3.1816 \(\int \frac {(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=332 \[ -\frac {21 e^2 (b d-a e)^{3/2} (-11 a B e+5 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}+\frac {21 e^2 \sqrt {d+e x} (b d-a e) (-11 a B e+5 A b e+6 b B d)}{8 b^6}+\frac {7 e^2 (d+e x)^{3/2} (-11 a B e+5 A b e+6 b B d)}{8 b^5}+\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+5 A b e+6 b B d)}{40 b^4 (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+5 A b e+6 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+5 A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

[Out]

7/8*e^2*(5*A*b*e-11*B*a*e+6*B*b*d)*(e*x+d)^(3/2)/b^5+21/40*e^2*(5*A*b*e-11*B*a*e+6*B*b*d)*(e*x+d)^(5/2)/b^4/(-
a*e+b*d)-3/8*e*(5*A*b*e-11*B*a*e+6*B*b*d)*(e*x+d)^(7/2)/b^3/(-a*e+b*d)/(b*x+a)-1/12*(5*A*b*e-11*B*a*e+6*B*b*d)
*(e*x+d)^(9/2)/b^2/(-a*e+b*d)/(b*x+a)^2-1/3*(A*b-B*a)*(e*x+d)^(11/2)/b/(-a*e+b*d)/(b*x+a)^3-21/8*e^2*(-a*e+b*d
)^(3/2)*(5*A*b*e-11*B*a*e+6*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(13/2)+21/8*e^2*(-a*e+b*d
)*(5*A*b*e-11*B*a*e+6*B*b*d)*(e*x+d)^(1/2)/b^6

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Rubi [A]  time = 0.33, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {21 e^2 (d+e x)^{5/2} (-11 a B e+5 A b e+6 b B d)}{40 b^4 (b d-a e)}+\frac {7 e^2 (d+e x)^{3/2} (-11 a B e+5 A b e+6 b B d)}{8 b^5}+\frac {21 e^2 \sqrt {d+e x} (b d-a e) (-11 a B e+5 A b e+6 b B d)}{8 b^6}-\frac {21 e^2 (b d-a e)^{3/2} (-11 a B e+5 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}-\frac {(d+e x)^{9/2} (-11 a B e+5 A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+5 A b e+6 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(21*e^2*(b*d - a*e)*(6*b*B*d + 5*A*b*e - 11*a*B*e)*Sqrt[d + e*x])/(8*b^6) + (7*e^2*(6*b*B*d + 5*A*b*e - 11*a*B
*e)*(d + e*x)^(3/2))/(8*b^5) + (21*e^2*(6*b*B*d + 5*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(40*b^4*(b*d - a*e)) -
(3*e*(6*b*B*d + 5*A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(8*b^3*(b*d - a*e)*(a + b*x)) - ((6*b*B*d + 5*A*b*e - 11*
a*B*e)*(d + e*x)^(9/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*b - a*B)*(d + e*x)^(11/2))/(3*b*(b*d - a*e)*(a
+ b*x)^3) - (21*e^2*(b*d - a*e)^(3/2)*(6*b*B*d + 5*A*b*e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(8*b^(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 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)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(A+B x) (d+e x)^{9/2}}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(6 b B d+5 A b e-11 a B e) \int \frac {(d+e x)^{9/2}}{(a+b x)^3} \, dx}{6 b (b d-a e)}\\ &=-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(3 e (6 b B d+5 A b e-11 a B e)) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{8 b^2 (b d-a e)}\\ &=-\frac {3 e (6 b B d+5 A b e-11 a B e) (d+e x)^{7/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (21 e^2 (6 b B d+5 A b e-11 a B e)\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{16 b^3 (b d-a e)}\\ &=\frac {21 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{5/2}}{40 b^4 (b d-a e)}-\frac {3 e (6 b B d+5 A b e-11 a B e) (d+e x)^{7/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (21 e^2 (6 b B d+5 A b e-11 a B e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{16 b^4}\\ &=\frac {7 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{3/2}}{8 b^5}+\frac {21 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{5/2}}{40 b^4 (b d-a e)}-\frac {3 e (6 b B d+5 A b e-11 a B e) (d+e x)^{7/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (21 e^2 (b d-a e) (6 b B d+5 A b e-11 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^5}\\ &=\frac {21 e^2 (b d-a e) (6 b B d+5 A b e-11 a B e) \sqrt {d+e x}}{8 b^6}+\frac {7 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{3/2}}{8 b^5}+\frac {21 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{5/2}}{40 b^4 (b d-a e)}-\frac {3 e (6 b B d+5 A b e-11 a B e) (d+e x)^{7/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (21 e^2 (b d-a e)^2 (6 b B d+5 A b e-11 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^6}\\ &=\frac {21 e^2 (b d-a e) (6 b B d+5 A b e-11 a B e) \sqrt {d+e x}}{8 b^6}+\frac {7 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{3/2}}{8 b^5}+\frac {21 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{5/2}}{40 b^4 (b d-a e)}-\frac {3 e (6 b B d+5 A b e-11 a B e) (d+e x)^{7/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (21 e (b d-a e)^2 (6 b B d+5 A b e-11 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 )}{8 b^6}\\ &=\frac {21 e^2 (b d-a e) (6 b B d+5 A b e-11 a B e) \sqrt {d+e x}}{8 b^6}+\frac {7 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{3/2}}{8 b^5}+\frac {21 e^2 (6 b B d+5 A b e-11 a B e) (d+e x)^{5/2}}{40 b^4 (b d-a e)}-\frac {3 e (6 b B d+5 A b e-11 a B e) (d+e x)^{7/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+5 A b e-11 a B e) (d+e x)^{9/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{11/2}}{3 b (b d-a e) (a+b x)^3}-\frac {21 e^2 (b d-a e)^{3/2} (6 b B d+5 A b e-11 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 100, normalized size = 0.30 \[ \frac {(d+e x)^{11/2} \left (\frac {11 (a B-A b)}{(a+b x)^3}-\frac {e^2 (-11 a B e+5 A b e+6 b B d) \, _2F_1\left (3,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}\right )}{33 b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

