3.14.66 \(\int \frac {(d+e x)^{15/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=253 \[ -\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}}+\frac {9009 e^5 \sqrt {d+e x} (b d-a e)^2}{128 b^8}+\frac {3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6} \]

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Rubi [A]  time = 0.18, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \begin {gather*} -\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}+\frac {3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}+\frac {9009 e^5 \sqrt {d+e x} (b d-a e)^2}{128 b^8}-\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(9009*e^5*(b*d - a*e)^2*Sqrt[d + e*x])/(128*b^8) + (3003*e^5*(b*d - a*e)*(d + e*x)^(3/2))/(128*b^7) + (9009*e^
5*(d + e*x)^(5/2))/(640*b^6) - (1287*e^4*(d + e*x)^(7/2))/(128*b^5*(a + b*x)) - (143*e^3*(d + e*x)^(9/2))/(64*
b^4*(a + b*x)^2) - (13*e^2*(d + e*x)^(11/2))/(16*b^3*(a + b*x)^3) - (3*e*(d + e*x)^(13/2))/(8*b^2*(a + b*x)^4)
- (d + e*x)^(15/2)/(5*b*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/(128*b^(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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{15/2}}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {(3 e) \int \frac {(d+e x)^{13/2}}{(a+b x)^5} \, dx}{2 b}\\ &=-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (39 e^2\right ) \int \frac {(d+e x)^{11/2}}{(a+b x)^4} \, dx}{16 b^2}\\ &=-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (143 e^3\right ) \int \frac {(d+e x)^{9/2}}{(a+b x)^3} \, dx}{32 b^3}\\ &=-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (1287 e^4\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{256 b^5}\\ &=\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{256 b^6}\\ &=\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^7}\\ &=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^8}\\ &=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^4 (b d-a e)^3\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^8}\\ &=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}-\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 52, normalized size = 0.21 \begin {gather*} \frac {2 e^5 (d+e x)^{17/2} \, _2F_1\left (6,\frac {17}{2};\frac {19}{2};-\frac {b (d+e x)}{a e-b d}\right )}{17 (a e-b d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(17/2)*Hypergeometric2F1[6, 17/2, 19/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(17*(-(b*d) + a*e)^
6)

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IntegrateAlgebraic [B]  time = 2.96, size = 675, normalized size = 2.67 \begin {gather*} \frac {e^5 \sqrt {d+e x} \left (45045 a^7 e^7+210210 a^6 b e^6 (d+e x)-315315 a^6 b d e^6+945945 a^5 b^2 d^2 e^5+384384 a^5 b^2 e^5 (d+e x)^2-1261260 a^5 b^2 d e^5 (d+e x)-1576575 a^4 b^3 d^3 e^4+3153150 a^4 b^3 d^2 e^4 (d+e x)+338910 a^4 b^3 e^4 (d+e x)^3-1921920 a^4 b^3 d e^4 (d+e x)^2+1576575 a^3 b^4 d^4 e^3-4204200 a^3 b^4 d^3 e^3 (d+e x)+3843840 a^3 b^4 d^2 e^3 (d+e x)^2+137995 a^3 b^4 e^3 (d+e x)^4-1355640 a^3 b^4 d e^3 (d+e x)^3-945945 a^2 b^5 d^5 e^2+3153150 a^2 b^5 d^4 e^2 (d+e x)-3843840 a^2 b^5 d^3 e^2 (d+e x)^2+2033460 a^2 b^5 d^2 e^2 (d+e x)^3+16640 a^2 b^5 e^2 (d+e x)^5-413985 a^2 b^5 d e^2 (d+e x)^4+315315 a b^6 d^6 e-1261260 a b^6 d^5 e (d+e x)+1921920 a b^6 d^4 e (d+e x)^2-1355640 a b^6 d^3 e (d+e x)^3+413985 a b^6 d^2 e (d+e x)^4-1280 a b^6 e (d+e x)^6-33280 a b^6 d e (d+e x)^5-45045 b^7 d^7+210210 b^7 d^6 (d+e x)-384384 b^7 d^5 (d+e x)^2+338910 b^7 d^4 (d+e x)^3-137995 b^7 d^3 (d+e x)^4+16640 b^7 d^2 (d+e x)^5+256 b^7 (d+e x)^7+1280 b^7 d (d+e x)^6\right )}{640 b^8 (a e+b (d+e x)-b d)^5}-\frac {9009 e^5 (b d-a e)^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{17/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(e^5*Sqrt[d + e*x]*(-45045*b^7*d^7 + 315315*a*b^6*d^6*e - 945945*a^2*b^5*d^5*e^2 + 1576575*a^3*b^4*d^4*e^3 - 1
576575*a^4*b^3*d^3*e^4 + 945945*a^5*b^2*d^2*e^5 - 315315*a^6*b*d*e^6 + 45045*a^7*e^7 + 210210*b^7*d^6*(d + e*x
) - 1261260*a*b^6*d^5*e*(d + e*x) + 3153150*a^2*b^5*d^4*e^2*(d + e*x) - 4204200*a^3*b^4*d^3*e^3*(d + e*x) + 31
53150*a^4*b^3*d^2*e^4*(d + e*x) - 1261260*a^5*b^2*d*e^5*(d + e*x) + 210210*a^6*b*e^6*(d + e*x) - 384384*b^7*d^
5*(d + e*x)^2 + 1921920*a*b^6*d^4*e*(d + e*x)^2 - 3843840*a^2*b^5*d^3*e^2*(d + e*x)^2 + 3843840*a^3*b^4*d^2*e^
3*(d + e*x)^2 - 1921920*a^4*b^3*d*e^4*(d + e*x)^2 + 384384*a^5*b^2*e^5*(d + e*x)^2 + 338910*b^7*d^4*(d + e*x)^
3 - 1355640*a*b^6*d^3*e*(d + e*x)^3 + 2033460*a^2*b^5*d^2*e^2*(d + e*x)^3 - 1355640*a^3*b^4*d*e^3*(d + e*x)^3
+ 338910*a^4*b^3*e^4*(d + e*x)^3 - 137995*b^7*d^3*(d + e*x)^4 + 413985*a*b^6*d^2*e*(d + e*x)^4 - 413985*a^2*b^
5*d*e^2*(d + e*x)^4 + 137995*a^3*b^4*e^3*(d + e*x)^4 + 16640*b^7*d^2*(d + e*x)^5 - 33280*a*b^6*d*e*(d + e*x)^5
+ 16640*a^2*b^5*e^2*(d + e*x)^5 + 1280*b^7*d*(d + e*x)^6 - 1280*a*b^6*e*(d + e*x)^6 + 256*b^7*(d + e*x)^7))/(
640*b^8*(-(b*d) + a*e + b*(d + e*x))^5) - (9009*e^5*(b*d - a*e)^3*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d +
e*x])/(b*d - a*e)])/(128*b^(17/2)*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.45, size = 1606, normalized size = 6.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(45045*(a^5*b^2*d^2*e^5 - 2*a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b^6*d*e^6 + a^2*b^5*e^7)*x^5 +
5*(a*b^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x
^3 + 10*(a^3*b^4*d^2*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + 5*(a^4*b^3*d^2*e^5 - 2*a^5*b^2*d*e^6 + a^6*b*e
^7)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(2
56*b^7*e^7*x^7 - 128*b^7*d^7 - 240*a*b^6*d^6*e - 520*a^2*b^5*d^5*e^2 - 1430*a^3*b^4*d^4*e^3 - 6435*a^4*b^3*d^3
*e^4 + 69069*a^5*b^2*d^2*e^5 - 105105*a^6*b*d*e^6 + 45045*a^7*e^7 + 256*(12*b^7*d*e^6 - 5*a*b^6*e^7)*x^6 + 256
*(116*b^7*d^2*e^5 - 160*a*b^6*d*e^6 + 65*a^2*b^5*e^7)*x^5 - 5*(5327*b^7*d^3*e^4 - 45677*a*b^6*d^2*e^5 + 66157*
a^2*b^5*d*e^6 - 27599*a^3*b^4*e^7)*x^4 - 10*(1211*b^7*d^4*e^3 + 5810*a*b^6*d^3*e^4 - 54392*a^2*b^5*d^2*e^5 + 8
0366*a^3*b^4*d*e^6 - 33891*a^4*b^3*e^7)*x^3 - 2*(2324*b^7*d^5*e^2 + 6545*a*b^6*d^4*e^3 + 30485*a^2*b^5*d^3*e^4
- 302445*a^3*b^4*d^2*e^5 + 452595*a^4*b^3*d*e^6 - 192192*a^5*b^2*e^7)*x^2 - 2*(568*b^7*d^6*e + 1240*a*b^6*d^5
*e^2 + 3445*a^2*b^5*d^4*e^3 + 15730*a^3*b^4*d^3*e^4 - 163020*a^4*b^3*d^2*e^5 + 246246*a^5*b^2*d*e^6 - 105105*a
