\(\int \frac {(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1875]

Optimal result

Integrand size = 35, antiderivative size = 489 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

35/64*e^3*(3*A*b*e-11*B*a*e+8*B*b*d)*(b*x+a)*(e*x+d)^(3/2)/b^5/(-a*e+b*d)/((b*x+a)^2)^(1/2)-21/64*e^2*(3*A*b*e
-11*B*a*e+8*B*b*d)*(e*x+d)^(5/2)/b^4/(-a*e+b*d)/((b*x+a)^2)^(1/2)-3/32*e*(3*A*b*e-11*B*a*e+8*B*b*d)*(e*x+d)^(7
/2)/b^3/(-a*e+b*d)/(b*x+a)/((b*x+a)^2)^(1/2)-1/24*(3*A*b*e-11*B*a*e+8*B*b*d)*(e*x+d)^(9/2)/b^2/(-a*e+b*d)/(b*x
+a)^2/((b*x+a)^2)^(1/2)-1/4*(A*b-B*a)*(e*x+d)^(11/2)/b/(-a*e+b*d)/(b*x+a)^3/((b*x+a)^2)^(1/2)-105/64*e^3*(3*A*
b*e-11*B*a*e+8*B*b*d)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(13/2)/((b*x+
a)^2)^(1/2)+105/64*e^3*(3*A*b*e-11*B*a*e+8*B*b*d)*(b*x+a)*(e*x+d)^(1/2)/b^6/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {784, 79, 43, 52, 65, 214} \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {105 e^3 (a+b x) \sqrt {b d-a e} (-11 a B e+3 A b e+8 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {105 e^3 (a+b x) \sqrt {d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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 65

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 79

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] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 784

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*} \text {integral}& = \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 ) \text {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 [A] (verified)

Time = 2.28 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3 A b \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )+B \left (3465 a^5 e^4+105 a^4 b e^3 (-35 d+121 e x)+21 a^3 b^2 e^2 \left (18 d^2-649 d e x+803 e^2 x^2\right )+9 a^2 b^3 e \left (8 d^3+164 d^2 e x-2041 d e^2 x^2+1023 e^3 x^3\right )+8 b^5 x \left (8 d^4+50 d^3 e x+165 d^2 e^2 x^2-208 d e^3 x^3-16 e^4 x^4\right )+a b^4 \left (16 d^4+280 d^3 e x+2130 d^2 e^2 x^2-10271 d e^3 x^3+1408 e^4 x^4\right )\right )\right )}{e^3 (a+b x)^4}-315 \sqrt {-b d+a e} (8 b B d+3 A b e-11 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \sqrt {(a+b x)^2}} \]

[In]

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

[Out]

(e^3*(a + b*x)*(-((Sqrt[b]*Sqrt[d + e*x]*(3*A*b*(-315*a^4*e^4 + 105*a^3*b*e^3*(d - 11*e*x) + 21*a^2*b^2*e^2*(2
*d^2 + 19*d*e*x - 73*e^2*x^2) + 3*a*b^3*e*(8*d^3 + 52*d^2*e*x + 185*d*e^2*x^2 - 279*e^3*x^3) + b^4*(16*d^4 + 8
8*d^3*e*x + 210*d^2*e^2*x^2 + 325*d*e^3*x^3 - 128*e^4*x^4)) + B*(3465*a^5*e^4 + 105*a^4*b*e^3*(-35*d + 121*e*x
) + 21*a^3*b^2*e^2*(18*d^2 - 649*d*e*x + 803*e^2*x^2) + 9*a^2*b^3*e*(8*d^3 + 164*d^2*e*x - 2041*d*e^2*x^2 + 10
23*e^3*x^3) + 8*b^5*x*(8*d^4 + 50*d^3*e*x + 165*d^2*e^2*x^2 - 208*d*e^3*x^3 - 16*e^4*x^4) + a*b^4*(16*d^4 + 28
0*d^3*e*x + 2130*d^2*e^2*x^2 - 10271*d*e^3*x^3 + 1408*e^4*x^4))))/(e^3*(a + b*x)^4)) - 315*Sqrt[-(b*d) + a*e]*
(8*b*B*d + 3*A*b*e - 11*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(192*b^(13/2)*Sqrt[(a + b*
x)^2])

