**2*(d + e*x)**3 + 96000*a**2*b**8*d**4*e**2*(d + e*x)**4 - 12800*a**2*b**8*d**3*e**2*(d + e*x
)**5 + 44800*a*b**9*d**9*e + 57600*a*b**9*d**8*e**2*x - 102400*a*b**9*d**7*e*(d + e*x)**2 + 89600*a*b**9*d**6*
e*(d + e*x)**3 - 38400*a*b**9*d**5*e*(d + e*x)**4 + 6400*a*b**9*d**4*e*(d + e*x)**5 - 5120*b**10*d**10 - 6400*
b**10*d**9*e*x + 12800*b**10*d**8*(d + e*x)**2 - 12800*b**10*d**7*(d + e*x)**3 + 6400*b**10*d**6*(d + e*x)**4
- 1280*b**10*d**5*(d + e*x)**5) - 4740*a**4*e**9*(d + e*x)**(3/2)/(1280*a**10*e**10 - 6400*a**9*b*d*e**9 + 640
0*a**9*b*e**10*x - 57600*a**8*b**2*d*e**9*x + 12800*a**8*b**2*e**8*(d + e*x)**2 + 76800*a**7*b**3*d**3*e**7 +
230400*a**7*b**3*d**2*e**8*x - 102400*a**7*b**3*d*e**7*(d + e*x)**2 + 12800*a**7*b**3*e**7*(d + e*x)**3 - 2688
00*a**6*b**4*d**4*e**6 - 537600*a**6*b**4*d**3*e**7*x + 358400*a**6*b**4*d**2*e**6*(d + e*x)**2 - 89600*a**6*b
**4*d*e**6*(d + e*x)**3 + 6400*a**6*b**4*e**6*(d + e*x)**4 + 483840*a**5*b**5*d**5*e**5 + 806400*a**5*b**5*d**
4*e**6*x - 716800*a**5*b**5*d**3*e**5*(d + e*x)**2 + 268800*a**5*b**5*d**2*e**5*(d + e*x)**3 - 38400*a**5*b**5
*d*e**5*(d + e*x)**4 + 1280*a**5*b**5*e**5*(d + e*x)**5 - 537600*a**4*b**6*d**6*e**4 - 806400*a**4*b**6*d**5*e
**5*x + 896000*a**4*b**6*d**4*e**4*(d + e*x)**2 - 448000*a**4*b**6*d**3*e**4*(d + e*x)**3 + 96000*a**4*b**6*d*
*2*e**4*(d + e*x)**4 - 6400*a**4*b**6*d*e**4*(d + e*x)**5 + 384000*a**3*b**7*d**7*e**3 + 537600*a**3*b**7*d**6
*e**4*x - 716800*a**3*b**7*d**5*e**3*(d + e*x)**2 + 448000*a**3*b**7*d**4*e**3*(d + e*x)**3 - 128000*a**3*b**7
*d**3*e**3*(d + e*x)**4 + 12800*a**3*b**7*d**2*e**3*(d + e*x)**5 - 172800*a**2*b**8*d**8*e**2 - 230400*a**2*b*
*8*d**7*e**3*x + 358400*a**2*b**8*d**6*e**2*(d + e*x)**2 - 268800*a**2*b**8*d**5*e**2*(d + e*x)**3 + 96000*a**
2*b**8*d**4*e**2*(d + e*x)**4 - 12800*a**2*b**8*d**3*e**2*(d + e*x)**5 + 44800*a*b**9*d**9*e + 57600*a*b**9*d*
*8*e**2*x - 102400*a*b**9*d**7*e*(d + e*x)**2 + 89600*a*b**9*d**6*e*(d + e*x)**3 - 38400*a*b**9*d**5*e*(d + e*
x)**4 + 6400*a*b**9*d**4*e*(d + e*x)**5 - 5120*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (195) = 390\).
time = 0.81, size = 432, normalized size = 1.98 \begin {gather*} \frac {7 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 490 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 896 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 790 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt {x e + d} b^{4} d^{4} e^{5} + 490 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt {x e + d} a b^{3} d^{3} e^{6} + 896 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{7} + 790 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{8} + 420 \, \sqrt {x e + d} a^{3} b d e^{8} - 105 \, \sqrt {x e + d} a^{4} e^{9}}{1920 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\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((e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

7/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b
^2*d*e^3 + a^4*b*e^4)*sqrt(-b^2*d + a*b*e)) + 1/1920*(105*(x*e + d)^(9/2)*b^4*e^5 - 490*(x*e + d)^(7/2)*b^4*d*
e^5 + 896*(x*e + d)^(5/2)*b^4*d^2*e^5 - 790*(x*e + d)^(3/2)*b^4*d^3*e^5 - 105*sqrt(x*e + d)*b^4*d^4*e^5 + 490*
(x*e + d)^(7/2)*a*b^3*e^6 - 1792*(x*e + d)^(5/2)*a*b^3*d*e^6 + 2370*(x*e + d)^(3/2)*a*b^3*d^2*e^6 + 420*sqrt(x
*e + d)*a*b^3*d^3*e^6 + 896*(x*e + d)^(5/2)*a^2*b^2*e^7 - 2370*(x*e + d)^(3/2)*a^2*b^2*d*e^7 - 630*sqrt(x*e +
d)*a^2*b^2*d^2*e^7 + 790*(x*e + d)^(3/2)*a^3*b*e^8 + 420*sqrt(x*e + d)*a^3*b*d*e^8 - 105*sqrt(x*e + d)*a^4*e^9
)/((b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*((x*e + d)*b - b*d + a*e)^5)

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Mupad [B]
time = 0.70, size = 401, normalized size = 1.84 \begin {gather*} \frac {\frac {79\,e^5\,{\left (d+e\,x\right )}^{3/2}}{192\,\left (a\,e-b\,d\right )}-\frac {7\,e^5\,\sqrt {d+e\,x}}{128\,b}+\frac {49\,b^2\,e^5\,{\left (d+e\,x\right )}^{7/2}}{192\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,{\left (a\,e-b\,d\right )}^4}+\frac {7\,b\,e^5\,{\left (d+e\,x\right )}^{5/2}}{15\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {7\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((79*e^5*(d + e*x)^(3/2))/(192*(a*e - b*d)) - (7*e^5*(d + e*x)^(1/2))/(128*b) + (49*b^2*e^5*(d + e*x)^(7/2))/(
192*(a*e - b*d)^3) + (7*b^3*e^5*(d + e*x)^(9/2))/(128*(a*e - b*d)^4) + (7*b*e^5*(d + e*x)^(5/2))/(15*(a*e - b*
d)^2))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) - (d + e*
x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*
e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3
*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (7*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)
^(1/2)))/(128*b^(3/2)*(a*e - b*d)^(9/2))

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