3.17.73 \(\int \frac {1}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [1673]

Optimal. Leaf size=239 \[ -\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}} \]

[Out]

693/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+b*d)^(13/2)-693/128*e^5/(-a*e+b*d)^6
/(e*x+d)^(1/2)-1/5/(-a*e+b*d)/(b*x+a)^5/(e*x+d)^(1/2)+11/40*e/(-a*e+b*d)^2/(b*x+a)^4/(e*x+d)^(1/2)-33/80*e^2/(
-a*e+b*d)^3/(b*x+a)^3/(e*x+d)^(1/2)+231/320*e^3/(-a*e+b*d)^4/(b*x+a)^2/(e*x+d)^(1/2)-231/128*e^4/(-a*e+b*d)^5/
(b*x+a)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \begin {gather*} -\frac {693 e^5}{128 \sqrt {d+e x} (b d-a e)^6}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}}-\frac {231 e^4}{128 (a+b x) \sqrt {d+e x} (b d-a e)^5}+\frac {231 e^3}{320 (a+b x)^2 \sqrt {d+e x} (b d-a e)^4}-\frac {33 e^2}{80 (a+b x)^3 \sqrt {d+e x} (b d-a e)^3}+\frac {11 e}{40 (a+b x)^4 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-693*e^5)/(128*(b*d - a*e)^6*Sqrt[d + e*x]) - 1/(5*(b*d - a*e)*(a + b*x)^5*Sqrt[d + e*x]) + (11*e)/(40*(b*d -
a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) - (33*e^2)/(80*(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) + (231*e^3)/(320*(b
*d - a*e)^4*(a + b*x)^2*Sqrt[d + e*x]) - (231*e^4)/(128*(b*d - a*e)^5*(a + b*x)*Sqrt[d + e*x]) + (693*Sqrt[b]*
e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(13/2))

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 53

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 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}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {(11 e) \int \frac {1}{(a+b x)^5 (d+e x)^{3/2}} \, dx}{10 (b d-a e)}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {\left (99 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{80 (b d-a e)^2}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {\left (231 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{160 (b d-a e)^3}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {\left (231 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 b e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^6}\\ &=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 b 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)^6}\\ &=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 1.54, size = 284, normalized size = 1.19 \begin {gather*} \frac {1}{640} \left (\frac {-1280 a^5 e^5-5 a^4 b e^4 (843 d+2123 e x)-10 a^3 b^2 e^3 \left (-359 d^2+968 d e x+2607 e^2 x^2\right )-2 a^2 b^3 e^2 \left (1124 d^3-2013 d^2 e x+5247 d e^2 x^2+14784 e^3 x^3\right )-2 a b^4 e \left (-408 d^4+616 d^3 e x-1089 d^2 e^2 x^2+2772 d e^3 x^3+8085 e^4 x^4\right )-b^5 \left (128 d^5-176 d^4 e x+264 d^3 e^2 x^2-462 d^2 e^3 x^3+1155 d e^4 x^4+3465 e^5 x^5\right )}{(b d-a e)^6 (a+b x)^5 \sqrt {d+e x}}-\frac {3465 \sqrt {b} e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1280*a^5*e^5 - 5*a^4*b*e^4*(843*d + 2123*e*x) - 10*a^3*b^2*e^3*(-359*d^2 + 968*d*e*x + 2607*e^2*x^2) - 2*a^
2*b^3*e^2*(1124*d^3 - 2013*d^2*e*x + 5247*d*e^2*x^2 + 14784*e^3*x^3) - 2*a*b^4*e*(-408*d^4 + 616*d^3*e*x - 108
9*d^2*e^2*x^2 + 2772*d*e^3*x^3 + 8085*e^4*x^4) - b^5*(128*d^5 - 176*d^4*e*x + 264*d^3*e^2*x^2 - 462*d^2*e^3*x^
3 + 1155*d*e^4*x^4 + 3465*e^5*x^5))/((b*d - a*e)^6*(a + b*x)^5*Sqrt[d + e*x]) - (3465*Sqrt[b]*e^5*ArcTan[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2))/640

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Maple [A]
time = 0.67, size = 270, normalized size = 1.13