((d + e*x)^(11/2)*((11*(-(A*b) + a*B))/(a + b*x)^3 - (e^2*(6*b*B*d + 5*A*b*e - 11*a*B*e)*Hypergeometric2F1[3,
11/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^3))/(33*b*(b*d - a*e))

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fricas [B]  time = 0.99, size = 1514, normalized size = 4.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

[1/240*(315*(6*B*a^3*b^2*d^2*e^2 - (17*B*a^4*b - 5*A*a^3*b^2)*d*e^3 + (11*B*a^5 - 5*A*a^4*b)*e^4 + (6*B*b^5*d^
2*e^2 - (17*B*a*b^4 - 5*A*b^5)*d*e^3 + (11*B*a^2*b^3 - 5*A*a*b^4)*e^4)*x^3 + 3*(6*B*a*b^4*d^2*e^2 - (17*B*a^2*
b^3 - 5*A*a*b^4)*d*e^3 + (11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 + 3*(6*B*a^2*b^3*d^2*e^2 - (17*B*a^3*b^2 - 5*A*
a^2*b^3)*d*e^3 + (11*B*a^4*b - 5*A*a^3*b^2)*e^4)*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*(48*B*b^5*e^4*x^5 - 20*(B*a*b^4 + 2*A*b^5)*d^4 - 90*(2*B*a^2*b^3 +
A*a*b^4)*d^3*e + 63*(51*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^2 - 210*(31*B*a^4*b - 10*A*a^3*b^2)*d*e^3 + 315*(11*B*a
^5 - 5*A*a^4*b)*e^4 + 16*(21*B*b^5*d*e^3 - (11*B*a*b^4 - 5*A*b^5)*e^4)*x^4 + 16*(108*B*b^5*d^2*e^2 - (197*B*a*
b^4 - 65*A*b^5)*d*e^3 + 9*(11*B*a^2*b^3 - 5*A*a*b^4)*e^4)*x^3 - 3*(170*B*b^5*d^3*e - (2513*B*a*b^4 - 275*A*b^5
)*d^2*e^2 + 6*(814*B*a^2*b^3 - 265*A*a*b^4)*d*e^3 - 231*(11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 - 2*(30*B*b^5*d^
4 + 5*(53*B*a*b^4 + 25*A*b^5)*d^3*e - 18*(244*B*a^2*b^3 - 25*A*a*b^4)*d^2*e^2 + 63*(139*B*a^3*b^2 - 45*A*a^2*b
^3)*d*e^3 - 420*(11*B*a^4*b - 5*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b
^6), -1/120*(315*(6*B*a^3*b^2*d^2*e^2 - (17*B*a^4*b - 5*A*a^3*b^2)*d*e^3 + (11*B*a^5 - 5*A*a^4*b)*e^4 + (6*B*b
^5*d^2*e^2 - (17*B*a*b^4 - 5*A*b^5)*d*e^3 + (11*B*a^2*b^3 - 5*A*a*b^4)*e^4)*x^3 + 3*(6*B*a*b^4*d^2*e^2 - (17*B
*a^2*b^3 - 5*A*a*b^4)*d*e^3 + (11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 + 3*(6*B*a^2*b^3*d^2*e^2 - (17*B*a^3*b^2 -
5*A*a^2*b^3)*d*e^3 + (11*B*a^4*b - 5*A*a^3*b^2)*e^4)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b
*d - a*e)/b)/(b*d - a*e)) - (48*B*b^5*e^4*x^5 - 20*(B*a*b^4 + 2*A*b^5)*d^4 - 90*(2*B*a^2*b^3 + A*a*b^4)*d^3*e
+ 63*(51*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^2 - 210*(31*B*a^4*b - 10*A*a^3*b^2)*d*e^3 + 315*(11*B*a^5 - 5*A*a^4*b)
*e^4 + 16*(21*B*b^5*d*e^3 - (11*B*a*b^4 - 5*A*b^5)*e^4)*x^4 + 16*(108*B*b^5*d^2*e^2 - (197*B*a*b^4 - 65*A*b^5)
*d*e^3 + 9*(11*B*a^2*b^3 - 5*A*a*b^4)*e^4)*x^3 - 3*(170*B*b^5*d^3*e - (2513*B*a*b^4 - 275*A*b^5)*d^2*e^2 + 6*(
814*B*a^2*b^3 - 265*A*a*b^4)*d*e^3 - 231*(11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 - 2*(30*B*b^5*d^4 + 5*(53*B*a*b
