3.17.55 \(\int \frac {(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=489 \[ -\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {105 e^3 (a+b x) \sqrt {b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 e^3 (a+b x) \sqrt {d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.42, antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {21 e^2 (d+e x)^{5/2} (-11 a B e+3 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 {35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {105 e^3 (a+b x) \sqrt {d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (a+b x) \sqrt {b d-a e} (-11 a B e+3 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^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2} (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)^{9/2} (-11 a B e+3 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 {3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 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)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(105*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*
e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(64*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (21*e^2*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(64*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (3*e*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(32*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(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)^(11/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*e^3*Sqrt[b*d - a*e]*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(64*b^(13/2)*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)^{9/2}}{\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)^{9/2}}{\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)^{11/2}}{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+3 A b e-11 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 )^4} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e (8 b B d+3 A b e-11 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 )^3} \, dx}{16 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 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+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\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 {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 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+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 (8 b B d+3 A b e-11 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^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 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+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (b^2 d-a b e\right ) (8 b B d+3 A b e-11 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^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 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+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 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^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 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+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^2 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 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^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 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+3 A b e-11 a B e) (d+e x)^{9/2}}{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)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.18, size = 117, normalized size = 0.24 \begin {gather*} \frac {(d+e x)^{11/2} \left (\frac {e^3 (a+b x)^4 (-11 a B e+3 A b e+8 b B d) \, _2F_1\left (4,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+11 (a B-A b)\right )}{44 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)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((d + e*x)^(11/2)*(11*(-(A*b) + a*B) + (e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)^4*Hypergeometric2F1[4, 11
/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(44*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 69.12, size = 726, normalized size = 1.48 \begin {gather*} \frac {(-a e-b e x) \left (\frac {105 \left (11 a^2 B e^5-3 a A b e^5-19 a b B d e^4+3 A b^2 d e^4+8 b^2 B d^2 e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{13/2} \sqrt {a e-b d}}-\frac {e^3 \sqrt {d+e x} \left (-3465 a^5 B e^5+945 a^4 A b e^5-12705 a^4 b B e^4 (d+e x)+16380 a^4 b B d e^4+3465 a^3 A b^2 e^4 (d+e x)-3780 a^3 A b^2 d e^4-30870 a^3 b^2 B d^2 e^3-16863 a^3 b^2 B e^3 (d+e x)^2+47355 a^3 b^2 B d e^3 (d+e x)+5670 a^2 A b^3 d^2 e^3+4599 a^2 A b^3 e^3 (d+e x)^2-10395 a^2 A b^3 d e^3 (d+e x)+28980 a^2 b^3 B d^3 e^2-65835 a^2 b^3 B d^2 e^2 (d+e x)-9207 a^2 b^3 B e^2 (d+e x)^3+45990 a^2 b^3 B d e^2 (d+e x)^2-3780 a A b^4 d^3 e^2+10395 a A b^4 d^2 e^2 (d+e x)+2511 a A b^4 e^2 (d+e x)^3-9198 a A b^4 d e^2 (d+e x)^2-13545 a b^4 B d^4 e+40425 a b^4 B d^3 e (d+e x)-41391 a b^4 B d^2 e (d+e x)^2-1408 a b^4 B e (d+e x)^4+15903 a b^4 B d e (d+e x)^3+945 A b^5 d^4 e-3465 A b^5 d^3 e (d+e x)+4599 A b^5 d^2 e (d+e x)^2+384 A b^5 e (d+e x)^4-2511 A b^5 d e (d+e x)^3+2520 b^5 B d^5-9240 b^5 B d^4 (d+e x)+12264 b^5 B d^3 (d+e x)^2-6696 b^5 B d^2 (d+e x)^3+128 b^5 B (d+e x)^5+1024 b^5 B d (d+e x)^4\right )}{192 b^6 (a e+b (d+e x)-b d)^4}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((-(a*e) - b*e*x)*(-1/192*(e^3*Sqrt[d + e*x]*(2520*b^5*B*d^5 + 945*A*b^5*d^4*e - 13545*a*b^4*B*d^4*e - 3780*a*
A*b^4*d^3*e^2 + 28980*a^2*b^3*B*d^3*e^2 + 5670*a^2*A*b^3*d^2*e^3 - 30870*a^3*b^2*B*d^2*e^3 - 3780*a^3*A*b^2*d*
e^4 + 16380*a^4*b*B*d*e^4 + 945*a^4*A*b*e^5 - 3465*a^5*B*e^5 - 9240*b^5*B*d^4*(d + e*x) - 3465*A*b^5*d^3*e*(d
+ e*x) + 40425*a*b^4*B*d^3*e*(d + e*x) + 10395*a*A*b^4*d^2*e^2*(d + e*x) - 65835*a^2*b^3*B*d^2*e^2*(d + e*x) -
10395*a^2*A*b^3*d*e^3*(d + e*x) + 47355*a^3*b^2*B*d*e^3*(d + e*x) + 3465*a^3*A*b^2*e^4*(d + e*x) - 12705*a^4*
b*B*e^4*(d + e*x) + 12264*b^5*B*d^3*(d + e*x)^2 + 4599*A*b^5*d^2*e*(d + e*x)^2 - 41391*a*b^4*B*d^2*e*(d + e*x)
^2 - 9198*a*A*b^4*d*e^2*(d + e*x)^2 + 45990*a^2*b^3*B*d*e^2*(d + e*x)^2 + 4599*a^2*A*b^3*e^3*(d + e*x)^2 - 168
63*a^3*b^2*B*e^3*(d + e*x)^2 - 6696*b^5*B*d^2*(d + e*x)^3 - 2511*A*b^5*d*e*(d + e*x)^3 + 15903*a*b^4*B*d*e*(d
+ e*x)^3 + 2511*a*A*b^4*e^2*(d + e*x)^3 - 9207*a^2*b^3*B*e^2*(d + e*x)^3 + 1024*b^5*B*d*(d + e*x)^4 + 384*A*b^
5*e*(d + e*x)^4 - 1408*a*b^4*B*e*(d + e*x)^4 + 128*b^5*B*(d + e*x)^5))/(b^6*(-(b*d) + a*e + b*(d + e*x))^4) +
(105*(8*b^2*B*d^2*e^3 + 3*A*b^2*d*e^4 - 19*a*b*B*d*e^4 - 3*a*A*b*e^5 + 11*a^2*B*e^5)*ArcTan[(Sqrt[b]*Sqrt[-(b*
d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*b^(13/2)*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 1414, normalized size = 2.89

