3.1825 \(\int \frac{(A+B x) (d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=393 \[ -\frac{33 e^2 (d+e x)^{7/2} (-13 a B e+3 A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac{231 e^3 (d+e x)^{5/2} (-13 a B e+3 A b e+10 b B d)}{640 b^5 (a+b x) (b d-a e)}+\frac{77 e^4 (d+e x)^{3/2} (-13 a B e+3 A b e+10 b B d)}{128 b^6 (b d-a e)}+\frac{231 e^4 \sqrt{d+e x} (-13 a B e+3 A b e+10 b B d)}{128 b^7}-\frac{231 e^4 \sqrt{b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}-\frac{(d+e x)^{11/2} (-13 a B e+3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{11 e (d+e x)^{9/2} (-13 a B e+3 A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
[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))
________________________________________________________________________________________
Rubi [A] time = 0.364393, antiderivative size = 393, normalized size of antiderivative = 1.,
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} \[ -\frac{33 e^2 (d+e x)^{7/2} (-13 a B e+3 A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac{231 e^3 (d+e x)^{5/2} (-13 a B e+3 A b e+10 b B d)}{640 b^5 (a+b x) (b d-a e)}+\frac{77 e^4 (d+e x)^{3/2} (-13 a B e+3 A b e+10 b B d)}{128 b^6 (b d-a e)}+\frac{231 e^4 \sqrt{d+e x} (-13 a B e+3 A b e+10 b B d)}{128 b^7}-\frac{231 e^4 \sqrt{b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}-\frac{(d+e x)^{11/2} (-13 a B e+3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{11 e (d+e x)^{9/2} (-13 a B e+3 A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
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 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)^{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*}
Mathematica [C] time = 0.0927383, size = 100, normalized size = 0.25 \[ \frac{(d+e x)^{13/2} \left (\frac{13 (a B-A b)}{(a+b x)^5}-\frac{e^4 (-13 a B e+3 A b e+10 b B d) \, _2F_1\left (5,\frac{13}{2};\frac{15}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}\right )}{65 b (b d-a e)} \]
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))
________________________________________________________________________________________
Maple [B] time = 0.05, size = 1633, normalized size = 4.2 \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)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
-12131/192*e^7/b^4/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^3+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+131/5*e^8/b^4/(b*e*x+a
*e)^5*(e*x+d)^(5/2)*A*a^3-131/5*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+977/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^4-96
29/192*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+4075/96*e^4/b
/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^3-693/128*e^6/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1
/2))*A*a+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+
1327/64*e^5/b/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d^2+693/128*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*d+3003/128*e^6/b^7/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^2*B-1
467/128*e^10/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^6+1155/64*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^2-765/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d^2-172/3*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+2*e^5
/b^6*A*(e*x+d)^(1/2)+2/3*e^4/b^6*B*(e*x+d)^(3/2)+10*e^4/b^6*B*d*(e*x+d)^(1/2)+4619/15*e^7/b^4/(b*e*x+a*e)^5*(e
*x+d)^(5/2)*B*a^3*d-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^2-977/16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^3+3903/128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)
*B*a*d-393/5*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2*d+3833/15*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^3-2
113/5*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d^2-17635/128*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^4+22
607/96*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^3-5313/128*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))*B*d*a-36421/192*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*d^4+6617/128*e^5/b^2/(b*e*x+a*e)^
5*(e*x+d)^(1/2)*B*a*d^5+12485/64*e^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*d^3-4955/32*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-2185/128*e^9/b^5/(b*e*x+a*e)^5*(e*
x+d)^(1/2)*A*a^4*d+2185/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^4-2185/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^
(1/2)*A*a^2*d^3+2185/64*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d^2+393/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2
)*A*a*d^2-1327/32*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-9477/64*e^5/b^2/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d^2-12*e^5/b^7*a*B*(e*x+d)^(1/2)
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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)^(11/2)/(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 = 1.59925, size = 4145, normalized size = 10.55 \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)^(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)]
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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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
Timed out
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Giac [B] time = 1.28621, size = 1439, normalized size = 3.66 \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)^(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