3.14.77 \(\int \frac {1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=295 \[ \frac {9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac {9009 b^2 e^5}{128 \sqrt {d+e x} (b d-a e)^8}-\frac {3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac {9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac {1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac {143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac {13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac {3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

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Rubi [A]  time = 0.27, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \begin {gather*} -\frac {9009 b^2 e^5}{128 \sqrt {d+e x} (b d-a e)^8}+\frac {9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac {3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac {9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac {1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac {143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac {13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac {3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-9009*e^5)/(640*(b*d - a*e)^6*(d + e*x)^(5/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) + (3*e)/(8*(b*
d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e^2)/(16*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) + (143*e^3)/
(64*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) - (1287*e^4)/(128*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) - (3
003*b*e^5)/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) - (9009*b^2*e^5)/(128*(b*d - a*e)^8*Sqrt[d + e*x]) + (9009*b^(5
/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

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}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^{7/2}} \, dx\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {(3 e) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 (b d-a e)}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {\left (39 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {\left (143 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{32 (b d-a e)^3}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {\left (1287 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 (b d-a e)^4}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {\left (9009 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {\left (9009 b e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {\left (9009 b^2 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {\left (9009 b^3 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^8}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {\left (9009 b^3 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)^8}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 52, normalized size = 0.18 \begin {gather*} -\frac {2 e^5 \, _2F_1\left (-\frac {5}{2},6;-\frac {3}{2};-\frac {b (d+e x)}{a e-b d}\right )}{5 (d+e x)^{5/2} (a e-b d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-2*e^5*Hypergeometric2F1[-5/2, 6, -3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^6*(d + e*x)^(5/2)
)