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)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4
+ 4*(8*B*a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^
3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*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*(128*B*b^5*e^4*x^5 - 16*(B*a*b^4 + 3*A*b^5)*d^4
- 72*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 105*(35*B*a^4*b - 3*A*a^3*b^2)*d*e
^3 - 315*(11*B*a^5 - 3*A*a^4*b)*e^4 + 128*(13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 - (1320*B*b^5*d^2*
e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 - (400*B*b^5*d^3*e + 30*(71*
B*a*b^4 + 21*A*b^5)*d^2*e^2 - 9*(2041*B*a^2*b^3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*
x^2 - (64*B*b^5*d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*a*b^4)*d^2*e^2 - 21*(649*B*a^3
*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 +
6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6), -1/192*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 + (8*B*b^5*d
*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2
*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*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)) - (128*B*b^5*e^4*x^5 - 16*(B*a
*b^4 + 3*A*b^5)*d^4 - 72*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 105*(35*B*a^4*b
- 3*A*a^3*b^2)*d*e^3 - 315*(11*B*a^5 - 3*A*a^4*b)*e^4 + 128*(13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4
- (1320*B*b^5*d^2*e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 - (400*B*
b^5*d^3*e + 30*(71*B*a*b^4 + 21*A*b^5)*d^2*e^2 - 9*(2041*B*a^2*b^3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 -
3*A*a^2*b^3)*e^4)*x^2 - (64*B*b^5*d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*a*b^4)*d^2*
e^2 - 21*(649*B*a^3*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^10*x
^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)]