^6*b*e^7)*x)*sqrt(e*x + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b
^8), -1/640*(45045*(a^5*b^2*d^2*e^5 - 2*a^6*b*d*e^6 + a^7*e^7 + (b^7*d^2*e^5 - 2*a*b^6*d*e^6 + a^2*b^5*e^7)*x^
5 + 5*(a*b^6*d^2*e^5 - 2*a^2*b^5*d*e^6 + a^3*b^4*e^7)*x^4 + 10*(a^2*b^5*d^2*e^5 - 2*a^3*b^4*d*e^6 + a^4*b^3*e^
7)*x^3 + 10*(a^3*b^4*d^2*e^5 - 2*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + 5*(a^4*b^3*d^2*e^5 - 2*a^5*b^2*d*e^6 + a^6
*b*e^7)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (256*b^7*e^7*x^7 -
128*b^7*d^7 - 240*a*b^6*d^6*e - 520*a^2*b^5*d^5*e^2 - 1430*a^3*b^4*d^4*e^3 - 6435*a^4*b^3*d^3*e^4 + 69069*a^5
*b^2*d^2*e^5 - 105105*a^6*b*d*e^6 + 45045*a^7*e^7 + 256*(12*b^7*d*e^6 - 5*a*b^6*e^7)*x^6 + 256*(116*b^7*d^2*e^
5 - 160*a*b^6*d*e^6 + 65*a^2*b^5*e^7)*x^5 - 5*(5327*b^7*d^3*e^4 - 45677*a*b^6*d^2*e^5 + 66157*a^2*b^5*d*e^6 -
27599*a^3*b^4*e^7)*x^4 - 10*(1211*b^7*d^4*e^3 + 5810*a*b^6*d^3*e^4 - 54392*a^2*b^5*d^2*e^5 + 80366*a^3*b^4*d*e
^6 - 33891*a^4*b^3*e^7)*x^3 - 2*(2324*b^7*d^5*e^2 + 6545*a*b^6*d^4*e^3 + 30485*a^2*b^5*d^3*e^4 - 302445*a^3*b^
4*d^2*e^5 + 452595*a^4*b^3*d*e^6 - 192192*a^5*b^2*e^7)*x^2 - 2*(568*b^7*d^6*e + 1240*a*b^6*d^5*e^2 + 3445*a^2*
b^5*d^4*e^3 + 15730*a^3*b^4*d^3*e^4 - 163020*a^4*b^3*d^2*e^5 + 246246*a^5*b^2*d*e^6 - 105105*a^6*b*e^7)*x)*sqr
t(e*x + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8)]

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giac [B]  time = 0.30, size = 785, normalized size = 3.10 \begin {gather*} \frac {9009 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{8}} - \frac {26635 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} d^{3} e^{5} - 94430 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d^{4} e^{5} + 128128 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{5} e^{5} - 78370 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{6} e^{5} + 18165 \, \sqrt {x e + d} b^{7} d^{7} e^{5} - 79905 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{6} d^{2} e^{6} + 377720 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} d^{3} e^{6} - 640640 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d^{4} e^{6} + 470220 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{5} e^{6} - 127155 \, \sqrt {x e + d} a b^{6} d^{6} e^{6} + 79905 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{2} b^{5} d e^{7} - 566580 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{5} d^{2} e^{7} + 1281280 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} d^{3} e^{7} - 1175550 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{4} e^{7} + 381465 \, \sqrt {x e + d} a^{2} b^{5} d^{5} e^{7} - 26635 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{3} b^{4} e^{8} + 377720 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{4} d e^{8} - 1281280 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{4} d^{2} e^{8} + 1567400 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} d^{3} e^{8} - 635775 \, \sqrt {x e + d} a^{3} b^{4} d^{4} e^{8} - 94430 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{4} b^{3} e^{9} + 640640 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{3} d e^{9} - 1175550 