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IntegrateAlgebraic [B]  time = 2.63, size = 683, normalized size = 2.32 \begin {gather*} \frac {e^5 \left (256 a^7 e^7-1280 a^6 b e^6 (d+e x)-1792 a^6 b d e^6+5376 a^5 b^2 d^2 e^5+16640 a^5 b^2 e^5 (d+e x)^2+7680 a^5 b^2 d e^5 (d+e x)-8960 a^4 b^3 d^3 e^4-19200 a^4 b^3 d^2 e^4 (d+e x)+137995 a^4 b^3 e^4 (d+e x)^3-83200 a^4 b^3 d e^4 (d+e x)^2+8960 a^3 b^4 d^4 e^3+25600 a^3 b^4 d^3 e^3 (d+e x)+166400 a^3 b^4 d^2 e^3 (d+e x)^2+338910 a^3 b^4 e^3 (d+e x)^4-551980 a^3 b^4 d e^3 (d+e x)^3-5376 a^2 b^5 d^5 e^2-19200 a^2 b^5 d^4 e^2 (d+e x)-166400 a^2 b^5 d^3 e^2 (d+e x)^2+827970 a^2 b^5 d^2 e^2 (d+e x)^3+384384 a^2 b^5 e^2 (d+e x)^5-1016730 a^2 b^5 d e^2 (d+e x)^4+1792 a b^6 d^6 e+7680 a b^6 d^5 e (d+e x)+83200 a b^6 d^4 e (d+e x)^2-551980 a b^6 d^3 e (d+e x)^3+1016730 a b^6 d^2 e (d+e x)^4+210210 a b^6 e (d+e x)^6-768768 a b^6 d e (d+e x)^5-256 b^7 d^7-1280 b^7 d^6 (d+e x)-16640 b^7 d^5 (d+e x)^2+137995 b^7 d^4 (d+e x)^3-338910 b^7 d^3 (d+e x)^4+384384 b^7 d^2 (d+e x)^5+45045 b^7 (d+e x)^7-210210 b^7 d (d+e x)^6\right )}{640 (d+e x)^{5/2} (b d-a e)^8 (-a e-b (d+e x)+b d)^5}+\frac {9009 b^{5/2} e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 (b d-a e)^8 \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(e^5*(-256*b^7*d^7 + 1792*a*b^6*d^6*e - 5376*a^2*b^5*d^5*e^2 + 8960*a^3*b^4*d^4*e^3 - 8960*a^4*b^3*d^3*e^4 + 5
376*a^5*b^2*d^2*e^5 - 1792*a^6*b*d*e^6 + 256*a^7*e^7 - 1280*b^7*d^6*(d + e*x) + 7680*a*b^6*d^5*e*(d + e*x) - 1
9200*a^2*b^5*d^4*e^2*(d + e*x) + 25600*a^3*b^4*d^3*e^3*(d + e*x) - 19200*a^4*b^3*d^2*e^4*(d + e*x) + 7680*a^5*
b^2*d*e^5*(d + e*x) - 1280*a^6*b*e^6*(d + e*x) - 16640*b^7*d^5*(d + e*x)^2 + 83200*a*b^6*d^4*e*(d + e*x)^2 - 1
66400*a^2*b^5*d^3*e^2*(d + e*x)^2 + 166400*a^3*b^4*d^2*e^3*(d + e*x)^2 - 83200*a^4*b^3*d*e^4*(d + e*x)^2 + 166
40*a^5*b^2*e^5*(d + e*x)^2 + 137995*b^7*d^4*(d + e*x)^3 - 551980*a*b^6*d^3*e*(d + e*x)^3 + 827970*a^2*b^5*d^2*
e^2*(d + e*x)^3 - 551980*a^3*b^4*d*e^3*(d + e*x)^3 + 137995*a^4*b^3*e^4*(d + e*x)^3 - 338910*b^7*d^3*(d + e*x)
^4 + 1016730*a*b^6*d^2*e*(d + e*x)^4 - 1016730*a^2*b^5*d*e^2*(d + e*x)^4 + 338910*a^3*b^4*e^3*(d + e*x)^4 + 38
4384*b^7*d^2*(d + e*x)^5 - 768768*a*b^6*d*e*(d + e*x)^5 + 384384*a^2*b^5*e^2*(d + e*x)^5 - 210210*b^7*d*(d + e
*x)^6 + 210210*a*b^6*e*(d + e*x)^6 + 45045*b^7*(d + e*x)^7))/(640*(b*d - a*e)^8*(d + e*x)^(5/2)*(b*d - a*e - b
*(d + e*x))^5) + (9009*b^(5/2)*e^5*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*(b*d -
a*e)^8*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.54, size = 4232, normalized size = 14.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(45045*(b^7*e^8*x^8 + a^5*b^2*d^3*e^5 + (3*b^7*d*e^7 + 5*a*b^6*e^8)*x^7 + (3*b^7*d^2*e^6 + 15*a*b^6*d*
e^7 + 10*a^2*b^5*e^8)*x^6 + (b^7*d^3*e^5 + 15*a*b^6*d^2*e^6 + 30*a^2*b^5*d*e^7 + 10*a^3*b^4*e^8)*x^5 + 5*(a*b^
6*d^3*e^5 + 6*a^2*b^5*d^2*e^6 + 6*a^3*b^4*d*e^7 + a^4*b^3*e^8)*x^4 + (10*a^2*b^5*d^3*e^5 + 30*a^3*b^4*d^2*e^6
+ 15*a^4*b^3*d*e^7 + a^5*b^2*e^8)*x^3 + (10*a^3*b^4*d^3*e^5 + 15*a^4*b^3*d^2*e^6 + 3*a^5*b^2*d*e^7)*x^2 + (5*a
^4*b^3*d^3*e^5 + 3*a^5*b^2*d^2*e^6)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x +
d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(45045*b^7*e^7*x^7 + 128*b^7*d^7 - 1136*a*b^6*d^6*e + 4648*a^2*b^5*d^5
*e^2 - 12110*a^3*b^4*d^4*e^3 + 26635*a^4*b^3*d^3*e^4 + 29696*a^5*b^2*d^2*e^5 - 3072*a^6*b*d*e^6 + 256*a^7*e^7
