3.17.5 \(\int \frac {(A+B x) (d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=393 \[ -\frac {231 e^4 \sqrt {b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}+\frac {231 e^4 \sqrt {d+e x} (-13 a B e+3 A b e+10 b B d)}{128 b^7}+\frac {77 e^4 (d+e x)^{3/2} (-13 a B e+3 A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac {231 e^3 (d+e x)^{5/2} (-13 a B e+3 A b e+10 b B d)}{640 b^5 (a+b x) (b d-a e)}-\frac {33 e^2 (d+e x)^{7/2} (-13 a B e+3 A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac {11 e (d+e x)^{9/2} (-13 a B e+3 A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{11/2} (-13 a B e+3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

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Rubi [A]  time = 0.36, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 50, 63, 208} \begin {gather*} -\frac {33 e^2 (d+e x)^{7/2} (-13 a B e+3 A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac {231 e^3 (d+e x)^{5/2} (-13 a B e+3 A b e+10 b B d)}{640 b^5 (a+b x) (b d-a e)}+\frac {77 e^4 (d+e x)^{3/2} (-13 a B e+3 A b e+10 b B d)}{128 b^6 (b d-a e)}+\frac {231 e^4 \sqrt {d+e x} (-13 a B e+3 A b e+10 b B d)}{128 b^7}-\frac {231 e^4 \sqrt {b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}-\frac {(d+e x)^{11/2} (-13 a B e+3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {11 e (d+e x)^{9/2} (-13 a B e+3 A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(11/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(231*e^4*(10*b*B*d + 3*A*b*e - 13*a*B*e)*Sqrt[d + e*x])/(128*b^7) + (77*e^4*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d
+ e*x)^(3/2))/(128*b^6*(b*d - a*e)) - (231*e^3*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(5/2))/(640*b^5*(b*d
- a*e)*(a + b*x)) - (33*e^2*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(7/2))/(320*b^4*(b*d - a*e)*(a + b*x)^2
) - (11*e*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(9/2))/(240*b^3*(b*d - a*e)*(a + b*x)^3) - ((10*b*B*d + 3*
A*b*e - 13*a*B*e)*(d + e*x)^(11/2))/(40*b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(13/2))/(5*b*(b*
d - a*e)*(a + b*x)^5) - (231*e^4*Sqrt[b*d - a*e]*(10*b*B*d + 3*A*b*e - 13*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/(128*b^(15/2))

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{11/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d+3 A b e-13 a B e) \int \frac {(d+e x)^{11/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(11 e (10 b B d+3 A b e-13 a B e)) \int \frac {(d+e x)^{9/2}}{(a+b x)^4} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (33 e^2 (10 b B d+3 A b e-13 a B e)\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^3} \, dx}{160 b^3 (b d-a e)}\\ &=-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (231 e^3 (10 b B d+3 A b e-13 a B e)\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{640 b^4 (b d-a e)}\\ &=-\frac {231 e^3 (10 b B d+3 A b e-13 a B e) (d+e x)^{5/2}}{640 b^5 (b d-a e) (a+b x)}-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (231 e^4 (10 b B d+3 A b e-13 a