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.90

method	result	size
risch	\(\frac {2 e^{3} \left (B b e x +3 A b e -15 B a e +13 B b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}-\frac {\left (2 a e -2 b d \right ) e^{3} \left (\frac {\left (-\frac {325}{128} A \,b^{4} e +\frac {765}{128} B e \,b^{3} a -\frac {55}{16} B \,b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {5 b^{2} \left (459 A a b \,e^{2}-459 A \,b^{2} d e -1171 a^{2} B \,e^{2}+1883 B a b d e -712 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {643}{128} A \,a^{2} b^{2} e^{3}+\frac {643}{64} A a \,b^{3} d \,e^{2}-\frac {643}{128} A \,b^{4} d^{2} e +\frac {5153}{384} B \,e^{3} b \,a^{3}-\frac {2255}{64} B \,a^{2} b^{2} d \,e^{2}+\frac {3867}{128} B a \,b^{3} d^{2} e -\frac {403}{48} B \,b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} A \,a^{3} b \,e^{4}+\frac {561}{128} A \,a^{2} b^{2} d \,e^{3}-\frac {561}{128} A a \,b^{3} d^{2} e^{2}+\frac {187}{128} A \,b^{4} d^{3} e +\frac {515}{128} B \,a^{4} e^{4}-\frac {1873}{128} B \,a^{3} b d \,e^{3}+\frac {2529}{128} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1499}{128} B a \,b^{3} d^{3} e +\frac {41}{16} b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {105 \left (3 A b e -11 B a e +8 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{6} \left (b x +a \right )}\)	\(439\)
default	\(\text {Expression too large to display}\)	\(2430\)

[In]

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

[Out]

2/3*e^3*(B*b*e*x+3*A*b*e-15*B*a*e+13*B*b*d)*(e*x+d)^(1/2)/b^6*((b*x+a)^2)^(1/2)/(b*x+a)-1/b^6*(2*a*e-2*b*d)*e^
3*(((-325/128*A*b^4*e+765/128*B*e*b^3*a-55/16*B*b^4*d)*(e*x+d)^(7/2)-5/384*b^2*(459*A*a*b*e^2-459*A*b^2*d*e-11
71*B*a^2*e^2+1883*B*a*b*d*e-712*B*b^2*d^2)*(e*x+d)^(5/2)+(-643/128*A*a^2*b^2*e^3+643/64*A*a*b^3*d*e^2-643/128*
A*b^4*d^2*e+5153/384*B*e^3*b*a^3-2255/64*B*a^2*b^2*d*e^2+3867/128*B*a*b^3*d^2*e-403/48*B*b^4*d^3)*(e*x+d)^(3/2
)+(-187/128*A*a^3*b*e^4+561/128*A*a^2*b^2*d*e^3-561/128*A*a*b^3*d^2*e^2+187/128*A*b^4*d^3*e+515/128*B*a^4*e^4-
1873/128*B*a^3*b*d*e^3+2529/128*B*a^2*b^2*d^2*e^2-1499/128*B*a*b^3*d^3*e+41/16*b^4*B*d^4)*(e*x+d)^(1/2))/(b*(e
*x+d)+a*e-b*d)^4+105/128*(3*A*b*e-11*B*a*e+8*B*b*d)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(
1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 1414, normalized size of antiderivative = 2.89 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 824 vs. \(2 (376) = 752\).