________________________________________________________________________________________

giac [B]  time = 0.52, size = 872, normalized size = 1.78 \begin {gather*} \frac {105 \, {\left (8 \, B b^{2} d^{2} e^{3} - 19 \, B a b d e^{4} + 3 \, A b^{2} d e^{4} + 11 \, B a^{2} e^{5} - 3 \, A a b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {1320 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{3} - 3560 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{3} + 3224 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{3} - 984 \, \sqrt {x e + d} B b^{5} d^{5} e^{3} - 3615 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{4} + 975 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{4} + 12975 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{4} - 2295 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{4} - 14825 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{4} + 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{4} + 5481 \, \sqrt {x e + d} B a b^{4} d^{4} e^{4} - 561 \, \sqrt {x e + d} A b^{5} d^{4} e^{4} + 2295 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{5} - 975 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{5} - 15270 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{5} + 4590 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{5} + 25131 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{5} - 5787 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{5} - 12084 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{5} + 2244 \, \sqrt {x e + d} A a b^{4} d^{3} e^{5} + 5855 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{6} - 2295 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{6} - 18683 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{6} + 5787 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{6} + 13206 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{6} - 3366 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{6} + 5153 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{7} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{7} - 7164 \, \sqrt {x e + d} B a^{4} b d e^{7} + 2244 \, \sqrt {x e + d} A a^{3} b^{2} d e^{7} + 1545 \, \sqrt {x e + d} B a^{5} e^{8} - 561 \, \sqrt {x e + d} A a^{4} b e^{8}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{10} e^{3} + 12 \, \sqrt {x e + d} B b^{10} d e^{3} - 15 \, \sqrt {x e + d} B a b^{9} e^{4} + 3 \, \sqrt {x e + d} A b^{10} e^{4}\right )}}{3 \, b^{15} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}

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)^(5/2),x, algorithm="giac")

[Out]

105/64*(8*B*b^2*d^2*e^3 - 19*B*a*b*d*e^4 + 3*A*b^2*d*e^4 + 11*B*a^2*e^5 - 3*A*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*sgn((x*e + d)*b*e - b*d*e + a*e^2)) - 1/192*(1320*(x*e + d)^(7
/2)*B*b^5*d^2*e^3 - 3560*(x*e + d)^(5/2)*B*b^5*d^3*e^3 + 3224*(x*e + d)^(3/2)*B*b^5*d^4*e^3 - 984*sqrt(x*e + d
)*B*b^5*d^5*e^3 - 3615*(x*e + d)^(7/2)*B*a*b^4*d*e^4 + 975*(x*e + d)^(7/2)*A*b^5*d*e^4 + 12975*(x*e + d)^(5/2)
*B*a*b^4*d^2*e^4 - 2295*(x*e + d)^(5/2)*A*b^5*d^2*e^4 - 14825*(x*e + d)^(3/2)*B*a*b^4*d^3*e^4 + 1929*(x*e + d)
^(3/2)*A*b^5*d^3*e^4 + 5481*sqrt(x*e + d)*B*a*b^4*d^4*e^4 - 561*sqrt(x*e + d)*A*b^5*d^4*e^4 + 2295*(x*e + d)^(
7/2)*B*a^2*b^3*e^5 - 975*(x*e + d)^(7/2)*A*a*b^4*e^5 - 15270*(x*e + d)^(5/2)*B*a^2*b^3*d*e^5 + 4590*(x*e + d)^
(5/2)*A*a*b^4*d*e^5 + 25131*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^5 - 5787*(x*e + d)^(3/2)*A*a*b^4*d^2*e^5 - 12084*s
qrt(x*e + d)*B*a^2*b^3*d^3*e^5 + 2244*sqrt(x*e + d)*A*a*b^4*d^3*e^5 + 5855*(x*e + d)^(5/2)*B*a^3*b^2*e^6 - 229
5*(x*e + d)^(5/2)*A*a^2*b^3*e^6 - 18683*(x*e + d)^(3/2)*B*a^3*b^2*d*e^6 + 5787*(x*e + d)^(3/2)*A*a^2*b^3*d*e^6
+ 13206*sqrt(x*e + d)*B*a^3*b^2*d^2*e^6 - 3366*sqrt(x*e + d)*A*a^2*b^3*d^2*e^6 + 5153*(x*e + d)^(3/2)*B*a^4*b
*e^7 - 1929*(x*e + d)^(3/2)*A*a^3*b^2*e^7 - 7164*sqrt(x*e + d)*B*a^4*b*d*e^7 + 2244*sqrt(x*e + d)*A*a^3*b^2*d*
e^7 + 1545*sqrt(x*e + d)*B*a^5*e^8 - 561*sqrt(x*e + d)*A*a^4*b*e^8)/(((x*e + d)*b - b*d + a*e)^4*b^6*sgn((x*e
+ d)*b*e - b*d*e + a*e^2)) + 2/3*((x*e + d)^(3/2)*B*b^10*e^3 + 12*sqrt(x*e + d)*B*b^10*d*e^3 - 15*sqrt(x*e + d
)*B*a*b^9*e^4 + 3*sqrt(x*e + d)*A*b^10*e^4)/(b^15*sgn((x*e + d)*b*e - b*d*e + a*e^2))

