3.17.54 \(\int \frac {(A+B x) (d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=557 \[ -\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{11/2} (-13 a B e+5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {231 e^3 (a+b x) (b d-a e)^{3/2} (-13 a B e+5 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (a+b x) \sqrt {d+e x} (b d-a e) (-13 a B e+5 A b e+8 b B d)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 e^3 (a+b x) (d+e x)^{3/2} (-13 a B e+5 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (a+b x) (d+e x)^{5/2} (-13 a B e+5 A b e+8 b B d)}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {33 e^2 (d+e x)^{7/2} (-13 a B e+5 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {11 e (d+e x)^{9/2} (-13 a B e+5 A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

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Rubi [A]  time = 0.52, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {770, 78, 47, 50, 63, 208} \begin {gather*} -\frac {33 e^2 (d+e x)^{7/2} (-13 a B e+5 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {231 e^3 (a+b x) (d+e x)^{5/2} (-13 a B e+5 A b e+8 b B d)}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {77 e^3 (a+b x) (d+e x)^{3/2} (-13 a B e+5 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (a+b x) \sqrt {d+e x} (b d-a e) (-13 a B e+5 A b e+8 b B d)}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x) (b d-a e)^{3/2} (-13 a B e+5 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{11/2} (-13 a B e+5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {11 e (d+e x)^{9/2} (-13 a B e+5 A b e+8 b B d)}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (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)^(5/2),x]

[Out]

(231*e^3*(b*d - a*e)*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) + (77*e^3*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) + (231*e^3*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(a + b*x)*(d + e*x)^(5/2))/(320*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2]) - (33*e^2*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(d + e*x)^(7/2))/(64*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (11*e*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(d + e*x)^(9/2))/(96*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) - ((8*b*B*d + 5*A*b*e - 13*a*B*e)*(d + e*x)^(11/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(13/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - (231*e^3*(b*d - a*e)^(3/2)*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[b*d - a*e]])/(64*b^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

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 )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{11/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (11 e (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 e^2 (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{128 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{5/2}}{320 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 \left (b^2 d-a b e\right ) (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {77 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{5/2}}{320 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 \left (b^2 d-a b e\right )^2 (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3 (b d-a e) (8 b B d+5 A b e-13 a B e) (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{5/2}}{320 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 \left (b^2 d-a b e\right )^3 (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^{10} (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3 (b d-a e) (8 b B d+5 A b e-13 a B e) (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{5/2}}{320 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^2 \left (b^2 d-a b e\right )^3 (8 b B d+5 A b e-13 a B e) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^{10} (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3 (b d-a e) (8 b B d+5 A b e-13 a B e) (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{5/2}}{320 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (b d-a e)^{3/2} (8 b B d+5 A b e-13 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.21, size = 117, normalized size = 0.21 \begin {gather*} \frac {(d+e x)^{13/2} \left (\frac {e^3 (a+b x)^4 (-13 a B e+5 A b e+8 b B d) \, _2F_1\left (4,\frac {13}{2};\frac {15}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+13 (a B-A b)\right )}{52 b (a+b x)^3 \sqrt {(a+b x)^2} (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)^(5/2),x]

[Out]