B e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{256 b^5 (b d-a e)}\\ &=\frac {77 e^4 (10 b B d+3 A b e-13 a B e) (d+e x)^{3/2}}{128 b^6 (b d-a e)}-\frac {231 e^3 (10 b B d+3 A b e-13 a B e) (d+e x)^{5/2}}{640 b^5 (b d-a e) (a+b x)}-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (231 e^4 (10 b B d+3 A b e-13 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^6}\\ &=\frac {231 e^4 (10 b B d+3 A b e-13 a B e) \sqrt {d+e x}}{128 b^7}+\frac {77 e^4 (10 b B d+3 A b e-13 a B e) (d+e x)^{3/2}}{128 b^6 (b d-a e)}-\frac {231 e^3 (10 b B d+3 A b e-13 a B e) (d+e x)^{5/2}}{640 b^5 (b d-a e) (a+b x)}-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (231 e^4 (b d-a e) (10 b B d+3 A b e-13 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^7}\\ &=\frac {231 e^4 (10 b B d+3 A b e-13 a B e) \sqrt {d+e x}}{128 b^7}+\frac {77 e^4 (10 b B d+3 A b e-13 a B e) (d+e x)^{3/2}}{128 b^6 (b d-a e)}-\frac {231 e^3 (10 b B d+3 A b e-13 a B e) (d+e x)^{5/2}}{640 b^5 (b d-a e) (a+b x)}-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (231 e^3 (b d-a e) (10 b B d+3 A b e-13 a B e)\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^7}\\ &=\frac {231 e^4 (10 b B d+3 A b e-13 a B e) \sqrt {d+e x}}{128 b^7}+\frac {77 e^4 (10 b B d+3 A b e-13 a B e) (d+e x)^{3/2}}{128 b^6 (b d-a e)}-\frac {231 e^3 (10 b B d+3 A b e-13 a B e) (d+e x)^{5/2}}{640 b^5 (b d-a e) (a+b x)}-\frac {33 e^2 (10 b B d+3 A b e-13 a B e) (d+e x)^{7/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {11 e (10 b B d+3 A b e-13 a B e) (d+e x)^{9/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+3 A b e-13 a B e) (d+e x)^{11/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{13/2}}{5 b (b d-a e) (a+b x)^5}-\frac {231 e^4 \sqrt {b d-a e} (10 b B d+3 A b e-13 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 100, normalized size = 0.25 \begin {gather*} \frac {(d+e x)^{13/2} \left (\frac {13 (a B-A b)}{(a+b x)^5}-\frac {e^4 (-13 a B e+3 A b e+10 b B d) \, _2F_1\left (5,\frac {13}{2};\frac {15}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}\right )}{65 b (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(11/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((d + e*x)^(13/2)*((13*(-(A*b) + a*B))/(a + b*x)^5 - (e^4*(10*b*B*d + 3*A*b*e - 13*a*B*e)*Hypergeometric2F1[5,
13/2, 15/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(65*b*(b*d - a*e))

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IntegrateAlgebraic [B]  time = 3.80, size = 1180, normalized size = 3.00 \begin {gather*} \frac {-10395 a^5 A b \sqrt {d+e x} e^{10}+45045 a^6 B \sqrt {d+e x} e^{10}-48510 a^4 A b^2 (d+e x)^{3/2} e^9+210210 a^5 b B (d+e x)^{3/2} e^9+51975 a^4 A b^2 d \sqrt {d+e x} e^9-259875 a^5 b B d \sqrt {d+e x} e^9-88704 a^3 A b^3 (d+e x)^{5/2} e^8+384384 a^4 b^2 B (d+e x)^{5/2} e^8+194040 a^3 A b^3 d (d+e x)^{3/2} e^8-1002540 a^4 b^2 B d (d+e x)^{3/2} e^8-103950 a^3 A b^3 d^2 \sqrt {d+e x} e^8+623700 a^4 b^2 B d^2 \sqrt {d+e x} e^8-78210 a^2 A b^4 (d+e x)^{7/2} e^7+338910 a^3 b^3 B (d+e