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {b \left (\frac {\frac {437 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {977 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {131}{10} a^{2} b^{2} e^{2}-\frac {131}{5} a \,b^{3} d e +\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} b \,e^{3}-\frac {3981}{128} a^{2} b^{2} d \,e^{2}+\frac {3981}{128} d^{2} e a \,b^{3}-\frac {1327}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {843}{256} e^{4} a^{4}-\frac {843}{64} a^{3} b d \,e^{3}+\frac {2529}{128} a^{2} b^{2} d^{2} e^{2}-\frac {843}{64} a \,b^{3} d^{3} e +\frac {843}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{6}}-\frac {1}{\left (a e -b d \right )^{6} \sqrt {e x +d}}\right )\)	\(270\)
default	\(2 e^{5} \left (-\frac {b \left (\frac {\frac {437 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {977 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {131}{10} a^{2} b^{2} e^{2}-\frac {131}{5} a \,b^{3} d e +\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} b \,e^{3}-\frac {3981}{128} a^{2} b^{2} d \,e^{2}+\frac {3981}{128} d^{2} e a \,b^{3}-\frac {1327}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {843}{256} e^{4} a^{4}-\frac {843}{64} a^{3} b d \,e^{3}+\frac {2529}{128} a^{2} b^{2} d^{2} e^{2}-\frac {843}{64} a \,b^{3} d^{3} e +\frac {843}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{6}}-\frac {1}{\left (a e -b d \right )^{6} \sqrt {e x +d}}\right )\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^5*(-1/(a*e-b*d)^6*b*((437/256*b^4*(e*x+d)^(9/2)+977/128*(a*e-b*d)*b^3*(e*x+d)^(7/2)+(131/10*a^2*b^2*e^2-13
1/5*a*b^3*d*e+131/10*b^4*d^2)*(e*x+d)^(5/2)+(1327/128*a^3*b*e^3-3981/128*a^2*b^2*d*e^2+3981/128*d^2*e*a*b^3-13
27/128*b^4*d^3)*(e*x+d)^(3/2)+(843/256*e^4*a^4-843/64*a^3*b*d*e^3+2529/128*a^2*b^2*d^2*e^2-843/64*a*b^3*d^3*e+
843/256*b^4*d^4)*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b*d)^5+693/256/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a
*e-b*d))^(1/2)))-1/(a*e-b*d)^6/(e*x+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)^(3/2)/(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. 1114 vs. \(2 (214) = 428\).
time = 2.77, size = 2239, normalized size = 9.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1280*(3465*((b^5*x^6 + 5*a*b^4*x^5 + 10*a^2*b^3*x^4 + 10*a^3*b^2*x^3 + 5*a^4*b*x^2 + a^5*x)*e^6 + (b^5*d*x^
5 + 5*a*b^4*d*x^4 + 10*a^2*b^3*d*x^3 + 10*a^3*b^2*d*x^2 + 5*a^4*b*d*x + a^5*d)*e^5)*sqrt(b/(b*d - a*e))*log((2
*b*d + 2*(b*d - a*e)*sqrt(x*e + d)*sqrt(b/(b*d - a*e)) + (b*x - a)*e)/(b*x + a)) - 2*(128*b^5*d^5 + (3465*b^5*
x^5 + 16170*a*b^4*x^4 + 29568*a^2*b^3*x^3 + 26070*a^3*b^2*x^2 + 10615*a^4*b*x + 1280*a^5)*e^5 + (1155*b^5*d*x^
4 + 5544*a*b^4*d*x^3 + 10494*a^2*b^3*d*x^2 + 9680*a^3*b^2*d*x + 4215*a^4*b*d)*e^4 - 2*(231*b^5*d^2*x^3 + 1089*
a*b^4*d^2*x^2 + 2013*a^2*b^3*d^2*x + 1795*a^3*b^2*d^2)*e^3 + 8*(33*b^5*d^3*x^2 + 154*a*b^4*d^3*x + 281*a^2*b^3
*d^3)*e^2 - 16*(11*b^5*d^4*x + 51*a*b^4*d^4)*e)*sqrt(x*e + d))/(b^11*d^7*x^5 + 5*a*b^10*d^7*x^4 + 10*a^2*b^9*d
^7*x^3 + 10*a^3*b^8*d^7*x^2 + 5*a^4*b^7*d^7*x + a^5*b^6*d^7 + (a^6*b^5*x^6 + 5*a^7*b^4*x^5 + 10*a^8*b^3*x^4 +
10*a^9*b^2*x^3 + 5*a^10*b*x^2 + a^11*x)*e^7 - (6*a^5*b^6*d*x^6 + 29*a^6*b^5*d*x^5 + 55*a^7*b^4*d*x^4 + 50*a^8*
b^3*d*x^3 + 20*a^9*b^2*d*x^2 + a^10*b*d*x - a^11*d)*e^6 + 3*(5*a^4*b^7*d^2*x^6 + 23*a^5*b^6*d^2*x^5 + 40*a^6*b
^5*d^2*x^4 + 30*a^7*b^4*d^2*x^3 + 5*a^8*b^3*d^2*x^2 - 5*a^9*b^2*d^2*x - 2*a^10*b*d^2)*e^5 - 5*(4*a^3*b^8*d^3*x
^6 + 17*a^4*b^7*d^3*x^5 + 25*a^5*b^6*d^3*x^4 + 10*a^6*b^5*d^3*x^3 - 10*a^7*b^4*d^3*x^2 - 11*a^8*b^3*d^3*x - 3*
a^9*b^2*d^3)*e^4 + 5*(3*a^2*b^9*d^4*x^6 + 11*a^3*b^8*d^4*x^5 + 10*a^4*b^7*d^4*x^4 - 10*a^5*b^6*d^4*x^3 - 25*a^
6*b^5*d^4*x^2 - 17*a^7*b^4*d^4*x - 4*a^8*b^3*d^4)*e^3 - 3*(2*a*b^10*d^5*x^6 + 5*a^2*b^9*d^5*x^5 - 5*a^3*b^8*d^
5*x^4 - 30*a^4*b^7*d^5*x^3 - 40*a^5*b^6*d^5*x^2 - 23*a^6*b^5*d^5*x - 5*a^7*b^4*d^5)*e^2 + (b^11*d^6*x^6 - a*b^
10*d^6*x^5 - 20*a^2*b^9*d^6*x^4 - 50*a^3*b^8*d^6*x^3 - 55*a^4*b^7*d^6*x^2 - 29*a^5*b^6*d^6*x - 6*a^6*b^5*d^6)*