+ 105105*(b^7*d*e^6 + 2*a*b^6*e^7)*x^6 + 3003*(23*b^7*d^2*e^5 + 164*a*b^6*d*e^6 + 128*a^2*b^5*e^7)*x^5 + 2145*
(3*b^7*d^3*e^4 + 152*a*b^6*d^2*e^5 + 422*a^2*b^5*d*e^6 + 158*a^3*b^4*e^7)*x^4 - 715*(2*b^7*d^4*e^3 - 44*a*b^6*
d^3*e^4 - 846*a^2*b^5*d^2*e^5 - 1124*a^3*b^4*d*e^6 - 193*a^4*b^3*e^7)*x^3 + 65*(8*b^7*d^5*e^2 - 106*a*b^6*d^4*
e^3 + 938*a^2*b^5*d^3*e^4 + 8368*a^3*b^4*d^2*e^5 + 5089*a^4*b^3*d*e^6 + 256*a^5*b^2*e^7)*x^2 - 5*(48*b^7*d^6*e
- 496*a*b^6*d^5*e^2 + 2618*a^2*b^5*d^4*e^3 - 11620*a^3*b^4*d^3*e^4 - 45677*a^4*b^3*d^2*e^5 - 8192*a^5*b^2*d*e
^6 + 256*a^6*b*e^7)*x)*sqrt(e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a^7*b^6*d^9*e^2 - 56*a^8*b^5*d^8*e
^3 + 70*a^9*b^4*d^7*e^4 - 56*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*b*d^4*e^7 + a^13*d^3*e^8 + (b^13*
d^8*e^3 - 8*a*b^12*d^7*e^4 + 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a^4*b^9*d^4*e^7 - 56*a^5*b^8*d^3*e
^8 + 28*a^6*b^7*d^2*e^9 - 8*a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*e^2 - 19*a*b^12*d^8*e^3 + 44*a^2*
b^11*d^7*e^4 - 28*a^3*b^10*d^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7 - 196*a^6*b^7*d^3*e^8 + 116*a^7*
b^6*d^2*e^9 - 37*a^8*b^5*d*e^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*b^12*d^9*e^2 - 26*a^2*b^11*d^8*e^
3 + 172*a^3*b^10*d^7*e^4 - 350*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6*b^7*d^4*e^7 - 164*a^7*b^6*d^3*e^
8 + 163*a^8*b^5*d^2*e^9 - 65*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d^11 + 7*a*b^12*d^10*e - 62*a^2*b^
11*d^9*e^2 + 134*a^3*b^10*d^8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5 + 728*a^6*b^7*d^5*e^6 - 568*a^7*b
^6*d^4*e^7 + 161*a^8*b^5*d^3*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 + 10*a^11*b^2*e^11)*x^5 + 5*(a*b^12
*d^11 - 2*a^2*b^11*d^10*e - 14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^5*b^8*d^7*e^4 + 56*a^6*b^7*d^6*e^
5 + 56*a^7*b^6*d^5*e^6 - 106*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b^3*d^2*e^9 - 2*a^11*b^2*d*e^10 +
a^12*b*e^11)*x^4 + (10*a^2*b^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^2 + 161*a^5*b^8*d^8*e^3 - 568*a^6
*b^7*d^7*e^4 + 728*a^7*b^6*d^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7 + 134*a^10*b^3*d^3*e^8 - 62*a^11
*b^2*d^2*e^9 + 7*a^12*b*d*e^10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*b^9*d^10*e + 163*a^5*b^8*d^9*e^2
- 164*a^6*b^7*d^8*e^3 - 56*a^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^4*d^5*e^6 + 172*a^10*b^3*d^4*e^7
- 26*a^11*b^2*d^3*e^8 - 9*a^12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^11 - 37*a^5*b^8*d^10*e + 116*a^6*
b^7*d^9*e^2 - 196*a^7*b^6*d^8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5 - 28*a^10*b^3*d^5*e^6 + 44*a^11*b
^2*d^4*e^7 - 19*a^12*b*d^3*e^8 + 3*a^13*d^2*e^9)*x), 1/640*(45045*(b^7*e^8*x^8 + a^5*b^2*d^3*e^5 + (3*b^7*d*e^
7 + 5*a*b^6*e^8)*x^7 + (3*b^7*d^2*e^6 + 15*a*b^6*d*e^7 + 10*a^2*b^5*e^8)*x^6 + (b^7*d^3*e^5 + 15*a*b^6*d^2*e^6
+ 30*a^2*b^5*d*e^7 + 10*a^3*b^4*e^8)*x^5 + 5*(a*b^6*d^3*e^5 + 6*a^2*b^5*d^2*e^6 + 6*a^3*b^4*d*e^7 + a^4*b^3*e
^8)*x^4 + (10*a^2*b^5*d^3*e^5 + 30*a^3*b^4*d^2*e^6 + 15*a^4*b^3*d*e^7 + a^5*b^2*e^8)*x^3 + (10*a^3*b^4*d^3*e^5
+ 15*a^4*b^3*d^2*e^6 + 3*a^5*b^2*d*e^7)*x^2 + (5*a^4*b^3*d^3*e^5 + 3*a^5*b^2*d^2*e^6)*x)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (45045*b^7*e^7*x^7 + 128*b^7*d^7 - 11