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{3} d^{2} e^{9} + 635775 \, \sqrt {x e + d} a^{4} b^{3} d^{3} e^{9} - 128128 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{5} b^{2} e^{10} + 470220 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b^{2} d e^{10} - 381465 \, \sqrt {x e + d} a^{5} b^{2} d^{2} e^{10} - 78370 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{6} b e^{11} + 127155 \, \sqrt {x e + d} a^{6} b d e^{11} - 18165 \, \sqrt {x e + d} a^{7} e^{12}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{8}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} b^{24} e^{5} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{24} d e^{5} + 105 \, \sqrt {x e + d} b^{24} d^{2} e^{5} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{23} e^{6} - 210 \, \sqrt {x e + d} a b^{23} d e^{6} + 105 \, \sqrt {x e + d} a^{2} b^{22} e^{7}\right )}}{5 \, b^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

9009/128*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e)
)/(sqrt(-b^2*d + a*b*e)*b^8) - 1/640*(26635*(x*e + d)^(9/2)*b^7*d^3*e^5 - 94430*(x*e + d)^(7/2)*b^7*d^4*e^5 +
128128*(x*e + d)^(5/2)*b^7*d^5*e^5 - 78370*(x*e + d)^(3/2)*b^7*d^6*e^5 + 18165*sqrt(x*e + d)*b^7*d^7*e^5 - 799
05*(x*e + d)^(9/2)*a*b^6*d^2*e^6 + 377720*(x*e + d)^(7/2)*a*b^6*d^3*e^6 - 640640*(x*e + d)^(5/2)*a*b^6*d^4*e^6
+ 470220*(x*e + d)^(3/2)*a*b^6*d^5*e^6 - 127155*sqrt(x*e + d)*a*b^6*d^6*e^6 + 79905*(x*e + d)^(9/2)*a^2*b^5*d
*e^7 - 566580*(x*e + d)^(7/2)*a^2*b^5*d^2*e^7 + 1281280*(x*e + d)^(5/2)*a^2*b^5*d^3*e^7 - 1175550*(x*e + d)^(3
/2)*a^2*b^5*d^4*e^7 + 381465*sqrt(x*e + d)*a^2*b^5*d^5*e^7 - 26635*(x*e + d)^(9/2)*a^3*b^4*e^8 + 377720*(x*e +
d)^(7/2)*a^3*b^4*d*e^8 - 1281280*(x*e + d)^(5/2)*a^3*b^4*d^2*e^8 + 1567400*(x*e + d)^(3/2)*a^3*b^4*d^3*e^8 -
635775*sqrt(x*e + d)*a^3*b^4*d^4*e^8 - 94430*(x*e + d)^(7/2)*a^4*b^3*e^9 + 640640*(x*e + d)^(5/2)*a^4*b^3*d*e^
9 - 1175550*(x*e + d)^(3/2)*a^4*b^3*d^2*e^9 + 635775*sqrt(x*e + d)*a^4*b^3*d^3*e^9 - 128128*(x*e + d)^(5/2)*a^
5*b^2*e^10 + 470220*(x*e + d)^(3/2)*a^5*b^2*d*e^10 - 381465*sqrt(x*e + d)*a^5*b^2*d^2*e^10 - 78370*(x*e + d)^(
3/2)*a^6*b*e^11 + 127155*sqrt(x*e + d)*a^6*b*d*e^11 - 18165*sqrt(x*e + d)*a^7*e^12)/(((x*e + d)*b - b*d + a*e)
^5*b^8) + 2/5*((x*e + d)^(5/2)*b^24*e^5 + 10*(x*e + d)^(3/2)*b^24*d*e^5 + 105*sqrt(x*e + d)*b^24*d^2*e^5 - 10*
(x*e + d)^(3/2)*a*b^23*e^6 - 210*sqrt(x*e + d)*a*b^23*d*e^6 + 105*sqrt(x*e + d)*a^2*b^22*e^7)/b^30

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maple [B]  time = 0.08, size = 1164, normalized size = 4.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2002*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3*d^2+15981/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a*d^2-1001*e^9/
b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^4*d-23511/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^5-25431/128*e^11/b^7/(b
*e*x+a*e)^5*(e*x+d)^(1/2)*a^6*d+28329/32*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2*d^2-9443/16*e^6/b^2/(b*e*x+a*
e)^5*(e*x+d)^(7/2)*a*d^3-15981/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a^2*d-2002*e^7/b^3/(b*e*x+a*e)^5*(e*x+d