^4 + 25*A*b^5)*d^3*e - 18*(244*B*a^2*b^3 - 25*A*a*b^4)*d^2*e^2 + 63*(139*B*a^3*b^2 - 45*A*a^2*b^3)*d*e^3 - 420
*(11*B*a^4*b - 5*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)]

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giac [B]  time = 0.30, size = 829, normalized size = 2.50 \[ \frac {21 \, {\left (6 \, B b^{3} d^{3} e^{2} - 23 \, B a b^{2} d^{2} e^{3} + 5 \, A b^{3} d^{2} e^{3} + 28 \, B a^{2} b d e^{4} - 10 \, A a b^{2} d e^{4} - 11 \, B a^{3} e^{5} + 5 \, A a^{2} b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {102 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{2} - 192 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{2} + 90 \, \sqrt {x e + d} B b^{5} d^{5} e^{2} - 471 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{3} + 165 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{3} + 1048 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{3} - 280 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{3} - 573 \, \sqrt {x e + d} B a b^{4} d^{4} e^{3} + 123 \, \sqrt {x e + d} A b^{5} d^{4} e^{3} + 636 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{4} - 330 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{4} - 1992 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{4} + 840 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{4} + 1392 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{4} - 492 \, \sqrt {x e + d} A a b^{4} d^{3} e^{4} - 267 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{5} + 165 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{5} + 1608 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{5} - 840 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{5} - 1638 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{5} + 738 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{5} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{6} + 280 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{6} + 942 \, \sqrt {x e + d} B a^{4} b d e^{6} - 492 \, \sqrt {x e + d} A a^{3} b^{2} d e^{6} - 213 \, \sqrt {x e + d} B a^{5} e^{7} + 123 \, \sqrt {x e + d} A a^{4} b e^{7}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{6}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{16} e^{2} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{16} d e^{2} + 90 \, \sqrt {x e + d} B b^{16} d^{2} e^{2} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{15} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{16} e^{3} - 240 \, \sqrt {x e + d} B a b^{15} d e^{3} + 60 \, \sqrt {x e + d} A b^{16} d e^{3} + 150 \, \sqrt {x e + d} B a^{2} b^{14} e^{4} - 60 \, \sqrt {x e + d} A a b^{15} e^{4}\right )}}{15 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