x)^{7/2} e^7+266112 a^2 A b^4 d (d+e x)^{5/2} e^7-1448832 a^3 b^3 B d (d+e x)^{5/2} e^7-291060 a^2 A b^4 d^2 (d+e x)^{3/2} e^7+1908060 a^3 b^3 B d^2 (d+e x)^{3/2} e^7+103950 a^2 A b^4 d^3 \sqrt {d+e x} e^7-796950 a^3 b^3 B d^3 \sqrt {d+e x} e^7-31845 a A b^5 (d+e x)^{9/2} e^6+137995 a^2 b^4 B (d+e x)^{9/2} e^6+156420 a A b^5 d (d+e x)^{7/2} e^6-938520 a^2 b^4 B d (d+e x)^{7/2} e^6-266112 a A b^5 d^2 (d+e x)^{5/2} e^6+2040192 a^2 b^4 B d^2 (d+e x)^{5/2} e^6+194040 a A b^5 d^3 (d+e x)^{3/2} e^6-1811040 a^2 b^4 B d^3 (d+e x)^{3/2} e^6-51975 a A b^5 d^4 \sqrt {d+e x} e^6+571725 a^2 b^4 B d^4 \sqrt {d+e x} e^6-3840 A b^6 (d+e x)^{11/2} e^5+16640 a b^5 B (d+e x)^{11/2} e^5+31845 A b^6 d (d+e x)^{9/2} e^5-244145 a b^5 B d (d+e x)^{9/2} e^5-78210 A b^6 d^2 (d+e x)^{7/2} e^5+860310 a b^5 B d^2 (d+e x)^{7/2} e^5+88704 A b^6 d^3 (d+e x)^{5/2} e^5-1271424 a b^5 B d^3 (d+e x)^{5/2} e^5-48510 A b^6 d^4 (d+e x)^{3/2} e^5+857010 a b^5 B d^4 (d+e x)^{3/2} e^5+10395 A b^6 d^5 \sqrt {d+e x} e^5-218295 a b^5 B d^5 \sqrt {d+e x} e^5-1280 b^6 B (d+e x)^{13/2} e^4-12800 b^6 B d (d+e x)^{11/2} e^4+106150 b^6 B d^2 (d+e x)^{9/2} e^4-260700 b^6 B d^3 (d+e x)^{7/2} e^4+295680 b^6 B d^4 (d+e x)^{5/2} e^4-161700 b^6 B d^5 (d+e x)^{3/2} e^4+34650 b^6 B d^6 \sqrt {d+e x} e^4}{1920 b^7 (b d-a e-b (d+e x))^5}-\frac {231 \left (-3 a A b e^6+13 a^2 B e^6+3 A b^2 d e^5-23 a b B d e^5+10 b^2 B d^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {a e-b d} \sqrt {d+e x}}{b d-a e}\right )}{128 b^{15/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(11/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(34650*b^6*B*d^6*e^4*Sqrt[d + e*x] + 10395*A*b^6*d^5*e^5*Sqrt[d + e*x] - 218295*a*b^5*B*d^5*e^5*Sqrt[d + e*x]
- 51975*a*A*b^5*d^4*e^6*Sqrt[d + e*x] + 571725*a^2*b^4*B*d^4*e^6*Sqrt[d + e*x] + 103950*a^2*A*b^4*d^3*e^7*Sqrt
[d + e*x] - 796950*a^3*b^3*B*d^3*e^7*Sqrt[d + e*x] - 103950*a^3*A*b^3*d^2*e^8*Sqrt[d + e*x] + 623700*a^4*b^2*B
*d^2*e^8*Sqrt[d + e*x] + 51975*a^4*A*b^2*d*e^9*Sqrt[d + e*x] - 259875*a^5*b*B*d*e^9*Sqrt[d + e*x] - 10395*a^5*
A*b*e^10*Sqrt[d + e*x] + 45045*a^6*B*e^10*Sqrt[d + e*x] - 161700*b^6*B*d^5*e^4*(d + e*x)^(3/2) - 48510*A*b^6*d
^4*e^5*(d + e*x)^(3/2) + 857010*a*b^5*B*d^4*e^5*(d + e*x)^(3/2) + 194040*a*A*b^5*d^3*e^6*(d + e*x)^(3/2) - 181
1040*a^2*b^4*B*d^3*e^6*(d + e*x)^(3/2) - 291060*a^2*A*b^4*d^2*e^7*(d + e*x)^(3/2) + 1908060*a^3*b^3*B*d^2*e^7*
(d + e*x)^(3/2) + 194040*a^3*A*b^3*d*e^8*(d + e*x)^(3/2) - 1002540*a^4*b^2*B*d*e^8*(d + e*x)^(3/2) - 48510*a^4
*A*b^2*e^9*(d + e*x)^(3/2) + 210210*a^5*b*B*e^9*(d + e*x)^(3/2) + 295680*b^6*B*d^4*e^4*(d + e*x)^(5/2) + 88704
*A*b^6*d^3*e^5*(d + e*x)^(5/2) - 1271424*a*b^5*B*d^3*e^5*(d + e*x)^(5/2) - 266112*a*A*b^5*d^2*e^6*(d + e*x)^(5
/2) + 2040192*a^2*b^4*B*d^2*e^6*(d + e*x)^(5/2) + 266112*a^2*A*b^4*d*e^7*(d + e*x)^(5/2) - 1448832*a^3*b^3*B*d
*e^7*(d + e*x)^(5/2) - 88704*a^3*A*b^3*e^8*(d + e*x)^(5/2) + 384384*a^4*b^2*B*e^8*(d + e*x)^(5/2) - 260700*b^6
*B*d^3*e^4*(d + e*x)^(7/2) - 78210*A*b^6*d^2*e^5*(d + e*x)^(7/2) + 860310*a*b^5*B*d^2*e^5*(d + e*x)^(7/2) + 15