36*a*b^6*d^6*e + 4648*a^2*b^5*d^5*e^2 - 12110*a^3*b^4*d^4*e^3 + 26635*a^4*b^3*d^3*e^4 + 29696*a^5*b^2*d^2*e^5
- 3072*a^6*b*d*e^6 + 256*a^7*e^7 + 105105*(b^7*d*e^6 + 2*a*b^6*e^7)*x^6 + 3003*(23*b^7*d^2*e^5 + 164*a*b^6*d*e
^6 + 128*a^2*b^5*e^7)*x^5 + 2145*(3*b^7*d^3*e^4 + 152*a*b^6*d^2*e^5 + 422*a^2*b^5*d*e^6 + 158*a^3*b^4*e^7)*x^4
- 715*(2*b^7*d^4*e^3 - 44*a*b^6*d^3*e^4 - 846*a^2*b^5*d^2*e^5 - 1124*a^3*b^4*d*e^6 - 193*a^4*b^3*e^7)*x^3 + 6
5*(8*b^7*d^5*e^2 - 106*a*b^6*d^4*e^3 + 938*a^2*b^5*d^3*e^4 + 8368*a^3*b^4*d^2*e^5 + 5089*a^4*b^3*d*e^6 + 256*a
^5*b^2*e^7)*x^2 - 5*(48*b^7*d^6*e - 496*a*b^6*d^5*e^2 + 2618*a^2*b^5*d^4*e^3 - 11620*a^3*b^4*d^3*e^4 - 45677*a
^4*b^3*d^2*e^5 - 8192*a^5*b^2*d*e^6 + 256*a^6*b*e^7)*x)*sqrt(e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a
^7*b^6*d^9*e^2 - 56*a^8*b^5*d^8*e^3 + 70*a^9*b^4*d^7*e^4 - 56*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*
b*d^4*e^7 + a^13*d^3*e^8 + (b^13*d^8*e^3 - 8*a*b^12*d^7*e^4 + 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a
^4*b^9*d^4*e^7 - 56*a^5*b^8*d^3*e^8 + 28*a^6*b^7*d^2*e^9 - 8*a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*
e^2 - 19*a*b^12*d^8*e^3 + 44*a^2*b^11*d^7*e^4 - 28*a^3*b^10*d^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7
- 196*a^6*b^7*d^3*e^8 + 116*a^7*b^6*d^2*e^9 - 37*a^8*b^5*d*e^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*
b^12*d^9*e^2 - 26*a^2*b^11*d^8*e^3 + 172*a^3*b^10*d^7*e^4 - 350*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6
*b^7*d^4*e^7 - 164*a^7*b^6*d^3*e^8 + 163*a^8*b^5*d^2*e^9 - 65*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d
^11 + 7*a*b^12*d^10*e - 62*a^2*b^11*d^9*e^2 + 134*a^3*b^10*d^8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5
+ 728*a^6*b^7*d^5*e^6 - 568*a^7*b^6*d^4*e^7 + 161*a^8*b^5*d^3*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 +
10*a^11*b^2*e^11)*x^5 + 5*(a*b^12*d^11 - 2*a^2*b^11*d^10*e - 14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^
5*b^8*d^7*e^4 + 56*a^6*b^7*d^6*e^5 + 56*a^7*b^6*d^5*e^6 - 106*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b
^3*d^2*e^9 - 2*a^11*b^2*d*e^10 + a^12*b*e^11)*x^4 + (10*a^2*b^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^
2 + 161*a^5*b^8*d^8*e^3 - 568*a^6*b^7*d^7*e^4 + 728*a^7*b^6*d^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7
+ 134*a^10*b^3*d^3*e^8 - 62*a^11*b^2*d^2*e^9 + 7*a^12*b*d*e^10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*
b^9*d^10*e + 163*a^5*b^8*d^9*e^2 - 164*a^6*b^7*d^8*e^3 - 56*a^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^
4*d^5*e^6 + 172*a^10*b^3*d^4*e^7 - 26*a^11*b^2*d^3*e^8 - 9*a^12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^
11 - 37*a^5*b^8*d^10*e + 116*a^6*b^7*d^9*e^2 - 196*a^7*b^6*d^8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5
- 28*a^10*b^3*d^5*e^6 + 44*a^11*b^2*d^4*e^7 - 19*a^12*b*d^3*e^8 + 3*a^13*d^2*e^9)*x)]