21/8*(6*B*b^3*d^3*e^2 - 23*B*a*b^2*d^2*e^3 + 5*A*b^3*d^2*e^3 + 28*B*a^2*b*d*e^4 - 10*A*a*b^2*d*e^4 - 11*B*a^3*
e^5 + 5*A*a^2*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) - 1/24*(102*(x*e
+ d)^(5/2)*B*b^5*d^3*e^2 - 192*(x*e + d)^(3/2)*B*b^5*d^4*e^2 + 90*sqrt(x*e + d)*B*b^5*d^5*e^2 - 471*(x*e + d)^
(5/2)*B*a*b^4*d^2*e^3 + 165*(x*e + d)^(5/2)*A*b^5*d^2*e^3 + 1048*(x*e + d)^(3/2)*B*a*b^4*d^3*e^3 - 280*(x*e +
d)^(3/2)*A*b^5*d^3*e^3 - 573*sqrt(x*e + d)*B*a*b^4*d^4*e^3 + 123*sqrt(x*e + d)*A*b^5*d^4*e^3 + 636*(x*e + d)^(
5/2)*B*a^2*b^3*d*e^4 - 330*(x*e + d)^(5/2)*A*a*b^4*d*e^4 - 1992*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^4 + 840*(x*e +
d)^(3/2)*A*a*b^4*d^2*e^4 + 1392*sqrt(x*e + d)*B*a^2*b^3*d^3*e^4 - 492*sqrt(x*e + d)*A*a*b^4*d^3*e^4 - 267*(x*
e + d)^(5/2)*B*a^3*b^2*e^5 + 165*(x*e + d)^(5/2)*A*a^2*b^3*e^5 + 1608*(x*e + d)^(3/2)*B*a^3*b^2*d*e^5 - 840*(x
*e + d)^(3/2)*A*a^2*b^3*d*e^5 - 1638*sqrt(x*e + d)*B*a^3*b^2*d^2*e^5 + 738*sqrt(x*e + d)*A*a^2*b^3*d^2*e^5 - 4
72*(x*e + d)^(3/2)*B*a^4*b*e^6 + 280*(x*e + d)^(3/2)*A*a^3*b^2*e^6 + 942*sqrt(x*e + d)*B*a^4*b*d*e^6 - 492*sqr
t(x*e + d)*A*a^3*b^2*d*e^6 - 213*sqrt(x*e + d)*B*a^5*e^7 + 123*sqrt(x*e + d)*A*a^4*b*e^7)/(((x*e + d)*b - b*d
+ a*e)^3*b^6) + 2/15*(3*(x*e + d)^(5/2)*B*b^16*e^2 + 15*(x*e + d)^(3/2)*B*b^16*d*e^2 + 90*sqrt(x*e + d)*B*b^16
*d^2*e^2 - 20*(x*e + d)^(3/2)*B*a*b^15*e^3 + 5*(x*e + d)^(3/2)*A*b^16*e^3 - 240*sqrt(x*e + d)*B*a*b^15*d*e^3 +
60*sqrt(x*e + d)*A*b^16*d*e^3 + 150*sqrt(x*e + d)*B*a^2*b^14*e^4 - 60*sqrt(x*e + d)*A*a*b^15*e^4)/b^20