((d + e*x)^(13/2)*(13*(-(A*b) + a*B) + (e^3*(8*b*B*d + 5*A*b*e - 13*a*B*e)*(a + b*x)^4*Hypergeometric2F1[4, 13
/2, 15/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(52*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [B]  time = 69.41, size = 1196, normalized size = 2.15 \begin {gather*} \frac {(-a e-b x e) \left (\frac {231 \left (5 A b e^4-13 a B e^4+8 b B d e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {a e-b d} \sqrt {d+e x}}{b d-a e}\right ) (b d-a e)^2}{64 b^{15/2} \sqrt {a e-b d}}+\frac {17325 a^5 A b \sqrt {d+e x} e^9-45045 a^6 B \sqrt {d+e x} e^9+63525 a^4 A b^2 (d+e x)^{3/2} e^8-165165 a^5 b B (d+e x)^{3/2} e^8-86625 a^4 A b^2 d \sqrt {d+e x} e^8+252945 a^5 b B d \sqrt {d+e x} e^8+84315 a^3 A b^3 (d+e x)^{5/2} e^7-219219 a^4 b^2 B (d+e x)^{5/2} e^7-254100 a^3 A b^3 d (d+e x)^{3/2} e^7+762300 a^4 b^2 B d (d+e x)^{3/2} e^7+173250 a^3 A b^3 d^2 \sqrt {d+e x} e^7-589050 a^4 b^2 B d^2 \sqrt {d+e x} e^7+46035 a^2 A b^4 (d+e x)^{7/2} e^6-119691 a^3 b^3 B (d+e x)^{7/2} e^6-252945 a^2 A b^4 d (d+e x)^{5/2} e^6+792561 a^3 b^3 B d (d+e x)^{5/2} e^6+381150 a^2 A b^4 d^2 (d+e x)^{3/2} e^6-1397550 a^3 b^3 B d^2 (d+e x)^{3/2} e^6-173250 a^2 A b^4 d^3 \sqrt {d+e x} e^6+727650 a^3 b^3 B d^3 \sqrt {d+e x} e^6+7040 a A b^5 (d+e x)^{9/2} e^5-18304 a^2 b^4 B (d+e x)^{9/2} e^5-92070 a A b^5 d (d+e x)^{7/2} e^5+313038 a^2 b^4 B d (d+e x)^{7/2} e^5+252945 a A b^5 d^2 (d+e x)^{5/2} e^5-1062369 a^2 b^4 B d^2 (d+e x)^{5/2} e^5-254100 a A b^5 d^3 (d+e x)^{3/2} e^5+1270500 a^2 b^4 B d^3 (d+e x)^{3/2} e^5+86625 a A b^5 d^4 \sqrt {d+e x} e^5-502425 a^2 b^4 B d^4 \sqrt {d+e x} e^5-640 A b^6 (d+e x)^{11/2} e^4+1664 a b^5 B (d+e x)^{11/2} e^4-7040 A b^6 d (d+e x)^{9/2} e^4+29568 a b^5 B d (d+e x)^{9/2} e^4+46035 A b^6 d^2 (d+e x)^{7/2} e^4-267003 a b^5 B d^2 (d+e x)^{7/2} e^4-84315 A b^6 d^3 (d+e x)^{5/2} e^4+623931 a b^5 B d^3 (d+e x)^{5/2} e^4+63525 A b^6 d^4 (d+e x)^{3/2} e^4-571725 a b^5 B d^4 (d+e x)^{3/2} e^4-17325 A b^6 d^5 \sqrt {d+e x} e^4+183645 a b^5 B d^5 \sqrt {d+e x} e^4-384 b^6 B (d+e x)^{13/2} e^3-1024 b^6 B d (d+e x)^{11/2} e^3-11264 b^6 B d^2 (d+e x)^{9/2} e^3+73656 b^6 B d^3 (d+e x)^{7/2} e^3-134904 b^6 B d^4 (d+e x)^{5/2} e^3+101640 b^6 B d^5 (d+e x)^{3/2} e^3-27720 b^6 B d^6 \sqrt {d+e x} e^3}{960 b^7 (b d-a e-b (d+e x))^4}\right )}{e \sqrt {\frac {(a e+b x e)^2}{e^2}}} \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)^(5/2),x]

[Out]