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giac [B]  time = 0.29, size = 884, normalized size = 3.00 \begin {gather*} -\frac {9009 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{8} d^{8} - 8 \, a b^{7} d^{7} e + 28 \, a^{2} b^{6} d^{6} e^{2} - 56 \, a^{3} b^{5} d^{5} e^{3} + 70 \, a^{4} b^{4} d^{4} e^{4} - 56 \, a^{5} b^{3} d^{3} e^{5} + 28 \, a^{6} b^{2} d^{2} e^{6} - 8 \, a^{7} b d e^{7} + a^{8} e^{8}\right )} \sqrt {-b^{2} d + a b e}} - \frac {45045 \, {\left (x e + d\right )}^{7} b^{7} e^{5} - 210210 \, {\left (x e + d\right )}^{6} b^{7} d e^{5} + 384384 \, {\left (x e + d\right )}^{5} b^{7} d^{2} e^{5} - 338910 \, {\left (x e + d\right )}^{4} b^{7} d^{3} e^{5} + 137995 \, {\left (x e + d\right )}^{3} b^{7} d^{4} e^{5} - 16640 \, {\left (x e + d\right )}^{2} b^{7} d^{5} e^{5} - 1280 \, {\left (x e + d\right )} b^{7} d^{6} e^{5} - 256 \, b^{7} d^{7} e^{5} + 210210 \, {\left (x e + d\right )}^{6} a b^{6} e^{6} - 768768 \, {\left (x e + d\right )}^{5} a b^{6} d e^{6} + 1016730 \, {\left (x e + d\right )}^{4} a b^{6} d^{2} e^{6} - 551980 \, {\left (x e + d\right )}^{3} a b^{6} d^{3} e^{6} + 83200 \, {\left (x e + d\right )}^{2} a b^{6} d^{4} e^{6} + 7680 \, {\left (x e + d\right )} a b^{6} d^{5} e^{6} + 1792 \, a b^{6} d^{6} e^{6} + 384384 \, {\left (x e + d\right )}^{5} a^{2} b^{5} e^{7} - 1016730 \, {\left (x e + d\right )}^{4} a^{2} b^{5} d e^{7} + 827970 \, {\left (x e + d\right )}^{3} a^{2} b^{5} d^{2} e^{7} - 166400 \, {\left (x e + d\right )}^{2} a^{2} b^{5} d^{3} e^{7} - 19200 \, {\left (x e + d\right )} a^{2} b^{5} d^{4} e^{7} - 5376 \, a^{2} b^{5} d^{5} e^{7} + 338910 \, {\left (x e + d\right )}^{4} a^{3} b^{4} e^{8} - 551980 \, {\left (x e + d\right )}^{3} a^{3} b^{4} d e^{8} + 166400 \, {\left (x e + d\right )}^{2} a^{3} b^{4} d^{2} e^{8} + 25600 \, {\left (x e + d\right )} a^{3} b^{4} d^{3} e^{8} + 8960 \, a^{3} b^{4} d^{4} e^{8} + 137995 \, {\left (x e + d\right )}^{3} a^{4} b^{3} e^{9} - 83200 \, {\left (x e + d\right )}^{2} a^{4} b^{3} d e^{9} - 19200 \, {\left (x e + d\right )} a^{4} b^{3} d^{2} e^{9} - 8960 \, a^{4} b^{3} d^{3} e^{9} + 16640 \, {\left (x e + d\right )}^{2} a^{5} b^{2} e^{10} + 7680 \, {\left (x e + d\right )} a^{5} b^{2} d e^{10} + 5376 \, a^{5} b^{2} d^{2} e^{10} - 1280 \, {\left (x e + d\right )} a^{6} b e^{11} - 1792 \, a^{6} b d e^{11} + 256 \, a^{7} e^{12}}{640 \, {\left (b^{8} d^{8} - 8 \, a b^{7} d^{7} e + 28 \, a^{2} b^{6} d^{6} e^{2} - 56 \, a^{3} b^{5} d^{5} e^{3} + 70 \, a^{4} b^{4} d^{4} e^{4} - 56 \, a^{5} b^{3} d^{3} e^{5} + 28 \, a^{6} b^{2} d^{2} e^{6} - 8 \, a^{7} b d e^{7} + a^{8} e^{8}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-9009/128*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2
- 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)
*sqrt(-b^2*d + a*b*e)) - 1/640*(45045*(x*e + d)^7*b^7*e^5 - 210210*(x*e + d)^6*b^7*d*e^5 + 384384*(x*e + d)^5*
b^7*d^2*e^5 - 338910*(x*e + d)^4*b^7*d^3*e^5 + 137995*(x*e + d)^3*b^7*d^4*e^5 - 16640*(x*e + d)^2*b^7*d^5*e^5
- 1280*(x*e + d)*b^7*d^6*e^5 - 256*b^7*d^7*e^5 + 210210*(x*e + d)^6*a*b^6*e^6 - 768768*(x*e + d)^5*a*b^6*d*e^6
+ 1016730*(x*e + d)^4*a*b^6*d^2*e^6 - 551980*(x*e + d)^3*a*b^6*d^3*e^6 + 83200*(x*e + d)^2*a*b^6*d^4*e^6 + 76
80*(x*e + d)*a*b^6*d^5*e^6 + 1792*a*b^6*d^6*e^6 + 384384*(x*e + d)^5*a^2*b^5*e^7 - 1016730*(x*e + d)^4*a^2*b^5
*d*e^7 + 827970*(x*e + d)^3*a^2*b^5*d^2*e^7 - 166400*(x*e + d)^2*a^2*b^5*d^3*e^7 - 19200*(x*e + d)*a^2*b^5*d^4
*e^7 - 5376*a^2*b^5*d^5*e^7 + 338910*(x*e + d)^4*a^3*b^4*e^8 - 551980*(x*e + d)^3*a^3*b^4*d*e^8 + 166400*(x*e
+ d)^2*a^3*b^4*d^2*e^8 + 25600*(x*e + d)*a^3*b^4*d^3*e^8 + 8960*a^3*b^4*d^4*e^8 + 137995*(x*e + d)^3*a^4*b^3*e