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maple [B]  time = 0.09, size = 1285, normalized size = 3.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/5*e^2/b^4*B*(e*x+d)^(5/2)+2/3*e^3/b^4*A*(e*x+d)^(3/2)-8/3*e^3/b^5*B*(e*x+d)^(3/2)*a-8*e^4/b^5*A*(e*x+d)^(1/2
)*a+8*e^3/b^4*A*(e*x+d)^(1/2)*d+20*e^4/b^6*B*(e*x+d)^(1/2)*a^2+2*e^2/b^4*B*(e*x+d)^(3/2)*d+12*e^2/b^4*B*(e*x+d
)^(1/2)*d^2-157/4*e^6/b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^4*d-483/8*e^3/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)
^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a*d^2+273/4*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3*d^2-58*e^4/b^3/(b*e*x+a*
e)^3*(e*x+d)^(1/2)*B*a^2*d^3+191/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^4+35*e^5/b^3/(b*e*x+a*e)^3*A*(e*x
+d)^(3/2)*a^2*d-123/4*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2*d^2+41/2*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A
*a*d^3+157/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a*d^2+41/2*e^6/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^3*d-35*e
^4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*d^2-105/4*e^4/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b
)^(1/2)*b)*A*a*d-131/3*e^3/b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*d^3+55/4*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*
a*d-53/2*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a^2*d+147/2*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*a^2*d-67*e^5/b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^3*d+83*e^4/b^3/(b*e*x+a*e)^3*B*(e*x+d)
^(3/2)*a^2*d^2-41/8*e^7/b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^4-41/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^4-35/
3*e^6/b^4/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a^3+35/3*e^3/b/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d^3+59/3*e^6/b^5/(b*e*x+a
*e)^3*B*(e*x+d)^(3/2)*a^4+105/8*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*a^2+
105/8*e^3/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*d^2-32*e^3/b^5*B*(e*x+d)^(1/2)
*a*d+63/4*e^2/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d^3+8*e^2/b/(b*e*x+a*e)^3*
B*(e*x+d)^(3/2)*d^4-17/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*d^3-15/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^5-
231/8*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^3+71/8*e^7/b^6/(b*e*x+a*e)^3
*(e*x+d)^(1/2)*B*a^5-55/8*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*a^2-55/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*d
^2+89/8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a^3

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^2,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 = 2.12, size = 790, normalized size = 2.38 \[ \left (\frac {\left (\frac {2\,A\,e^3-2\,B\,d\,e^2}{b^4}+\frac {2\,B\,e^2\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{b^8}\right )\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{b^4}-\frac {12\,B\,e^2\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-\frac {89\,B\,a^3\,b^2\,e^5}{8}+\frac {53\,B\,a^2\,b^3\,d\,e^4}{2}+\frac {55\,A\,a^2\,b^3\,e^5}{8}-\frac {157\,B\,a\,b^4\,d^2\,e^3}{8}-\frac {55\,A\,a\,b^4\,d\,e^4}{4}+\frac {17\,B\,b^5\,d^3\,e^2}{4}+\frac {55\,A\,b^5\,d^2\,e^3}{8}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {59\,B\,a^4\,b\,e^6}{3}-67\,B\,a^3\,b^2\,d\,e^5-\frac {35\,A\,a^3\,b^2\,e^6}{3}+83\,B\,a^2\,b^3\,d^2\,e^4+35\,A\,a^2\,b^3\,d\,e^5-\frac {131\,B\,a\,b^4\,d^3\,e^3}{3}-35\,A\,a\,b^4\,d^2\,e^4+8\,B\,b^5\,d^4\,e^2+\frac {35\,A\,b^5\,d^3\,e^3}{3}\right )+\sqrt {d+e\,x}\,\left (-\frac {71\,B\,a^5\,e^7}{8}+\frac {157\,B\,a^4\,b\,d\,e^6}{4}+\frac {41\,A\,a^4\,b\,e^7}{8}-\frac {273\,B\,a^3\,b^2\,d^2\,e^5}{4}-\frac {41\,A\,a^3\,b^2\,d\,e^6}{2}+58\,B\,a^2\,b^3\,d^3\,e^4+\frac {123\,A\,a^2\,b^3\,d^2\,e^5}{4}-\frac {191\,B\,a\,b^4\,d^4\,e^3}{8}-\frac {41\,A\,a\,b^4\,d^3\,e^4}{2}+\frac {15\,B\,b^5\,d^5\,e^2}{4}+\frac {41\,A\,b^5\,d^4\,e^3}{8}\right )}{b^9\,{\left (d+e\,x\right )}^3-\left (3\,b^9\,d-3\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^7\,e^2-6\,a\,b^8\,d\,e+3\,b^9\,d^2\right )-b^9\,d^3+a^3\,b^6\,e^3-3\,a^2\,b^7\,d\,e^2+3\,a\,b^8\,d^2\,e}+\left (\frac {2\,A\,e^3-2\,B\,d\,e^2}{3\,b^4}+\frac {2\,B\,e^2\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{3\,b^8}\right )\,{\left (d+e\,x\right )}^{3/2}+\frac {2\,B\,e^2\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {21\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e-11\,B\,a\,e+6\,B\,b\,d\right )}{-11\,B\,a^3\,e^5+28\,B\,a^2\,b\,d\,e^4+5\,A\,a^2\,b\,e^5-23\,B\,a\,b^2\,d^2\,e^3-10\,A\,a\,b^2\,d\,e^4+6\,B\,b^3\,d^3\,e^2+5\,A\,b^3\,d^2\,e^3}\right )\,{\left (a\,e-b\,d\right )}^{3/2}\,\left (5\,A\,b\,e-11\,B\,a\,e+6\,B\,b\,d\right )}{8\,b^{13/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(9/2))/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

