3.1826 \(\int \frac{(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=352 \[ -\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac{21 e^3 (d+e x)^{3/2} (-11 a B e+A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}+\frac{63 e^4 \sqrt{d+e x} (-11 a B e+A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac{63 e^4 (-11 a B e+A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2} \sqrt{b d-a e}}-\frac{(d+e x)^{9/2} (-11 a B e+A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+A b e+10 b B d)}{80 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{11/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

(63*e^4*(10*b*B*d + A*b*e - 11*a*B*e)*Sqrt[d + e*x])/(128*b^6*(b*d - a*e)) - (21*e^3*(10*b*B*d + A*b*e - 11*a*
B*e)*(d + e*x)^(3/2))/(128*b^5*(b*d - a*e)*(a + b*x)) - (21*e^2*(10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(5/2))
/(320*b^4*(b*d - a*e)*(a + b*x)^2) - (3*e*(10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(80*b^3*(b*d - a*e)*(
a + b*x)^3) - ((10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(40*b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b - a*B)*
(d + e*x)^(11/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (63*e^4*(10*b*B*d + A*b*e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[
d + e*x])/Sqrt[b*d - a*e]])/(128*b^(13/2)*Sqrt[b*d - a*e])

________________________________________________________________________________________

Rubi [A]  time = 0.295702, antiderivative size = 352, 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} \[ -\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac{21 e^3 (d+e x)^{3/2} (-11 a B e+A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}+\frac{63 e^4 \sqrt{d+e x} (-11 a B e+A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac{63 e^4 (-11 a B e+A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2} \sqrt{b d-a e}}-\frac{(d+e x)^{9/2} (-11 a B e+A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+A b e+10 b B d)}{80 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{11/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(63*e^4*(10*b*B*d + A*b*e - 11*a*B*e)*Sqrt[d + e*x])/(128*b^6*(b*d - a*e)) - (21*e^3*(10*b*B*d + A*b*e - 11*a*
B*e)*(d + e*x)^(3/2))/(128*b^5*(b*d - a*e)*(a + b*x)) - (21*e^2*(10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(5/2))
/(320*b^4*(b*d - a*e)*(a + b*x)^2) - (3*e*(10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(80*b^3*(b*d - a*e)*(
a + b*x)^3) - ((10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(40*b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b - a*B)*
(d + e*x)^(11/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (63*e^4*(10*b*B*d + A*b*e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[
d + e*x])/Sqrt[b*d - a*e]])/(128*b^(13/2)*Sqrt[b*d - a*e])

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 )^3} \, dx &=\int \frac{(A+B x) (d+e x)^{9/2}}{(a+b x)^6} \, dx\\ &=-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{(10 b B d+A b e-11 a B e) \int \frac{(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{(9 e (10 b B d+A b e-11 a B e)) \int \frac{(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac{3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (21 e^2 (10 b B d+A b e-11 a B e)\right ) \int \frac{(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3 (b d-a e)}\\ &=-\frac{21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac{3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (21 e^3 (10 b B d+A b e-11 a B e)\right ) \int \frac{(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac{21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac{21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac{3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (63 e^4 (10 b B d+A b e-11 a B e)\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{256 b^5 (b d-a e)}\\ &=\frac{63 e^4 (10 b B d+A b e-11 a B e) \sqrt{d+e x}}{128 b^6 (b d-a e)}-\frac{21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac{21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac{3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (63 e^4 (10 b B d+A b e-11 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^6}\\ &=\frac{63 e^4 (10 b B d+A b e-11 a B e) \sqrt{d+e x}}{128 b^6 (b d-a e)}-\frac{21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac{21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac{3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (63 e^3 (10 b B d+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 )}{128 b^6}\\ &=\frac{63 e^4 (10 b B d+A b e-11 a B e) \sqrt{d+e x}}{128 b^6 (b d-a e)}-\frac{21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac{21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac{3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}-\frac{63 e^4 (10 b B d+A b e-11 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2} \sqrt{b d-a e}}\\ \end{align*}

Mathematica [C]  time = 0.0842329, size = 99, normalized size = 0.28 \[ \frac{(d+e x)^{11/2} \left (\frac{11 (a B-A b)}{(a+b x)^5}-\frac{e^4 (-11 a B e+A b e+10 b B d) \, _2F_1\left (5,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}\right )}{55 b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((d + e*x)^(11/2)*((11*(-(A*b) + a*B))/(a + b*x)^5 - (e^4*(10*b*B*d + A*b*e - 11*a*B*e)*Hypergeometric2F1[5, 1
1/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(55*b*(b*d - a*e))

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Maple [B]  time = 0.032, size = 1173, normalized size = 3.3 \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)^3,x)

[Out]