6420*a*A*b^5*d*e^6*(d + e*x)^(7/2) - 938520*a^2*b^4*B*d*e^6*(d + e*x)^(7/2) - 78210*a^2*A*b^4*e^7*(d + e*x)^(7
/2) + 338910*a^3*b^3*B*e^7*(d + e*x)^(7/2) + 106150*b^6*B*d^2*e^4*(d + e*x)^(9/2) + 31845*A*b^6*d*e^5*(d + e*x
)^(9/2) - 244145*a*b^5*B*d*e^5*(d + e*x)^(9/2) - 31845*a*A*b^5*e^6*(d + e*x)^(9/2) + 137995*a^2*b^4*B*e^6*(d +
e*x)^(9/2) - 12800*b^6*B*d*e^4*(d + e*x)^(11/2) - 3840*A*b^6*e^5*(d + e*x)^(11/2) + 16640*a*b^5*B*e^5*(d + e*
x)^(11/2) - 1280*b^6*B*e^4*(d + e*x)^(13/2))/(1920*b^7*(b*d - a*e - b*(d + e*x))^5) - (231*(10*b^2*B*d^2*e^4 +
3*A*b^2*d*e^5 - 23*a*b*B*d*e^5 - 3*a*A*b*e^6 + 13*a^2*B*e^6)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x]
)/(b*d - a*e)])/(128*b^(15/2)*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.49, size = 1862, normalized size = 4.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(3465*(10*B*a^5*b*d*e^4 - (13*B*a^6 - 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^5)*
x^5 + 5*(10*B*a*b^5*d*e^4 - (13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (13*B*a^3*b^3 - 3*A
*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (13*B*a^4*b^2 - 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 -
(13*B*a^5*b - 3*A*a^4*b^2)*e^5)*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*(1280*B*b^6*e^5*x^6 - 96*(B*a*b^5 + 4*A*b^6)*d^5 - 176*(2*B*a^2*b^4 + 3*A*a*b^5)*d^4
*e - 396*(3*B*a^3*b^3 + 2*A*a^2*b^4)*d^3*e^2 - 1386*(4*B*a^4*b^2 + A*a^3*b^3)*d^2*e^3 + 1155*(43*B*a^5*b - 3*A
*a^4*b^2)*d*e^4 - 3465*(13*B*a^6 - 3*A*a^5*b)*e^5 + 1280*(16*B*b^6*d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 - 5
*(4590*B*b^6*d^2*e^3 - (32189*B*a*b^5 - 2529*A*b^6)*d*e^4 + 2123*(13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 - 10*(103
0*B*b^6*d^3*e^2 + 3*(1671*B*a*b^5 + 359*A*b^6)*d^2*e^3 - 22*(1757*B*a^2*b^4 - 132*A*a*b^5)*d*e^4 + 2607*(13*B*
a^3*b^3 - 3*A*a^2*b^4)*e^5)*x^3 - 2*(1640*B*b^6*d^4*e + 2*(2759*B*a*b^5 + 1686*A*b^6)*d^3*e^2 + 33*(797*B*a^2*
b^4 + 183*A*a*b^5)*d^2*e^3 - 33*(6547*B*a^3*b^3 - 477*A*a^2*b^4)*d*e^4 + 14784*(13*B*a^4*b^2 - 3*A*a^3*b^3)*e^
5)*x^2 - 2*(240*B*b^6*d^5 + 8*(107*B*a*b^5 + 153*A*b^6)*d^4*e + 22*(131*B*a^2*b^4 + 84*A*a*b^5)*d^3*e^2 + 99*(
137*B*a^3*b^3 + 33*A*a^2*b^4)*d^2*e^3 - 462*(253*B*a^4*b^2 - 18*A*a^3*b^3)*d*e^4 + 8085*(13*B*a^5*b - 3*A*a^4*
b^2)*e^5)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^
7), -1/1920*(3465*(10*B*a^5*b*d*e^4 - (13*B*a^6 - 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^
5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (13*B*a^3*b^3 -
3*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (13*B*a^4*b^2 - 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4
- (13*B*a^5*b - 3*A*a^4*b^2)*e^5)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d -
a*e)) - (1280*B*b^6*e^5*x^6 - 96*(B*a*b^5 + 4*A*b^6)*d^5 - 176*(2*B*a^2*b^4 + 3*A*a*b^5)*d^4*e - 396*(3*B*a^3
*b^3 + 2*A*a^2*b^4)*d^3*e^2 - 1386*(4*B*a^4*b^2 + A*a^3*b^3)*d^2*e^3 + 1155*(43*B*a^5*b - 3*A*a^4*b^2)*d*e^4 -