________________________________________________________________________________________

maple [B]  time = 0.14, size = 2430, normalized size = 4.97

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)^(5/2),x)

[Out]

1/192*(-3780*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^4*b^2*e^6+6144*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1
/2)*x^3*a*b^4*d*e^4+3465*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^6*e^6+9216*B*((a*e-b*d)*b)^(1/2)*(e*x
+d)^(1/2)*x^2*a^2*b^3*d*e^4+6144*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b^2*d*e^4+3560*B*((a*e-b*d)*b)^(1/2
)*(e*x+d)^(5/2)*b^5*d^3-945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^5*b*e^6-3224*B*((a*e-b*d)*b)^(1/2)
*(e*x+d)^(3/2)*b^5*d^4-3465*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^5*e^5+984*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2
)*b^5*d^5-1320*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^5*d^2+1536*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a*b^4*
e^5+5670*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^4*d*e^5+3615*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7
/2)*a*b^4*d*e+768*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*a^2*b^3*e^4-7680*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)
*x^3*a^2*b^3*e^5-35910*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^3*b^3*d*e^5+15120*B*arctan((e*x+d)^
(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^4*d^2*e^4-4590*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^4*d*e^2-5985*B*a
rctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a*b^5*d*e^5+3780*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*
x^3*a*b^5*d*e^5+512*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*a*b^4*e^4-1920*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)
*x^4*a*b^4*e^5+1536*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*b^5*d*e^4-23940*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*
b)^(1/2)*b)*x^3*a^2*b^4*d*e^5+10080*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^5*d^2*e^4+8700*B*((a
*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*b*d*e^4-13206*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b^2*d^2*e^3+12084*B*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^3*e^2-5481*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^4*d^4*e+2304*A*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^3*e^5+3780*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^3*
d*e^5+15270*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b^3*d*e^2+18683*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b^
2*d*e^3-25131*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^3*d^2*e^2+14825*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*
b^4*d^3*e-7680*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^4*b*e^5-2244*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b^
2*d*e^4-23940*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^4*b^2*d*e^5+10080*B*arctan((e*x+d)^(1/2)/((a*e
-b*d)*b)^(1/2)*b)*x*a^3*b^3*d^2*e^4-5787*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^3*d*e^3+5787*A*((a*e-b*d)*b
)^(1/2)*(e*x+d)^(3/2)*a*b^4*d^2*e^2+1536*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b^2*e^5+3366*A*((a*e-b*d)*b
)^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^2*e^3-2244*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^4*d^3*e^2-5855*B*((a*e-b*d)
*b)^(1/2)*(e*x+d)^(5/2)*a^3*b^2*e^3+13860*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^5*b*e^6+1929*A*((a
*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b^2*e^4-1929*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^5*d^3*e+945*A*arctan((e*
x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b^2*d*e^5-5025*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^4*b*e^4-5985*B*arct
an((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^5*b*d*e^5+2520*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b^2
*d^2*e^4+945*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*b*e^5+561*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^5*d^4*e-2
295*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a^2*b^3*e^2+20790*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^
4*b^2*e^6+2295*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b^3*e^3+2295*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^5*d^
2*e-975*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^5*d*e-5670*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^3
*b^3*e^6-945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a*b^5*e^6+945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2)*b)*x^4*b^6*d*e^5+128*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^4*b^5*e^4+3465*B*arctan((e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2)*b)*x^4*a^2*b^4*e^6+2520*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*b^6*d^2*e^4+384*A*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*b^5*e^5-3780*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^2*b^4*e^
6+13860*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^3*b^3*e^6+975*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*
a*b^4*e^2-12975*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^4*d^2*e+512*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a^3*
b^2*e^4-11520*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^3*b^2*e^5)/e*(b*x+a)/((a*e-b*d)*b)^(1/2)/b^6/((b*x+a)^
2)^(5/2)

________________________________________________________________________________________

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 {9}{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)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

________________________________________________________________________________________

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 )}^{9/2}}{{\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)^(9/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(9/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

________________________________________________________________________________________

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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________