((-(a*e) - b*e*x)*((-27720*b^6*B*d^6*e^3*Sqrt[d + e*x] - 17325*A*b^6*d^5*e^4*Sqrt[d + e*x] + 183645*a*b^5*B*d^
5*e^4*Sqrt[d + e*x] + 86625*a*A*b^5*d^4*e^5*Sqrt[d + e*x] - 502425*a^2*b^4*B*d^4*e^5*Sqrt[d + e*x] - 173250*a^
2*A*b^4*d^3*e^6*Sqrt[d + e*x] + 727650*a^3*b^3*B*d^3*e^6*Sqrt[d + e*x] + 173250*a^3*A*b^3*d^2*e^7*Sqrt[d + e*x
] - 589050*a^4*b^2*B*d^2*e^7*Sqrt[d + e*x] - 86625*a^4*A*b^2*d*e^8*Sqrt[d + e*x] + 252945*a^5*b*B*d*e^8*Sqrt[d
+ e*x] + 17325*a^5*A*b*e^9*Sqrt[d + e*x] - 45045*a^6*B*e^9*Sqrt[d + e*x] + 101640*b^6*B*d^5*e^3*(d + e*x)^(3/
2) + 63525*A*b^6*d^4*e^4*(d + e*x)^(3/2) - 571725*a*b^5*B*d^4*e^4*(d + e*x)^(3/2) - 254100*a*A*b^5*d^3*e^5*(d
+ e*x)^(3/2) + 1270500*a^2*b^4*B*d^3*e^5*(d + e*x)^(3/2) + 381150*a^2*A*b^4*d^2*e^6*(d + e*x)^(3/2) - 1397550*
a^3*b^3*B*d^2*e^6*(d + e*x)^(3/2) - 254100*a^3*A*b^3*d*e^7*(d + e*x)^(3/2) + 762300*a^4*b^2*B*d*e^7*(d + e*x)^
(3/2) + 63525*a^4*A*b^2*e^8*(d + e*x)^(3/2) - 165165*a^5*b*B*e^8*(d + e*x)^(3/2) - 134904*b^6*B*d^4*e^3*(d + e
*x)^(5/2) - 84315*A*b^6*d^3*e^4*(d + e*x)^(5/2) + 623931*a*b^5*B*d^3*e^4*(d + e*x)^(5/2) + 252945*a*A*b^5*d^2*
e^5*(d + e*x)^(5/2) - 1062369*a^2*b^4*B*d^2*e^5*(d + e*x)^(5/2) - 252945*a^2*A*b^4*d*e^6*(d + e*x)^(5/2) + 792
561*a^3*b^3*B*d*e^6*(d + e*x)^(5/2) + 84315*a^3*A*b^3*e^7*(d + e*x)^(5/2) - 219219*a^4*b^2*B*e^7*(d + e*x)^(5/
2) + 73656*b^6*B*d^3*e^3*(d + e*x)^(7/2) + 46035*A*b^6*d^2*e^4*(d + e*x)^(7/2) - 267003*a*b^5*B*d^2*e^4*(d + e
*x)^(7/2) - 92070*a*A*b^5*d*e^5*(d + e*x)^(7/2) + 313038*a^2*b^4*B*d*e^5*(d + e*x)^(7/2) + 46035*a^2*A*b^4*e^6
*(d + e*x)^(7/2) - 119691*a^3*b^3*B*e^6*(d + e*x)^(7/2) - 11264*b^6*B*d^2*e^3*(d + e*x)^(9/2) - 7040*A*b^6*d*e
^4*(d + e*x)^(9/2) + 29568*a*b^5*B*d*e^4*(d + e*x)^(9/2) + 7040*a*A*b^5*e^5*(d + e*x)^(9/2) - 18304*a^2*b^4*B*
e^5*(d + e*x)^(9/2) - 1024*b^6*B*d*e^3*(d + e*x)^(11/2) - 640*A*b^6*e^4*(d + e*x)^(11/2) + 1664*a*b^5*B*e^4*(d
+ e*x)^(11/2) - 384*b^6*B*e^3*(d + e*x)^(13/2))/(960*b^7*(b*d - a*e - b*(d + e*x))^4) + (231*(b*d - a*e)^2*(8
*b*B*d*e^3 + 5*A*b*e^4 - 13*a*B*e^4)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*b^(15
/2)*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.47, size = 2006, normalized size = 3.60

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)^(5/2),x, algorithm="fricas")

[Out]