3465*(13*B*a^6 - 3*A*a^5*b)*e^5 + 1280*(16*B*b^6*d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 - 5*(4590*B*b^6*d^2*
e^3 - (32189*B*a*b^5 - 2529*A*b^6)*d*e^4 + 2123*(13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 - 10*(1030*B*b^6*d^3*e^2 +
3*(1671*B*a*b^5 + 359*A*b^6)*d^2*e^3 - 22*(1757*B*a^2*b^4 - 132*A*a*b^5)*d*e^4 + 2607*(13*B*a^3*b^3 - 3*A*a^2
*b^4)*e^5)*x^3 - 2*(1640*B*b^6*d^4*e + 2*(2759*B*a*b^5 + 1686*A*b^6)*d^3*e^2 + 33*(797*B*a^2*b^4 + 183*A*a*b^5
)*d^2*e^3 - 33*(6547*B*a^3*b^3 - 477*A*a^2*b^4)*d*e^4 + 14784*(13*B*a^4*b^2 - 3*A*a^3*b^3)*e^5)*x^2 - 2*(240*B
*b^6*d^5 + 8*(107*B*a*b^5 + 153*A*b^6)*d^4*e + 22*(131*B*a^2*b^4 + 84*A*a*b^5)*d^3*e^2 + 99*(137*B*a^3*b^3 + 3
3*A*a^2*b^4)*d^2*e^3 - 462*(253*B*a^4*b^2 - 18*A*a^3*b^3)*d*e^4 + 8085*(13*B*a^5*b - 3*A*a^4*b^2)*e^5)*x)*sqrt
(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)]

________________________________________________________________________________________

giac [B]  time = 0.36, size = 1066, normalized size = 2.71

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

231/128*(10*B*b^2*d^2*e^4 - 23*B*a*b*d*e^5 + 3*A*b^2*d*e^5 + 13*B*a^2*e^6 - 3*A*a*b*e^6)*arctan(sqrt(x*e + d)*
b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^7) - 1/1920*(22950*(x*e + d)^(9/2)*B*b^6*d^2*e^4 - 81500*(x*e
+ d)^(7/2)*B*b^6*d^3*e^4 + 110080*(x*e + d)^(5/2)*B*b^6*d^4*e^4 - 66980*(x*e + d)^(3/2)*B*b^6*d^5*e^4 + 15450*
sqrt(x*e + d)*B*b^6*d^6*e^4 - 58545*(x*e + d)^(9/2)*B*a*b^5*d*e^5 + 12645*(x*e + d)^(9/2)*A*b^6*d*e^5 + 284310
*(x*e + d)^(7/2)*B*a*b^5*d^2*e^5 - 39810*(x*e + d)^(7/2)*A*b^6*d^2*e^5 - 490624*(x*e + d)^(5/2)*B*a*b^5*d^3*e^
5 + 50304*(x*e + d)^(5/2)*A*b^6*d^3*e^5 + 364210*(x*e + d)^(3/2)*B*a*b^5*d^4*e^5 - 29310*(x*e + d)^(3/2)*A*b^6
*d^4*e^5 - 99255*sqrt(x*e + d)*B*a*b^5*d^5*e^5 + 6555*sqrt(x*e + d)*A*b^6*d^5*e^5 + 35595*(x*e + d)^(9/2)*B*a^
2*b^4*e^6 - 12645*(x*e + d)^(9/2)*A*a*b^5*e^6 - 324120*(x*e + d)^(7/2)*B*a^2*b^4*d*e^6 + 79620*(x*e + d)^(7/2)
*A*a*b^5*d*e^6 + 811392*(x*e + d)^(5/2)*B*a^2*b^4*d^2*e^6 - 150912*(x*e + d)^(5/2)*A*a*b^5*d^2*e^6 - 787040*(x
*e + d)^(3/2)*B*a^2*b^4*d^3*e^6 + 117240*(x*e + d)^(3/2)*A*a*b^5*d^3*e^6 + 264525*sqrt(x*e + d)*B*a^2*b^4*d^4*
e^6 - 32775*sqrt(x*e + d)*A*a*b^5*d^4*e^6 + 121310*(x*e + d)^(7/2)*B*a^3*b^3*e^7 - 39810*(x*e + d)^(7/2)*A*a^2
*b^4*e^7 - 591232*(x*e + d)^(5/2)*B*a^3*b^3*d*e^7 + 150912*(x*e + d)^(5/2)*A*a^2*b^4*d*e^7 + 845660*(x*e + d)^
(3/2)*B*a^3*b^3*d^2*e^7 - 175860*(x*e + d)^(3/2)*A*a^2*b^4*d^2*e^7 - 374550*sqrt(x*e + d)*B*a^3*b^3*d^3*e^7 +
65550*sqrt(x*e + d)*A*a^2*b^4*d^3*e^7 + 160384*(x*e + d)^(5/2)*B*a^4*b^2*e^8 - 50304*(x*e + d)^(5/2)*A*a^3*b^3
*e^8 - 452140*(x*e + d)^(3/2)*B*a^4*b^2*d*e^8 + 117240*(x*e + d)^(3/2)*A*a^3*b^3*d*e^8 + 297300*sqrt(x*e + d)*
B*a^4*b^2*d^2*e^8 - 65550*sqrt(x*e + d)*A*a^3*b^3*d^2*e^8 + 96290*(x*e + d)^(3/2)*B*a^5*b*e^9 - 29310*(x*e + d
)^(3/2)*A*a^4*b^2*e^9 - 125475*sqrt(x*e + d)*B*a^5*b*d*e^9 + 32775*sqrt(x*e + d)*A*a^4*b^2*d*e^9 + 22005*sqrt(
x*e + d)*B*a^6*e^10 - 6555*sqrt(x*e + d)*A*a^5*b*e^10)/(((x*e + d)*b - b*d + a*e)^5*b^7) + 2/3*((x*e + d)^(3/2
