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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.333252, antiderivative size = 332, normalized size of antiderivative = 1., 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 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 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{(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*}

Mathematica [C]  time = 0.115489, size = 100, normalized size = 0.3 \[ \frac{(d+e x)^{11/2} \left (\frac{11 (a B-A b)}{(a+b x)^3}-\frac{e^2 (-11 a B e+5 A b e+6 b B d) \, _2F_1\left (3,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}\right )}{33 b (b d-a e)} \]

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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Maple [B]  time = 0.033, size = 1285, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

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]

147/2*e^4/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^2*d-483/8*e^3/b^4/((a*e-b*d)
*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a*d^2+59/3*e^6/b^5/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^4-1
7/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*d^3+63/4*e^2/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)
*b)^(1/2))*B*d^3+105/8*e^5/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a^2+105/8*e^3
/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*d^2+35/3*e^3/b/(b*e*x+a*e)^3*A*(e*x+d)^
(3/2)*d^3-41/8*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^3-231/8*e^5/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^
(1/2))*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+8*e^2/b/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^4-15/4*e^2/b/(b*e*x+a*e)^3*(
e*x+d)^(1/2)*B*d^5-32*e^3/b^5*B*a*d*(e*x+d)^(1/2)+89/8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a^3+2/3*e^3/b^4*A
*(e*x+d)^(3/2)+2/5*e^2/b^4*B*(e*x+d)^(5/2)+12*e^2/b^4*B*d^2*(e*x+d)^(1/2)+8*e^3/b^4*A*d*(e*x+d)^(1/2)+20*e^4/b
^6*a^2*B*(e*x+d)^(1/2)-8/3*e^3/b^5*B*(e*x+d)^(3/2)*a-8*e^4/b^5*A*a*(e*x+d)^(1/2)+2*e^2/b^4*B*(e*x+d)^(3/2)*d-1
57/4*e^6/b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^4*d+273/4*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3*d^2-123/4*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+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+157/8*e^3/b^2/(b*e*x+a*e)^3*(e*x
+d)^(5/2)*B*a*d^2-35*e^4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*d^2-131/3*e^3/b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a
*d^3+191/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^4+41/2*e^6/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^3*d+83*e^4
/b^3/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2*d^2-67*e^5/b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^3*d-105/4*e^4/b^4/((a*e-
b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a*d+35*e^5/b^3/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a^2*d
-58*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 1.39067, size = 3258, normalized size = 9.81 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.247, size = 1119, normalized size = 3.37 \begin{align*} \text{result too large to display} \end{align*}

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