e), 1/640*(3465*((b^5*x^6 + 5*a*b^4*x^5 + 10*a^2*b^3*x^4 + 10*a^3*b^2*x^3 + 5*a^4*b*x^2 + a^5*x)*e^6 + (b^5*d*
x^5 + 5*a*b^4*d*x^4 + 10*a^2*b^3*d*x^3 + 10*a^3*b^2*d*x^2 + 5*a^4*b*d*x + a^5*d)*e^5)*sqrt(-b/(b*d - a*e))*arc
tan(-(b*d - a*e)*sqrt(x*e + d)*sqrt(-b/(b*d - a*e))/(b*x*e + b*d)) - (128*b^5*d^5 + (3465*b^5*x^5 + 16170*a*b^
4*x^4 + 29568*a^2*b^3*x^3 + 26070*a^3*b^2*x^2 + 10615*a^4*b*x + 1280*a^5)*e^5 + (1155*b^5*d*x^4 + 5544*a*b^4*d
*x^3 + 10494*a^2*b^3*d*x^2 + 9680*a^3*b^2*d*x + 4215*a^4*b*d)*e^4 - 2*(231*b^5*d^2*x^3 + 1089*a*b^4*d^2*x^2 +
2013*a^2*b^3*d^2*x + 1795*a^3*b^2*d^2)*e^3 + 8*(33*b^5*d^3*x^2 + 154*a*b^4*d^3*x + 281*a^2*b^3*d^3)*e^2 - 16*(
11*b^5*d^4*x + 51*a*b^4*d^4)*e)*sqrt(x*e + d))/(b^11*d^7*x^5 + 5*a*b^10*d^7*x^4 + 10*a^2*b^9*d^7*x^3 + 10*a^3*
b^8*d^7*x^2 + 5*a^4*b^7*d^7*x + a^5*b^6*d^7 + (a^6*b^5*x^6 + 5*a^7*b^4*x^5 + 10*a^8*b^3*x^4 + 10*a^9*b^2*x^3 +
5*a^10*b*x^2 + a^11*x)*e^7 - (6*a^5*b^6*d*x^6 + 29*a^6*b^5*d*x^5 + 55*a^7*b^4*d*x^4 + 50*a^8*b^3*d*x^3 + 20*a
^9*b^2*d*x^2 + a^10*b*d*x - a^11*d)*e^6 + 3*(5*a^4*b^7*d^2*x^6 + 23*a^5*b^6*d^2*x^5 + 40*a^6*b^5*d^2*x^4 + 30*
a^7*b^4*d^2*x^3 + 5*a^8*b^3*d^2*x^2 - 5*a^9*b^2*d^2*x - 2*a^10*b*d^2)*e^5 - 5*(4*a^3*b^8*d^3*x^6 + 17*a^4*b^7*
d^3*x^5 + 25*a^5*b^6*d^3*x^4 + 10*a^6*b^5*d^3*x^3 - 10*a^7*b^4*d^3*x^2 - 11*a^8*b^3*d^3*x - 3*a^9*b^2*d^3)*e^4
+ 5*(3*a^2*b^9*d^4*x^6 + 11*a^3*b^8*d^4*x^5 + 10*a^4*b^7*d^4*x^4 - 10*a^5*b^6*d^4*x^3 - 25*a^6*b^5*d^4*x^2 -
17*a^7*b^4*d^4*x - 4*a^8*b^3*d^4)*e^3 - 3*(2*a*b^10*d^5*x^6 + 5*a^2*b^9*d^5*x^5 - 5*a^3*b^8*d^5*x^4 - 30*a^4*b
^7*d^5*x^3 - 40*a^5*b^6*d^5*x^2 - 23*a^6*b^5*d^5*x - 5*a^7*b^4*d^5)*e^2 + (b^11*d^6*x^6 - a*b^10*d^6*x^5 - 20*
a^2*b^9*d^6*x^4 - 50*a^3*b^8*d^6*x^3 - 55*a^4*b^7*d^6*x^2 - 29*a^5*b^6*d^6*x - 6*a^6*b^5*d^6)*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)**(3/2)/(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. 571 vs. \(2 (214) = 428\).
time = 2.19, size = 571, normalized size = 2.39 \begin {gather*} -\frac {693 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, e^{5}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {x e + d}} - \frac {2185 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{5} e^{5} - 9770 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d e^{5} + 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{5} - 13270 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{5} + 4215 \, \sqrt {x e + d} b^{5} d^{4} e^{5} + 9770 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} e^{6} - 33536 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d e^{6} + 39810 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{6} - 16860 \, \sqrt {x e + d} a b^{4} d^{3} e^{6} + 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{7} - 39810 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{7} + 25290 \, \sqrt {x e + d} a^{2} b^{3} d^{2} e^{7} + 13270 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{8} - 16860 \, \sqrt {x e + d} a^{3} b^{2} d e^{8} + 4215 \, \sqrt {x e + d} a^{4} b e^{9}}{640 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-693/128*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 2
0*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(-b^2*d + a*b*e)) - 2*e^5/((b^6*d^6 - 6*
a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(x*e
+ d)) - 1/640*(2185*(x*e + d)^(9/2)*b^5*e^5 - 9770*(x*e + d)^(7/2)*b^5*d*e^5 + 16768*(x*e + d)^(5/2)*b^5*d^2*
e^5 - 13270*(x*e + d)^(3/2)*b^5*d^3*e^5 + 4215*sqrt(x*e + d)*b^5*d^4*e^5 + 9770*(x*e + d)^(7/2)*a*b^4*e^6 - 33
536*(x*e + d)^(5/2)*a*b^4*d*e^6 + 39810*(x*e + d)^(3/2)*a*b^4*d^2*e^6 - 16860*sqrt(x*e + d)*a*b^4*d^3*e^6 + 16
768*(x*e + d)^(5/2)*a^2*b^3*e^7 - 39810*(x*e + d)^(3/2)*a^2*b^3*d*e^7 + 25290*sqrt(x*e + d)*a^2*b^3*d^2*e^7 +
13270*(x*e + d)^(3/2)*a^3*b^2*e^8 - 16860*sqrt(x*e + d)*a^3*b^2*d*e^8 + 4215*sqrt(x*e + d)*a^4*b*e^9)/((b^6*d^
6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*((
x*e + d)*b - b*d + a*e)^5)