)*B*b^12*e^4 + 15*sqrt(x*e + d)*B*b^12*d*e^4 - 18*sqrt(x*e + d)*B*a*b^11*e^5 + 3*sqrt(x*e + d)*A*b^12*e^5)/b^1
8

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maple [B]  time = 0.09, size = 1633, normalized size = 4.16

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2/3*e^4/b^6*B*(e*x+d)^(3/2)+2*e^5/b^6*A*(e*x+d)^(1/2)-12*e^5/b^7*a*B*(e*x+d)^(1/2)+10*e^4/b^6*B*d*(e*x+d)^(1/2
)+977/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^4-9629/192*e^9/b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^5-437/128*e^
5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*d^5+437/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^5+1327/64*e^7/b^3/(b*e*
x+a*e)^5*A*(e*x+d)^(7/2)*a^2-131/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^3-1253/15*e^8/b^5/(b*e*x+a*e)^5*(e*x+
d)^(5/2)*B*a^4+977/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^4+3003/128*e^6/b^7/((a*e-b*d)*b)^(1/2)*arctan((e
*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a^2-1467/128*e^10/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^6+131/5*e^8/b^4/(b*
e*x+a*e)^5*(e*x+d)^(5/2)*A*a^3-693/128*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*a+693/128*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)*A*d-172/3*e^4/b/(b*e*x+a*
e)^5*(e*x+d)^(5/2)*B*d^4+3349/96*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^5-515/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(1
/2)*B*d^6+4075/96*e^4/b/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^3+1327/64*e^5/b/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d^2+843/
128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A*a-843/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A*d-2373/128*e^6/b^3/(b*
e*x+a*e)^5*(e*x+d)^(9/2)*B*a^2-12131/192*e^7/b^4/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^3+1155/64*e^4/b^5/((a*e-b*d)*
b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d^2-765/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d^2-977/
16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^3+22607/96*e^8/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4*d-42283/96*e
^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d^2+4919/12*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^2*d^3-36421/192*e
^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*d^4+2185/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^4+12485/64*e^7/b
^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*d^3-17635/128*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^4+6617/128*e^5/
b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^5-5313/128*e^5/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b
)^(1/2)*b)*B*a*d+3903/128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a*d-2185/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*A*a^2*d^3-2185/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4*d+2185/64*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A
*a^3*d^2-4955/32*e^8/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d^2+8365/128*e^9/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*