[1/1920*(3465*(8*B*a^4*b^2*d^2*e^3 - (21*B*a^5*b - 5*A*a^4*b^2)*d*e^4 + (13*B*a^6 - 5*A*a^5*b)*e^5 + (8*B*b^6*
d^2*e^3 - (21*B*a*b^5 - 5*A*b^6)*d*e^4 + (13*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 + 4*(8*B*a*b^5*d^2*e^3 - (21*B*a^
2*b^4 - 5*A*a*b^5)*d*e^4 + (13*B*a^3*b^3 - 5*A*a^2*b^4)*e^5)*x^3 + 6*(8*B*a^2*b^4*d^2*e^3 - (21*B*a^3*b^3 - 5*
A*a^2*b^4)*d*e^4 + (13*B*a^4*b^2 - 5*A*a^3*b^3)*e^5)*x^2 + 4*(8*B*a^3*b^3*d^2*e^3 - (21*B*a^4*b^2 - 5*A*a^3*b^
3)*d*e^4 + (13*B*a^5*b - 5*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*(384*B*b^6*e^5*x^6 - 80*(B*a*b^5 + 3*A*b^6)*d^5 - 440*(B*a^2*b^4 + A*a*b^
5)*d^4*e - 990*(3*B*a^3*b^3 + A*a^2*b^4)*d^3*e^2 + 231*(199*B*a^4*b^2 - 15*A*a^3*b^3)*d^2*e^3 - 4620*(19*B*a^5
*b - 5*A*a^4*b^2)*d*e^4 + 3465*(13*B*a^6 - 5*A*a^5*b)*e^5 + 128*(26*B*b^6*d*e^4 - (13*B*a*b^5 - 5*A*b^6)*e^5)*
x^5 + 128*(173*B*b^6*d^2*e^3 - 8*(37*B*a*b^5 - 10*A*b^6)*d*e^4 + 11*(13*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 - (106
80*B*b^6*d^3*e^2 - (132091*B*a*b^5 - 11475*A*b^6)*d^2*e^3 + 22*(10901*B*a^2*b^4 - 2905*A*a*b^5)*d*e^4 - 9207*(
13*B*a^3*b^3 - 5*A*a^2*b^4)*e^5)*x^3 - (2480*B*b^6*d^4*e + 10*(1697*B*a*b^5 + 515*A*b^6)*d^3*e^2 - 33*(7063*B*
a^2*b^4 - 575*A*a*b^5)*d^2*e^3 + 264*(1642*B*a^3*b^3 - 435*A*a^2*b^4)*d*e^4 - 16863*(13*B*a^4*b^2 - 5*A*a^3*b^
3)*e^5)*x^2 - (320*B*b^6*d^5 + 40*(43*B*a*b^5 + 41*A*b^6)*d^4*e + 220*(53*B*a^2*b^4 + 17*A*a*b^5)*d^3*e^2 - 33
*(5197*B*a^3*b^3 - 405*A*a^2*b^4)*d^2*e^3 + 462*(701*B*a^4*b^2 - 185*A*a^3*b^3)*d*e^4 - 12705*(13*B*a^5*b - 5*
A*a^4*b^2)*e^5)*x)*sqrt(e*x + d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7), -1/960*(3
465*(8*B*a^4*b^2*d^2*e^3 - (21*B*a^5*b - 5*A*a^4*b^2)*d*e^4 + (13*B*a^6 - 5*A*a^5*b)*e^5 + (8*B*b^6*d^2*e^3 -
(21*B*a*b^5 - 5*A*b^6)*d*e^4 + (13*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 + 4*(8*B*a*b^5*d^2*e^3 - (21*B*a^2*b^4 - 5*
A*a*b^5)*d*e^4 + (13*B*a^3*b^3 - 5*A*a^2*b^4)*e^5)*x^3 + 6*(8*B*a^2*b^4*d^2*e^3 - (21*B*a^3*b^3 - 5*A*a^2*b^4)
*d*e^4 + (13*B*a^4*b^2 - 5*A*a^3*b^3)*e^5)*x^2 + 4*(8*B*a^3*b^3*d^2*e^3 - (21*B*a^4*b^2 - 5*A*a^3*b^3)*d*e^4 +
(13*B*a^5*b - 5*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)) - (384*B*b^6*e^5*x^6 - 80*(B*a*b^5 + 3*A*b^6)*d^5 - 440*(B*a^2*b^4 + A*a*b^5)*d^4*e - 990*(3*B*a^3*b^3 +
A*a^2*b^4)*d^3*e^2 + 231*(199*B*a^4*b^2 - 15*A*a^3*b^3)*d^2*e^3 - 4620*(19*B*a^5*b - 5*A*a^4*b^2)*d*e^4 + 3465
*(13*B*a^6 - 5*A*a^5*b)*e^5 + 128*(26*B*b^6*d*e^4 - (13*B*a*b^5 - 5*A*b^6)*e^5)*x^5 + 128*(173*B*b^6*d^2*e^3 -
8*(37*B*a*b^5 - 10*A*b^6)*d*e^4 + 11*(13*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 - (10680*B*b^6*d^3*e^2 - (132091*B*a
*b^5 - 11475*A*b^6)*d^2*e^3 + 22*(10901*B*a^2*b^4 - 2905*A*a*b^5)*d*e^4 - 9207*(13*B*a^3*b^3 - 5*A*a^2*b^4)*e^
5)*x^3 - (2480*B*b^6*d^4*e + 10*(1697*B*a*b^5 + 515*A*b^6)*d^3*e^2 - 33*(7063*B*a^2*b^4 - 575*A*a*b^5)*d^2*e^3
+ 264*(1642*B*a^3*b^3 - 435*A*a^2*b^4)*d*e^4 - 16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)*e^5)*x^2 - (320*B*b^6*d^5 +
40*(43*B*a*b^5 + 41*A*b^6)*d^4*e + 220*(53*B*a^2*b^4 + 17*A*a*b^5)*d^3*e^2 - 33*(5197*B*a^3*b^3 - 405*A*a^2*b
^4)*d^2*e^3 + 462*(701*B*a^4*b^2 - 185*A*a^3*b^3)*d*e^4 - 12705*(13*B*a^5*b - 5*A*a^4*b^2)*e^5)*x)*sqrt(e*x +
d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)]