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Mupad [B]
time = 1.10, size = 515, normalized size = 2.15 \begin {gather*} -\frac {\frac {2\,e^5}{a\,e-b\,d}+\frac {2607\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {231\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{5\,{\left (a\,e-b\,d\right )}^4}+\frac {1617\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {693\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{128\,{\left (a\,e-b\,d\right )}^6}+\frac {2123\,b\,e^5\,\left (d+e\,x\right )}{128\,{\left (a\,e-b\,d\right )}^2}}{\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{5/2}\,\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 )+{\left (d+e\,x\right )}^{3/2}\,\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 )+b^5\,{\left (d+e\,x\right )}^{11/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {693\,\sqrt {b}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*e^5)/(a*e - b*d) + (2607*b^2*e^5*(d + e*x)^2)/(64*(a*e - b*d)^3) + (231*b^3*e^5*(d + e*x)^3)/(5*(a*e - b
*d)^4) + (1617*b^4*e^5*(d + e*x)^4)/(64*(a*e - b*d)^5) + (693*b^5*e^5*(d + e*x)^5)/(128*(a*e - b*d)^6) + (2123
*b*e^5*(d + e*x))/(128*(a*e - b*d)^2))/((d + e*x)^(1/2)*(a^5*e^5 - b^5*d^5 - 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) - (d + e*x)^(5/2)*(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) + (d + e*x)^(3/2)*(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) + b^5*(d + e*x)^(11/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(9/2) + (d + e*x)^(7/2)*(10*b^5*d^2 + 10*a^2*b^3
*e^2 - 20*a*b^4*d*e)) - (693*b^(1/2)*e^5*atan((b^(1/2)*(d + e*x)^(1/2)*(a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2
- 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a*b^5*d^5*e - 6*a^5*b*d*e^5))/(a*e - b*d)^(13/2)))/(128*(a*e -
b*d)^(13/2))

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