((((2*A*e^3 - 2*B*d*e^2)/b^4 + (2*B*e^2*(4*b^4*d - 4*a*b^3*e))/b^8)*(4*b^4*d - 4*a*b^3*e))/b^4 - (12*B*e^2*(a*
e - b*d)^2)/b^6)*(d + e*x)^(1/2) - ((d + e*x)^(5/2)*((55*A*a^2*b^3*e^5)/8 - (89*B*a^3*b^2*e^5)/8 + (55*A*b^5*d
^2*e^3)/8 + (17*B*b^5*d^3*e^2)/4 - (157*B*a*b^4*d^2*e^3)/8 + (53*B*a^2*b^3*d*e^4)/2 - (55*A*a*b^4*d*e^4)/4) -
(d + e*x)^(3/2)*((59*B*a^4*b*e^6)/3 - (35*A*a^3*b^2*e^6)/3 + (35*A*b^5*d^3*e^3)/3 + 8*B*b^5*d^4*e^2 - 35*A*a*b
^4*d^2*e^4 + 35*A*a^2*b^3*d*e^5 - (131*B*a*b^4*d^3*e^3)/3 - 67*B*a^3*b^2*d*e^5 + 83*B*a^2*b^3*d^2*e^4) + (d +
e*x)^(1/2)*((41*A*a^4*b*e^7)/8 - (71*B*a^5*e^7)/8 + (41*A*b^5*d^4*e^3)/8 + (15*B*b^5*d^5*e^2)/4 - (41*A*a*b^4*
d^3*e^4)/2 - (41*A*a^3*b^2*d*e^6)/2 - (191*B*a*b^4*d^4*e^3)/8 + (123*A*a^2*b^3*d^2*e^5)/4 + 58*B*a^2*b^3*d^3*e
^4 - (273*B*a^3*b^2*d^2*e^5)/4 + (157*B*a^4*b*d*e^6)/4))/(b^9*(d + e*x)^3 - (3*b^9*d - 3*a*b^8*e)*(d + e*x)^2
+ (d + e*x)*(3*b^9*d^2 + 3*a^2*b^7*e^2 - 6*a*b^8*d*e) - b^9*d^3 + a^3*b^6*e^3 - 3*a^2*b^7*d*e^2 + 3*a*b^8*d^2*
e) + ((2*A*e^3 - 2*B*d*e^2)/(3*b^4) + (2*B*e^2*(4*b^4*d - 4*a*b^3*e))/(3*b^8))*(d + e*x)^(3/2) + (2*B*e^2*(d +
e*x)^(5/2))/(5*b^4) + (21*e^2*atan((b^(1/2)*e^2*(a*e - b*d)^(3/2)*(d + e*x)^(1/2)*(5*A*b*e - 11*B*a*e + 6*B*b
*d))/(5*A*a^2*b*e^5 - 11*B*a^3*e^5 + 5*A*b^3*d^2*e^3 + 6*B*b^3*d^3*e^2 - 23*B*a*b^2*d^2*e^3 - 10*A*a*b^2*d*e^4
+ 28*B*a^2*b*d*e^4))*(a*e - b*d)^(3/2)*(5*A*b*e - 11*B*a*e + 6*B*b*d))/(8*b^(13/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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