^9 - 83200*(x*e + d)^2*a^4*b^3*d*e^9 - 19200*(x*e + d)*a^4*b^3*d^2*e^9 - 8960*a^4*b^3*d^3*e^9 + 16640*(x*e + d
)^2*a^5*b^2*e^10 + 7680*(x*e + d)*a^5*b^2*d*e^10 + 5376*a^5*b^2*d^2*e^10 - 1280*(x*e + d)*a^6*b*e^11 - 1792*a^
6*b*d*e^11 + 256*a^7*e^12)/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^
4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*
b*d + sqrt(x*e + d)*a*e)^5)

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maple [B]  time = 0.07, size = 693, normalized size = 2.35 \begin {gather*} -\frac {5327 \sqrt {e x +d}\, a^{4} b^{3} e^{9}}{128 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}+\frac {5327 \sqrt {e x +d}\, a^{3} b^{4} d \,e^{8}}{32 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {15981 \sqrt {e x +d}\, a^{2} b^{5} d^{2} e^{7}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}+\frac {5327 \sqrt {e x +d}\, a \,b^{6} d^{3} e^{6}}{32 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {5327 \sqrt {e x +d}\, b^{7} d^{4} e^{5}}{128 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {9443 \left (e x +d \right )^{\frac {3}{2}} a^{3} b^{4} e^{8}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}+\frac {28329 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{5} d \,e^{7}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {28329 \left (e x +d \right )^{\frac {3}{2}} a \,b^{6} d^{2} e^{6}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}+\frac {9443 \left (e x +d \right )^{\frac {3}{2}} b^{7} d^{3} e^{5}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {1001 \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{5} e^{7}}{5 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}+\frac {2002 \left (e x +d \right )^{\frac {5}{2}} a \,b^{6} d \,e^{6}}{5 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {1001 \left (e x +d \right )^{\frac {5}{2}} b^{7} d^{2} e^{5}}{5 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {7837 \left (e x +d \right )^{\frac {7}{2}} a \,b^{6} e^{6}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}+\frac {7837 \left (e x +d \right )^{\frac {7}{2}} b^{7} d \,e^{5}}{64 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {3633 \left (e x +d \right )^{\frac {9}{2}} b^{7} e^{5}}{128 \left (a e -b d \right )^{8} \left (b e x +a e \right )^{5}}-\frac {9009 b^{3} 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 )^{8} \sqrt {\left (a e -b d \right ) b}}-\frac {42 b^{2} e^{5}}{\left (a e -b d \right )^{8} \sqrt {e x +d}}+\frac {4 b \,e^{5}}{\left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 e^{5}}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-3633/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)-7837/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(7
/2)*a+7837/64*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d-1001/5*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+
d)^(5/2)*a^2+2002/5*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d-1001/5*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)
^5*(e*x+d)^(5/2)*d^2-9443/64*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3+28329/64*e^7/(a*e-b*d)^8*b^5/
(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d-28329/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^2+9443/64*e^5/(
a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3-5327/128*e^9/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4+53