437/128*e^9/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5-693/128*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))*a*B+315/64*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*d-325
/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d-22*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^3-187/64*e^4/b/(b*e*x+a*e)^
5*(e*x+d)^(1/2)*B*d^5+545/32*e^4/b/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2-63/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*A*a^4+147/64*e^5/b/(b*e*x+a*e)^5*A*(e*x+d)^(3/2)*d^3+977/64*e^8/b^5/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*a^4+843/12
8*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a-237/64*e^6/b^2/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a+237/64*e^5/b/(b*e*x+a
*e)^5*A*(e*x+d)^(7/2)*d+415/32*e^4/b/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*d^4-63/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*A*d^4+131/5*e^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3-21/5*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2-21/5*e^
5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^2+1327/64*e^6/b^3/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2-147/64*e^8/b^4/(b*e*x+
a*e)^5*A*(e*x+d)^(3/2)*a^3+2*e^4*B/b^6*(e*x+d)^(1/2)-193/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A+63/128*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-1061/64*e^8/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*B*a^4*d+2059/64*e^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*d^2-499/16*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a
^2*d^3-3761/64*e^7/b^4/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*a^3*d+42/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a*d+5421
/64*e^6/b^3/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*a^2*d^2-3467/64*e^5/b^2/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*a*d^3+63/32*e^
8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d+1933/128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^4-2417/64*e^5/b^2
/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d+441/64*e^7/b^3/(b*e*x+a*e)^5*A*(e*x+d)^(3/2)*a^2*d-441/64*e^6/b^2/(b*e*x+a*
e)^5*A*(e*x+d)^(3/2)*a*d^2-372/5*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d+351/5*e^5/b^2/(b*e*x+a*e)^5*(e*x+
d)^(5/2)*B*a*d^2-189/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^2+63/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*A*a*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.61933, size = 4158, normalized size = 11.81 \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)^3,x, algorithm="fricas")

[Out]

[1/1280*(315*(10*B*a^5*b*d*e^4 - (11*B*a^6 - A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (11*B*a*b^5 - A*b^6)*e^5)*x^5 +
5*(10*B*a*b^5*d*e^4 - (11*B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (11*B*a^3*b^3 - A*a^2*b^4)*
e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (11*B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (11*B*a^5*b
- A*a^4*b^2)*e^5)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x
+ a)) - 2*(32*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(4*B*a^2*b^5 + A*a*b^6)*d^4*e + 12*(13*B*a^3*b^4 + 2*A*a^2*b^5)*d^3
*e^2 + 42*(14*B*a^4*b^3 + A*a^3*b^4)*d^2*e^3 - 105*(41*B*a^5*b^2 - A*a^4*b^3)*d*e^4 + 315*(11*B*a^6*b - A*a^5*
b^2)*e^5 - 1280*(B*b^7*d*e^4 - B*a*b^6*e^5)*x^5 + 5*(650*B*b^7*d^2*e^3 - (2773*B*a*b^6 - 193*A*b^7)*d*e^4 + 19
3*(11*B*a^2*b^5 - A*a*b^6)*e^5)*x^4 + 10*(210*B*b^7*d^3*e^2 + (521*B*a*b^6 + 149*A*b^7)*d^2*e^3 - 2*(1669*B*a^
2*b^5 - 44*A*a*b^6)*d*e^4 + 237*(11*B*a^3*b^4 - A*a^2*b^5)*e^5)*x^3 + 2*(440*B*b^7*d^4*e + 2*(353*B*a*b^6 + 34
2*A*b^7)*d^3*e^2 + 3*(919*B*a^2*b^5 + 61*A*a*b^6)*d^2*e^3 - 3*(6229*B*a^3*b^4 - 159*A*a^2*b^5)*d*e^4 + 1344*(1
1*B*a^4*b^3 - A*a^3*b^4)*e^5)*x^2 + 2*(80*B*b^7*d^5 + 8*(19*B*a*b^6 + 41*A*b^7)*d^4*e + 2*(187*B*a^2*b^5 + 28*
A*a*b^6)*d^3*e^2 + 9*(159*B*a^3*b^4 + 11*A*a^2*b^5)*d^2*e^3 - 42*(241*B*a^4*b^3 - 6*A*a^3*b^4)*d*e^4 + 735*(11
*B*a^5*b^2 - A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d - a^6*b^7*e + (b^13*d - a*b^12*e)*x^5 + 5*(a*b^12*d
- a^2*b^11*e)*x^4 + 10*(a^2*b^11*d - a^3*b^10*e)*x^3 + 10*(a^3*b^10*d - a^4*b^9*e)*x^2 + 5*(a^4*b^9*d - a^5*b^
8*e)*x), 1/640*(315*(10*B*a^5*b*d*e^4 - (11*B*a^6 - A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (11*B*a*b^5 - A*b^6)*e^5)
*x^5 + 5*(10*B*a*b^5*d*e^4 - (11*B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (11*B*a^3*b^3 - A*a^
2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (11*B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (11*B
*a^5*b - A*a^4*b^2)*e^5)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (3
2*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(4*B*a^2*b^5 + A*a*b^6)*d^4*e + 12*(13*B*a^3*b^4 + 2*A*a^2*b^5)*d^3*e^2 + 42*(1
4*B*a^4*b^3 + A*a^3*b^4)*d^2*e^3 - 105*(41*B*a^5*b^2 - A*a^4*b^3)*d*e^4 + 315*(11*B*a^6*b - A*a^5*b^2)*e^5 - 1
280*(B*b^7*d*e^4 - B*a*b^6*e^5)*x^5 + 5*(650*B*b^7*d^2*e^3 - (2773*B*a*b^6 - 193*A*b^7)*d*e^4 + 193*(11*B*a^2*
b^5 - A*a*b^6)*e^5)*x^4 + 10*(210*B*b^7*d^3*e^2 + (521*B*a*b^6 + 149*A*b^7)*d^2*e^3 - 2*(1669*B*a^2*b^5 - 44*A
*a*b^6)*d*e^4 + 237*(11*B*a^3*b^4 - A*a^2*b^5)*e^5)*x^3 + 2*(440*B*b^7*d^4*e + 2*(353*B*a*b^6 + 342*A*b^7)*d^3
*e^2 + 3*(919*B*a^2*b^5 + 61*A*a*b^6)*d^2*e^3 - 3*(6229*B*a^3*b^4 - 159*A*a^2*b^5)*d*e^4 + 1344*(11*B*a^4*b^3
- A*a^3*b^4)*e^5)*x^2 + 2*(80*B*b^7*d^5 + 8*(19*B*a*b^6 + 41*A*b^7)*d^4*e + 2*(187*B*a^2*b^5 + 28*A*a*b^6)*d^3
*e^2 + 9*(159*B*a^3*b^4 + 11*A*a^2*b^5)*d^2*e^3 - 42*(241*B*a^4*b^3 - 6*A*a^3*b^4)*d*e^4 + 735*(11*B*a^5*b^2 -
A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d - a^6*b^7*e + (b^13*d - a*b^12*e)*x^5 + 5*(a*b^12*d - a^2*b^11*e
)*x^4 + 10*(a^2*b^11*d - a^3*b^10*e)*x^3 + 10*(a^3*b^10*d - a^4*b^9*e)*x^2 + 5*(a^4*b^9*d - a^5*b^8*e)*x)]