)^(5/2)*a^2*d^3-9443/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^3*d+25431/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*a*d^6+117555/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d^4+9443/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^4+
117555/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^4*d^2-39185/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d^3-762
93/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^2*d^5+27027/128*e^7/b^7/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/
((a*e-b*d)*b)^(1/2)*b)*a^2*d-27027/128*e^6/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)
*a*d^2-127155/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4*d^3+127155/128*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a
^3*d^4+76293/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^5*d^2+4*e^5/b^6*(e*x+d)^(3/2)*d+42*e^5/b^6*d^2*(e*x+d)
^(1/2)-4*e^6/b^7*(e*x+d)^(3/2)*a+42*e^7/b^8*a^2*(e*x+d)^(1/2)+2/5*e^5*(e*x+d)^(5/2)/b^6+1001*e^6/b^2/(b*e*x+a*
e)^5*(e*x+d)^(5/2)*a*d^4-23511/32*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^5*d-84*e^6/b^7*a*d*(e*x+d)^(1/2)+9009
/128*e^5/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*d^3-3633/128*e^5/b/(b*e*x+a*e)^5*
(e*x+d)^(1/2)*d^7+9443/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^4-5327/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*d^3
-1001/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^5+7837/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^6-9009/128*e^8/b^8/(
(a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^3+3633/128*e^12/b^8/(b*e*x+a*e)^5*(e*x+d)^(1/
2)*a^7+5327/128*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a^3+1001/5*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^5+7837/6
4*e^11/b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^6

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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)^(15/2)/(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 = 0.83, size = 846, normalized size = 3.34 \begin {gather*} \left (\frac {2\,e^5\,{\left (6\,b^6\,d-6\,a\,b^5\,e\right )}^2}{b^{18}}-\frac {30\,e^5\,{\left (a\,e-b\,d\right )}^2}{b^8}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (\frac {3633\,a^7\,e^{12}}{128}-\frac {25431\,a^6\,b\,d\,e^{11}}{128}+\frac {76293\,a^5\,b^2\,d^2\,e^{10}}{128}-\frac {127155\,a^4\,b^3\,d^3\,e^9}{128}+\frac {127155\,a^3\,b^4\,d^4\,e^8}{128}-\frac {76293\,a^2\,b^5\,d^5\,e^7}{128}+\frac {25431\,a\,b^6\,d^6\,e^6}{128}-\frac {3633\,b^7\,d^7\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1001\,a^5\,b^2\,e^{10}}{5}-1001\,a^4\,b^3\,d\,e^9+2002\,a^3\,b^4\,d^2\,e^8-2002\,a^2\,b^5\,d^3\,e^7+1001\,a\,b^6\,d^4\,e^6-\frac {1001\,b^7\,d^5\,e^5}{5}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {7837\,a^6\,b\,e^{11}}{64}-\frac {23511\,a^5\,b^2\,d\,e^{10}}{32}+\frac {117555\,a^4\,b^3\,d^2\,e^9}{64}-\frac {39185\,a^3\,b^4\,d^3\,e^8}{16}+\frac {117555\,a^2\,b^5\,d^4\,e^7}{64}-\frac {23511\,a\,b^6\,d^5\,e^6}{32}+\frac {7837\,b^7\,d^6\,e^5}{64}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {5327\,a^3\,b^4\,e^8}{128}-\frac {15981\,a^2\,b^5\,d\,e^7}{128}+\frac {15981\,a\,b^6\,d^2\,e^6}{128}-\frac {5327\,b^7\,d^3\,e^5}{128}\right )+{\left (d+e\,x\right )}^{7/2}\,\left (\frac {9443\,a^4\,b^3\,e^9}{64}-\frac {9443\,a^3\,b^4\,d\,e^8}{16}+\frac {28329\,a^2\,b^5\,d^2\,e^7}{32}-\frac {9443\,a\,b^6\,d^3\,e^6}{16}+\frac {9443\,b^7\,d^4\,e^5}{64}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^9\,e^4-20\,a^3\,b^{10}\,d\,e^3+30\,a^2\,b^{11}\,d^2\,e^2-20\,a\,b^{12}\,d^3\,e+5\,b^{13}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^{10}\,e^3+30\,a^2\,b^{11}\,d\,e^2-30\,a\,b^{12}\,d^2\,e+10\,b^{13}\,d^3\right )+b^{13}\,{\left (d+e\,x\right )}^5-\left (5\,b^{13}\,d-5\,a\,b^{12}\,e\right )\,{\left (d+e\,x\right )}^4-b^{13}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{11}\,e^2-20\,a\,b^{12}\,d\,e+10\,b^{13}\,d^2\right )+a^5\,b^8\,e^5-5\,a^4\,b^9\,d\,e^4-10\,a^2\,b^{11}\,d^3\,e^2+10\,a^3\,b^{10}\,d^2\,e^3+5\,a\,b^{12}\,d^4\,e}+\frac {2\,e^5\,{\left (d+e\,x\right )}^{5/2}}{5\,b^6}+\frac {2\,e^5\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^{12}}-\frac {9009\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^8-3\,a^2\,b\,d\,e^7+3\,a\,b^2\,d^2\,e^6-b^3\,d^3\,e^5}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{128\,b^{17/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(15/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((2*e^5*(6*b^6*d - 6*a*b^5*e)^2)/b^18 - (30*e^5*(a*e - b*d)^2)/b^8)*(d + e*x)^(1/2) + ((d + e*x)^(1/2)*((3633*
a^7*e^12)/128 - (3633*b^7*d^7*e^5)/128 + (25431*a*b^6*d^6*e^6)/128 - (76293*a^2*b^5*d^5*e^7)/128 + (127155*a^3
*b^4*d^4*e^8)/128 - (127155*a^4*b^3*d^3*e^9)/128 + (76293*a^5*b^2*d^2*e^10)/128 - (25431*a^6*b*d*e^11)/128) +
(d + e*x)^(5/2)*((1001*a^5*b^2*e^10)/5 - (1001*b^7*d^5*e^5)/5 + 1001*a*b^6*d^4*e^6 - 1001*a^4*b^3*d*e^9 - 2002
*a^2*b^5*d^3*e^7 + 2002*a^3*b^4*d^2*e^8) + (d + e*x)^(3/2)*((7837*a^6*b*e^11)/64 + (7837*b^7*d^6*e^5)/64 - (23
511*a*b^6*d^5*e^6)/32 - (23511*a^5*b^2*d*e^10)/32 + (117555*a^2*b^5*d^4*e^7)/64 - (39185*a^3*b^4*d^3*e^8)/16 +
(117555*a^4*b^3*d^2*e^9)/64) + (d + e*x)^(9/2)*((5327*a^3*b^4*e^8)/128 - (5327*b^7*d^3*e^5)/128 + (15981*a*b^
6*d^2*e^6)/128 - (15981*a^2*b^5*d*e^7)/128) + (d + e*x)^(7/2)*((9443*a^4*b^3*e^9)/64 + (9443*b^7*d^4*e^5)/64 -
(9443*a*b^6*d^3*e^6)/16 - (9443*a^3*b^4*d*e^8)/16 + (28329*a^2*b^5*d^2*e^7)/32))/((d + e*x)*(5*b^13*d^4 + 5*a
^4*b^9*e^4 - 20*a^3*b^10*d*e^3 + 30*a^2*b^11*d^2*e^2 - 20*a*b^12*d^3*e) - (d + e*x)^2*(10*b^13*d^3 - 10*a^3*b^
10*e^3 + 30*a^2*b^11*d*e^2 - 30*a*b^12*d^2*e) + b^13*(d + e*x)^5 - (5*b^13*d - 5*a*b^12*e)*(d + e*x)^4 - b^13*
d^5 + (d + e*x)^3*(10*b^13*d^2 + 10*a^2*b^11*e^2 - 20*a*b^12*d*e) + a^5*b^8*e^5 - 5*a^4*b^9*d*e^4 - 10*a^2*b^1
1*d^3*e^2 + 10*a^3*b^10*d^2*e^3 + 5*a*b^12*d^4*e) + (2*e^5*(d + e*x)^(5/2))/(5*b^6) + (2*e^5*(6*b^6*d - 6*a*b^
5*e)*(d + e*x)^(3/2))/(3*b^12) - (9009*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(5/2)*(d + e*x)^(1/2))/(a^3*e^8 - b^3
*d^3*e^5 + 3*a*b^2*d^2*e^6 - 3*a^2*b*d*e^7))*(a*e - b*d)^(5/2))/(128*b^(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((e*x+d)**(15/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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