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giac [B]  time = 0.61, size = 1167, normalized size = 2.10

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)^(5/2),x, algorithm="giac")

[Out]

231/64*(8*B*b^3*d^3*e^3 - 29*B*a*b^2*d^2*e^4 + 5*A*b^3*d^2*e^4 + 34*B*a^2*b*d*e^5 - 10*A*a*b^2*d*e^5 - 13*B*a^
3*e^6 + 5*A*a^2*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*sgn((x*e + d)*b*
e - b*d*e + a*e^2)) - 1/192*(2136*(x*e + d)^(7/2)*B*b^6*d^3*e^3 - 5912*(x*e + d)^(5/2)*B*b^6*d^4*e^3 + 5480*(x
*e + d)^(3/2)*B*b^6*d^5*e^3 - 1704*sqrt(x*e + d)*B*b^6*d^6*e^3 - 8703*(x*e + d)^(7/2)*B*a*b^5*d^2*e^4 + 2295*(
x*e + d)^(7/2)*A*b^6*d^2*e^4 + 29503*(x*e + d)^(5/2)*B*a*b^5*d^3*e^4 - 5855*(x*e + d)^(5/2)*A*b^6*d^3*e^4 - 32
553*(x*e + d)^(3/2)*B*a*b^5*d^4*e^4 + 5153*(x*e + d)^(3/2)*A*b^6*d^4*e^4 + 11769*sqrt(x*e + d)*B*a*b^5*d^5*e^4
- 1545*sqrt(x*e + d)*A*b^6*d^5*e^4 + 10998*(x*e + d)^(7/2)*B*a^2*b^4*d*e^5 - 4590*(x*e + d)^(7/2)*A*a*b^5*d*e
^5 - 53037*(x*e + d)^(5/2)*B*a^2*b^4*d^2*e^5 + 17565*(x*e + d)^(5/2)*A*a*b^5*d^2*e^5 + 75412*(x*e + d)^(3/2)*B
*a^2*b^4*d^3*e^5 - 20612*(x*e + d)^(3/2)*A*a*b^5*d^3*e^5 - 33285*sqrt(x*e + d)*B*a^2*b^4*d^4*e^5 + 7725*sqrt(x
*e + d)*A*a*b^5*d^4*e^5 - 4431*(x*e + d)^(7/2)*B*a^3*b^3*e^6 + 2295*(x*e + d)^(7/2)*A*a^2*b^4*e^6 + 41213*(x*e
+ d)^(5/2)*B*a^3*b^3*d*e^6 - 17565*(x*e + d)^(5/2)*A*a^2*b^4*d*e^6 - 85718*(x*e + d)^(3/2)*B*a^3*b^3*d^2*e^6
+ 30918*(x*e + d)^(3/2)*A*a^2*b^4*d^2*e^6 + 49530*sqrt(x*e + d)*B*a^3*b^3*d^3*e^6 - 15450*sqrt(x*e + d)*A*a^2*
b^4*d^3*e^6 - 11767*(x*e + d)^(5/2)*B*a^4*b^2*e^7 + 5855*(x*e + d)^(5/2)*A*a^3*b^3*e^7 + 48012*(x*e + d)^(3/2)
*B*a^4*b^2*d*e^7 - 20612*(x*e + d)^(3/2)*A*a^3*b^3*d*e^7 - 41010*sqrt(x*e + d)*B*a^4*b^2*d^2*e^7 + 15450*sqrt(
x*e + d)*A*a^3*b^3*d^2*e^7 - 10633*(x*e + d)^(3/2)*B*a^5*b*e^8 + 5153*(x*e + d)^(3/2)*A*a^4*b^2*e^8 + 17949*sq
rt(x*e + d)*B*a^5*b*d*e^8 - 7725*sqrt(x*e + d)*A*a^4*b^2*d*e^8 - 3249*sqrt(x*e + d)*B*a^6*e^9 + 1545*sqrt(x*e
+ d)*A*a^5*b*e^9)/(((x*e + d)*b - b*d + a*e)^4*b^7*sgn((x*e + d)*b*e - b*d*e + a*e^2)) + 2/15*(3*(x*e + d)^(5/
2)*B*b^20*e^3 + 20*(x*e + d)^(3/2)*B*b^20*d*e^3 + 150*sqrt(x*e + d)*B*b^20*d^2*e^3 - 25*(x*e + d)^(3/2)*B*a*b^
19*e^4 + 5*(x*e + d)^(3/2)*A*b^20*e^4 - 375*sqrt(x*e + d)*B*a*b^19*d*e^4 + 75*sqrt(x*e + d)*A*b^20*d*e^4 + 225
*sqrt(x*e + d)*B*a^2*b^18*e^5 - 75*sqrt(x*e + d)*A*a*b^19*e^5)/(b^25*sgn((x*e + d)*b*e - b*d*e + a*e^2))