27/32*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d-15981/64*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)
^(1/2)*a^2*d^2+5327/32*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3-5327/128*e^5/(a*e-b*d)^8*b^7/(b*e
*x+a*e)^5*(e*x+d)^(1/2)*d^4-9009/128*e^5/(a*e-b*d)^8*b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b
)^(1/2)*b)-2/5*e^5/(a*e-b*d)^6/(e*x+d)^(5/2)-42*e^5/(a*e-b*d)^8*b^2/(e*x+d)^(1/2)+4*e^5/(a*e-b*d)^7*b/(e*x+d)^
(3/2)

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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)^(7/2)/(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 = 1.51, size = 594, normalized size = 2.01 \begin {gather*} -\frac {\frac {2\,e^5}{5\,\left (a\,e-b\,d\right )}+\frac {26\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{{\left (a\,e-b\,d\right )}^3}+\frac {27599\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{128\,{\left (a\,e-b\,d\right )}^4}+\frac {33891\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {3003\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{5\,{\left (a\,e-b\,d\right )}^6}+\frac {21021\,b^6\,e^5\,{\left (d+e\,x\right )}^6}{64\,{\left (a\,e-b\,d\right )}^7}+\frac {9009\,b^7\,e^5\,{\left (d+e\,x\right )}^7}{128\,{\left (a\,e-b\,d\right )}^8}-\frac {2\,b\,e^5\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{5/2}\,\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 )}^{9/2}\,\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 )}^{7/2}\,\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 )}^{15/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{13/2}+{\left (d+e\,x\right )}^{11/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {9009\,b^{5/2}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^8\,e^8-8\,a^7\,b\,d\,e^7+28\,a^6\,b^2\,d^2\,e^6-56\,a^5\,b^3\,d^3\,e^5+70\,a^4\,b^4\,d^4\,e^4-56\,a^3\,b^5\,d^5\,e^3+28\,a^2\,b^6\,d^6\,e^2-8\,a\,b^7\,d^7\,e+b^8\,d^8\right )}{{\left (a\,e-b\,d\right )}^{17/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{17/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

- ((2*e^5)/(5*(a*e - b*d)) + (26*b^2*e^5*(d + e*x)^2)/(a*e - b*d)^3 + (27599*b^3*e^5*(d + e*x)^3)/(128*(a*e -
b*d)^4) + (33891*b^4*e^5*(d + e*x)^4)/(64*(a*e - b*d)^5) + (3003*b^5*e^5*(d + e*x)^5)/(5*(a*e - b*d)^6) + (210
21*b^6*e^5*(d + e*x)^6)/(64*(a*e - b*d)^7) + (9009*b^7*e^5*(d + e*x)^7)/(128*(a*e - b*d)^8) - (2*b*e^5*(d + e*
x))/(a*e - b*d)^2)/((d + e*x)^(5/2)*(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)^(9/2)*(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)^(7/2)*(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)
^(15/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(13/2) + (d + e*x)^(11/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*
e)) - (9009*b^(5/2)*e^5*atan((b^(1/2)*(d + e*x)^(1/2)*(a^8*e^8 + b^8*d^8 + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5
*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a*b^7*d^7*e - 8*a^7*b*d*e^7))/(a*e - b
*d)^(17/2)))/(128*(a*e - b*d)^(17/2))

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sympy [F(-1)]  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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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