a^5*d-1327/32*e^6/b^2/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a*d+2701/16*e^6/b^3/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2*d-94
77/64*e^5/b^2/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d^2-393/5*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2*d+393/5*e^6/
b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a*d^2+4619/15*e^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3*d-2113/5*e^6/b^3/(b*
e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d^2+3833/15*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^3-977/16*e^8/b^4/(b*e*x+a
*e)^5*(e*x+d)^(3/2)*A*a^3*d+2931/32*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*d^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((B*x+A)*(e*x+d)^(11/2)/(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 = 0.48, size = 996, normalized size = 2.53 \begin {gather*} \left (\frac {2\,A\,e^5-2\,B\,d\,e^4}{b^6}+\frac {2\,B\,e^4\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )}{b^{12}}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1253\,B\,a^4\,b^2\,e^8}{15}-\frac {4619\,B\,a^3\,b^3\,d\,e^7}{15}-\frac {131\,A\,a^3\,b^3\,e^8}{5}+\frac {2113\,B\,a^2\,b^4\,d^2\,e^6}{5}+\frac {393\,A\,a^2\,b^4\,d\,e^7}{5}-\frac {3833\,B\,a\,b^5\,d^3\,e^5}{15}-\frac {393\,A\,a\,b^5\,d^2\,e^6}{5}+\frac {172\,B\,b^6\,d^4\,e^4}{3}+\frac {131\,A\,b^6\,d^3\,e^5}{5}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {9629\,B\,a^5\,b\,e^9}{192}+\frac {22607\,B\,a^4\,b^2\,d\,e^8}{96}+\frac {977\,A\,a^4\,b^2\,e^9}{64}-\frac {42283\,B\,a^3\,b^3\,d^2\,e^7}{96}-\frac {977\,A\,a^3\,b^3\,d\,e^8}{16}+\frac {4919\,B\,a^2\,b^4\,d^3\,e^6}{12}+\frac {2931\,A\,a^2\,b^4\,d^2\,e^7}{32}-\frac {36421\,B\,a\,b^5\,d^4\,e^5}{192}-\frac {977\,A\,a\,b^5\,d^3\,e^6}{16}+\frac {3349\,B\,b^6\,d^5\,e^4}{96}+\frac {977\,A\,b^6\,d^4\,e^5}{64}\right )-{\left (d+e\,x\right )}^{7/2}\,\left (-\frac {12131\,B\,a^3\,b^3\,e^7}{192}+\frac {2701\,B\,a^2\,b^4\,d\,e^6}{16}+\frac {1327\,A\,a^2\,b^4\,e^7}{64}-\frac {9477\,B\,a\,b^5\,d^2\,e^5}{64}-\frac {1327\,A\,a\,b^5\,d\,e^6}{32}+\frac {4075\,B\,b^6\,d^3\,e^4}{96}+\frac {1327\,A\,b^6\,d^2\,e^5}{64}\right )+\sqrt {d+e\,x}\,\left (\frac {1467\,B\,a^6\,e^{10}}{128}-\frac {8365\,B\,a^5\,b\,d\,e^9}{128}-\frac {437\,A\,a^5\,b\,e^{10}}{128}+\frac {4955\,B\,a^4\,b^2\,d^2\,e^8}{32}+\frac {2185\,A\,a^4\,b^2\,d\,e^9}{128}-\frac {12485\,B\,a^3\,b^3\,d^3\,e^7}{64}-\frac {2185\,A\,a^3\,b^3\,d^2\,e^8}{64}+\frac {17635\,B\,a^2\,b^4\,d^4\,e^6}{128}+\frac {2185\,A\,a^2\,b^4\,d^3\,e^7}{64}-\frac {6617\,B\,a\,b^5\,d^5\,e^5}{128}-\frac {2185\,A\,a\,b^5\,d^4\,e^6}{128}+\frac {515\,B\,b^6\,d^6\,e^4}{64}+\frac {437\,A\,b^6\,d^5\,e^5}{128}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {2373\,B\,a^2\,b^4\,e^6}{128}-\frac {3903\,B\,a\,b^5\,d\,e^5}{128}-\frac {843\,A\,a\,b^5\,e^6}{128}+\frac {765\,B\,b^6\,d^2\,e^4}{64}+\frac {843\,A\,b^6\,d\,e^5}{128}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^8\,e^4-20\,a^3\,b^9\,d\,e^3+30\,a^2\,b^{10}\,d^2\,e^2-20\,a\,b^{11}\,d^3\,e+5\,b^{12}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^9\,e^3+30\,a^2\,b^{10}\,d\,e^2-30\,a\,b^{11}\,d^2\,e+10\,b^{12}\,d^3\right )+b^{12}\,{\left (d+e\,x\right )}^5-\left (5\,b^{12}\,d-5\,a\,b^{11}\,e\right )\,{\left (d+e\,x\right )}^4-b^{12}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{10}\,e^2-20\,a\,b^{11}\,d\,e+10\,b^{12}\,d^2\right )+a^5\,b^7\,e^5-5\,a^4\,b^8\,d\,e^4-10\,a^2\,b^{10}\,d^3\,e^2+10\,a^3\,b^9\,d^2\,e^3+5\,a\,b^{11}\,d^4\,e}+\frac {2\,B\,e^4\,{\left (d+e\,x\right )}^{3/2}}{3\,b^6}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (3\,A\,b\,e-13\,B\,a\,e+10\,B\,b\,d\right )\,231{}\mathrm {i}}{128\,b^{15/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(11/2))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((2*A*e^5 - 2*B*d*e^4)/b^6 + (2*B*e^4*(6*b^6*d - 6*a*b^5*e))/b^12)*(d + e*x)^(1/2) - ((d + e*x)^(5/2)*((1253*B
*a^4*b^2*e^8)/15 - (131*A*a^3*b^3*e^8)/5 + (131*A*b^6*d^3*e^5)/5 + (172*B*b^6*d^4*e^4)/3 - (393*A*a*b^5*d^2*e^
6)/5 + (393*A*a^2*b^4*d*e^7)/5 - (3833*B*a*b^5*d^3*e^5)/15 - (4619*B*a^3*b^3*d*e^7)/15 + (2113*B*a^2*b^4*d^2*e
^6)/5) - (d + e*x)^(3/2)*((977*A*a^4*b^2*e^9)/64 - (9629*B*a^5*b*e^9)/192 + (977*A*b^6*d^4*e^5)/64 + (3349*B*b
^6*d^5*e^4)/96 - (977*A*a*b^5*d^3*e^6)/16 - (977*A*a^3*b^3*d*e^8)/16 - (36421*B*a*b^5*d^4*e^5)/192 + (22607*B*
a^4*b^2*d*e^8)/96 + (2931*A*a^2*b^4*d^2*e^7)/32 + (4919*B*a^2*b^4*d^3*e^6)/12 - (42283*B*a^3*b^3*d^2*e^7)/96)
- (d + e*x)^(7/2)*((1327*A*a^2*b^4*e^7)/64 - (12131*B*a^3*b^3*e^7)/192 + (1327*A*b^6*d^2*e^5)/64 + (4075*B*b^6
*d^3*e^4)/96 - (9477*B*a*b^5*d^2*e^5)/64 + (2701*B*a^2*b^4*d*e^6)/16 - (1327*A*a*b^5*d*e^6)/32) + (d + e*x)^(1
/2)*((1467*B*a^6*e^10)/128 - (437*A*a^5*b*e^10)/128 + (437*A*b^6*d^5*e^5)/128 + (515*B*b^6*d^6*e^4)/64 - (2185
*A*a*b^5*d^4*e^6)/128 + (2185*A*a^4*b^2*d*e^9)/128 - (6617*B*a*b^5*d^5*e^5)/128 + (2185*A*a^2*b^4*d^3*e^7)/64
- (2185*A*a^3*b^3*d^2*e^8)/64 + (17635*B*a^2*b^4*d^4*e^6)/128 - (12485*B*a^3*b^3*d^3*e^7)/64 + (4955*B*a^4*b^2
*d^2*e^8)/32 - (8365*B*a^5*b*d*e^9)/128) + (d + e*x)^(9/2)*((843*A*b^6*d*e^5)/128 - (843*A*a*b^5*e^6)/128 + (2
373*B*a^2*b^4*e^6)/128 + (765*B*b^6*d^2*e^4)/64 - (3903*B*a*b^5*d*e^5)/128))/((d + e*x)*(5*b^12*d^4 + 5*a^4*b^
8*e^4 - 20*a^3*b^9*d*e^3 + 30*a^2*b^10*d^2*e^2 - 20*a*b^11*d^3*e) - (d + e*x)^2*(10*b^12*d^3 - 10*a^3*b^9*e^3
+ 30*a^2*b^10*d*e^2 - 30*a*b^11*d^2*e) + b^12*(d + e*x)^5 - (5*b^12*d - 5*a*b^11*e)*(d + e*x)^4 - b^12*d^5 + (
d + e*x)^3*(10*b^12*d^2 + 10*a^2*b^10*e^2 - 20*a*b^11*d*e) + a^5*b^7*e^5 - 5*a^4*b^8*d*e^4 - 10*a^2*b^10*d^3*e
^2 + 10*a^3*b^9*d^2*e^3 + 5*a*b^11*d^4*e) + (2*B*e^4*(d + e*x)^(3/2))/(3*b^6) + (e^4*atan((b^(1/2)*(d + e*x)^(
1/2)*1i)/(b*d - a*e)^(1/2))*(b*d - a*e)^(1/2)*(3*A*b*e - 13*B*a*e + 10*B*b*d)*231i)/(128*b^(15/2))

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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((B*x+A)*(e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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