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maple [B]  time = 0.16, size = 3768, normalized size = 6.76 \begin {gather*} \text {output too large to display} \end {gather*}

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)^(5/2),x)

[Out]

-1/960*(-10240*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a^3*b^3*d*e^4-76800*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x
^3*a*b^5*d^2*e^4-57600*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^4*d*e^5-117810*B*arctan((e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2)*b)*a^6*b*d*e^6+288000*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^3*b^3*d*e^5-115200*B*((a*e-b*
d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^4*d^2*e^4-38400*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b^3*d*e^5-76800*
B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b^3*d^2*e^4+192000*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^4*b^2*d*e
^5-10240*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*a*b^5*d*e^4+48000*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*a*b
^5*d*e^5-38400*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a*b^5*d*e^5-15360*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x
^2*a^2*b^4*d*e^4+192000*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a^2*b^4*d*e^5+45045*B*arctan((e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2)*b)*a^7*e^7+10680*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^6*d^3-29560*B*((a*e-b*d)*b)^(1/2)*(e*x
+d)^(5/2)*b^6*d^4-17325*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^6*b*e^7+27400*B*((a*e-b*d)*b)^(1/2)*(e
*x+d)^(3/2)*b^6*d^5-45045*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^6*e^6-8520*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)
*b^6*d^6-27720*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*b^7*d^3*e^4-69300*A*arctan((e*x+d)^(1/2)/((a*
e-b*d)*b)^(1/2)*b)*x^3*a^3*b^4*e^7+180180*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^4*b^3*e^7+11475*
A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a^2*b^4*e^3+11475*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^6*d^2*e-103950*A*a
rctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^4*b^3*e^7-22155*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a^3*b^3*e
^3+270270*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^5*b^2*e^7+29275*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5
/2)*a^3*b^3*e^4-29275*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^6*d^3*e-69300*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)
^(1/2)*b)*x*a^5*b^2*e^7-59219*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^4*b^2*e^4+180180*B*arctan((e*x+d)^(1/2)/((
a*e-b*d)*b)^(1/2)*b)*x*a^6*b*e^7+25125*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^4*b^2*e^5+25765*A*((a*e-b*d)*b)^(
1/2)*(e*x+d)^(3/2)*b^6*d^4*e+34650*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^5*b^2*d*e^6-17325*A*arctan(
(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b^3*d^2*e^5-49965*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^5*b*e^5+10048
5*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^5*b^2*d^2*e^5-27720*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/
2)*b)*a^4*b^3*d^3*e^4+17325*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^5*b*e^6-7725*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(
1/2)*b^6*d^5*e-384*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x^4*b^6*e^4-640*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^4
*b^6*e^5-17325*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a^2*b^5*e^7-17325*A*arctan((e*x+d)^(1/2)/((a*
e-b*d)*b)^(1/2)*b)*x^4*b^7*d^2*e^5+45045*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a^3*b^4*e^7+34650*A
*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a*b^6*d*e^6-1536*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x^3*a*b^
5*e^4+3200*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^4*a*b^5*e^5-2560*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^4*b^6*
d*e^4-117810*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a^2*b^5*d*e^6+100485*B*arctan((e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2)*b)*x^4*a*b^6*d^2*e^5-2560*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*a*b^5*e^5+9600*A*((a*e-b*d)
*b)^(1/2)*(e*x+d)^(1/2)*x^4*a*b^5*e^6-9600*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*b^6*d*e^5-115200*B*((a*e-b*
d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^5*b*e^6-48225*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*b^2*d*e^5+77250*A*((a*e-b*