Time = 0.32 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.69 \[ \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 {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 {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {1320 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{3} - 3560 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{3} + 3224 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{3} - 984 \, \sqrt {e x + d} B b^{5} d^{5} e^{3} - 3615 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{4} + 975 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{4} + 12975 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{4} - 2295 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{4} - 14825 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{4} + 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{4} + 5481 \, \sqrt {e x + d} B a b^{4} d^{4} e^{4} - 561 \, \sqrt {e x + d} A b^{5} d^{4} e^{4} + 2295 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{5} - 975 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{5} - 15270 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{5} + 4590 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{5} + 25131 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{5} - 5787 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{5} - 12084 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{5} + 2244 \, \sqrt {e x + d} A a b^{4} d^{3} e^{5} + 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{6} - 2295 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{6} - 18683 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{6} + 5787 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{6} + 13206 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{6} - 3366 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{6} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{7} - 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{7} - 7164 \, \sqrt {e x + d} B a^{4} b d e^{7} + 2244 \, \sqrt {e x + d} A a^{3} b^{2} d e^{7} + 1545 \, \sqrt {e x + d} B a^{5} e^{8} - 561 \, \sqrt {e x + d} A a^{4} b e^{8}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{10} e^{3} + 12 \, \sqrt {e x + d} B b^{10} d e^{3} - 15 \, \sqrt {e x + d} B a b^{9} e^{4} + 3 \, \sqrt {e x + d} A b^{10} e^{4}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \]

[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(e*x + d)*b/
sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6*sgn(b*x + a)) - 1/192*(1320*(e*x + d)^(7/2)*B*b^5*d^2*e^3 - 35
60*(e*x + d)^(5/2)*B*b^5*d^3*e^3 + 3224*(e*x + d)^(3/2)*B*b^5*d^4*e^3 - 984*sqrt(e*x + d)*B*b^5*d^5*e^3 - 3615
*(e*x + d)^(7/2)*B*a*b^4*d*e^4 + 975*(e*x + d)^(7/2)*A*b^5*d*e^4 + 12975*(e*x + d)^(5/2)*B*a*b^4*d^2*e^4 - 229
5*(e*x + d)^(5/2)*A*b^5*d^2*e^4 - 14825*(e*x + d)^(3/2)*B*a*b^4*d^3*e^4 + 1929*(e*x + d)^(3/2)*A*b^5*d^3*e^4 +
5481*sqrt(e*x + d)*B*a*b^4*d^4*e^4 - 561*sqrt(e*x + d)*A*b^5*d^4*e^4 + 2295*(e*x + d)^(7/2)*B*a^2*b^3*e^5 - 9
75*(e*x + d)^(7/2)*A*a*b^4*e^5 - 15270*(e*x + d)^(5/2)*B*a^2*b^3*d*e^5 + 4590*(e*x + d)^(5/2)*A*a*b^4*d*e^5 +
25131*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^5 - 5787*(e*x + d)^(3/2)*A*a*b^4*d^2*e^5 - 12084*sqrt(e*x + d)*B*a^2*b^3
*d^3*e^5 + 2244*sqrt(e*x + d)*A*a*b^4*d^3*e^5 + 5855*(e*x + d)^(5/2)*B*a^3*b^2*e^6 - 2295*(e*x + d)^(5/2)*A*a^
2*b^3*e^6 - 18683*(e*x + d)^(3/2)*B*a^3*b^2*d*e^6 + 5787*(e*x + d)^(3/2)*A*a^2*b^3*d*e^6 + 13206*sqrt(e*x + d)
*B*a^3*b^2*d^2*e^6 - 3366*sqrt(e*x + d)*A*a^2*b^3*d^2*e^6 + 5153*(e*x + d)^(3/2)*B*a^4*b*e^7 - 1929*(e*x + d)^
(3/2)*A*a^3*b^2*e^7 - 7164*sqrt(e*x + d)*B*a^4*b*d*e^7 + 2244*sqrt(e*x + d)*A*a^3*b^2*d*e^7 + 1545*sqrt(e*x +
d)*B*a^5*e^8 - 561*sqrt(e*x + d)*A*a^4*b*e^8)/(((e*x + d)*b - b*d + a*e)^4*b^6*sgn(b*x + a)) + 2/3*((e*x + d)^
(3/2)*B*b^10*e^3 + 12*sqrt(e*x + d)*B*b^10*d*e^3 - 15*sqrt(e*x + d)*B*a*b^9*e^4 + 3*sqrt(e*x + d)*A*b^10*e^4)/
(b^15*sgn(b*x + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

[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)