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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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.25043, size = 1040, normalized size = 2.95 \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)^3,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^4/b^6 + 63/128*(10*B*b*d*e^4 - 11*B*a*e^5 + A*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) - 1/640*(3250*(x*e + d)^(9/2)*B*b^5*d*e^4 - 10900*(x*e + d)^(7/2)*B*b^5*d^2
*e^4 + 14080*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 8300*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 1870*sqrt(x*e + d)*B*b^5*d^5
*e^4 - 4215*(x*e + d)^(9/2)*B*a*b^4*e^5 + 965*(x*e + d)^(9/2)*A*b^5*e^5 + 24170*(x*e + d)^(7/2)*B*a*b^4*d*e^5
- 2370*(x*e + d)^(7/2)*A*b^5*d*e^5 - 44928*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 + 2688*(x*e + d)^(5/2)*A*b^5*d^2*e^
5 + 34670*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 - 1470*(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 9665*sqrt(x*e + d)*B*a*b^4*d^
4*e^5 + 315*sqrt(x*e + d)*A*b^5*d^4*e^5 - 13270*(x*e + d)^(7/2)*B*a^2*b^3*e^6 + 2370*(x*e + d)^(7/2)*A*a*b^4*e
^6 + 47616*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 - 5376*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 54210*(x*e + d)^(3/2)*B*a^2*
b^3*d^2*e^6 + 4410*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 + 19960*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 - 1260*sqrt(x*e + d
)*A*a*b^4*d^3*e^6 - 16768*(x*e + d)^(5/2)*B*a^3*b^2*e^7 + 2688*(x*e + d)^(5/2)*A*a^2*b^3*e^7 + 37610*(x*e + d)
^(3/2)*B*a^3*b^2*d*e^7 - 4410*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 - 20590*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 + 1890*s
qrt(x*e + d)*A*a^2*b^3*d^2*e^7 - 9770*(x*e + d)^(3/2)*B*a^4*b*e^8 + 1470*(x*e + d)^(3/2)*A*a^3*b^2*e^8 + 10610
*sqrt(x*e + d)*B*a^4*b*d*e^8 - 1260*sqrt(x*e + d)*A*a^3*b^2*d*e^8 - 2185*sqrt(x*e + d)*B*a^5*e^9 + 315*sqrt(x*
e + d)*A*a^4*b*e^9)/(((x*e + d)*b - b*d + a*e)^5*b^6)