d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b^3*d^2*e^4-77250*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d^3*e^3+38625*A*((
a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^5*d^4*e^2+137745*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^5*b*d*e^5-224250*B*
((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*b^2*d^2*e^4+247650*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b^3*d^3*e^3-1
66425*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d^4*e^2+377060*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^4*d
^3*e^2-162765*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^5*d^4*e-265185*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b
^4*d^2*e^2+147515*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^5*d^3*e+12800*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*
a^4*b^2*e^5-1536*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x*a^3*b^3*e^4+19200*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x
^2*a^3*b^3*e^5-115200*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a^3*b^3*e^6-706860*B*arctan((e*x+d)^(1/2)/((a*e-
b*d)*b)^(1/2)*b)*x^2*a^4*b^3*d*e^6+602910*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^3*b^4*d^2*e^5-16
6320*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^5*d^3*e^4-87825*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/
2)*a^2*b^4*d*e^3+87825*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^5*d^2*e^2-2560*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3
/2)*x*a^3*b^3*e^5+57600*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^3*b^3*e^6+138600*A*arctan((e*x+d)^(1/2)/((a*
e-b*d)*b)^(1/2)*b)*x*a^4*b^3*d*e^6-69300*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^4*d^2*e^5+20606
5*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^3*b^3*d*e^3-172800*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^4*b^2*e^6
-471240*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^5*b^2*d*e^6+401940*B*arctan((e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2)*b)*x*a^4*b^3*d^2*e^5-110880*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^4*d^3*e^4-103060*A
*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b^3*d*e^4+154590*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^4*d^2*e^3-10
3060*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^5*d^3*e^2+38400*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^4*b^2*e^6
+237500*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^4*b^2*d*e^4-428590*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b^3*d
^2*e^3+58845*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^5*d^5*e+138600*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)
*b)*x^3*a^2*b^5*d*e^6-69300*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^6*d^2*e^5-2304*B*((a*e-b*d)*
b)^(1/2)*(e*x+d)^(5/2)*x^2*a^2*b^4*e^4+12800*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*a^2*b^4*e^5-28800*B*((a*e
-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*a^2*b^4*e^6-19200*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*b^6*d^2*e^4-471240*
B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^3*b^4*d*e^6+401940*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1
/2)*b)*x^3*a^2*b^5*d^2*e^5-110880*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^6*d^3*e^4-22950*A*((a*
e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^5*d*e^2-3840*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*a^2*b^4*e^5+38400*A*((a
*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a^2*b^4*e^6+207900*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^3*b^
4*d*e^6-103950*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^5*d^2*e^5+54990*B*((a*e-b*d)*b)^(1/2)*(
e*x+d)^(7/2)*a^2*b^4*d*e^2-43515*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^5*d^2*e)/e*(b*x+a)/((a*e-b*d)*b)^(1/2
)/b^7/((b*x+a)^2)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {11}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \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)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{11/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \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)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(11/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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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)**(5/2),x)

[Out]

Timed out

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