\(\int \frac {(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [494]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [B] (verification not implemented)
Mupad [F(-1)]
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 35, antiderivative size = 424 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 (712 b B d+315 A b e-1027 a B e) (d+e x)^{3/2}}{192 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B e^3 (a+b x) (d+e x)^{3/2}}{3 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (104 b B d+63 A b e-167 a B e) (d+e x)^{5/2}}{96 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+9 A b e-17 a B e) (d+e x)^{7/2}}{24 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{9/2}}{4 b^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:

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)-1/192*e^2*(315*A*b*e-1027*B*a*e+712*B*b*d)*(e*x+d)^(3/2)/b^5/((b*x 
+a)^2)^(1/2)+2/3*B*e^3*(b*x+a)*(e*x+d)^(3/2)/b^5/((b*x+a)^2)^(1/2)-1/96*e* 
(63*A*b*e-167*B*a*e+104*B*b*d)*(e*x+d)^(5/2)/b^4/(b*x+a)/((b*x+a)^2)^(1/2) 
-1/24*(9*A*b*e-17*B*a*e+8*B*b*d)*(e*x+d)^(7/2)/b^3/(b*x+a)^2/((b*x+a)^2)^( 
1/2)-1/4*(A*b-B*a)*(e*x+d)^(9/2)/b^2/(b*x+a)^3/((b*x+a)^2)^(1/2)-105/64*e^ 
3*(-a*e+b*d)^(1/2)*(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))/b^(13/2)/((b*x+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 3.83 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3 A b \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )+B \left (3465 a^5 e^4+105 a^4 b e^3 (-35 d+121 e x)+21 a^3 b^2 e^2 \left (18 d^2-649 d e x+803 e^2 x^2\right )+9 a^2 b^3 e \left (8 d^3+164 d^2 e x-2041 d e^2 x^2+1023 e^3 x^3\right )+8 b^5 x \left (8 d^4+50 d^3 e x+165 d^2 e^2 x^2-208 d e^3 x^3-16 e^4 x^4\right )+a b^4 \left (16 d^4+280 d^3 e x+2130 d^2 e^2 x^2-10271 d e^3 x^3+1408 e^4 x^4\right )\right )\right )}{e^3 (a+b x)^4}-315 \sqrt {-b d+a e} (8 b B d+3 A b e-11 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \sqrt {(a+b x)^2}} \] Input:

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

Output:

(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 + 88* 
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 + 1023*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 + 280*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])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.67, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1187, 27, 87, 51, 51, 51, 60, 60, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1187

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \int \frac {(d+e x)^{9/2}}{(a+b x)^4}dx}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \int \frac {(d+e x)^{7/2}}{(a+b x)^3}dx}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

Input:

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

Output:

((a + b*x)*(-1/4*((A*b - a*B)*(d + e*x)^(11/2))/(b*(b*d - a*e)*(a + b*x)^4 
) + ((8*b*B*d + 3*A*b*e - 11*a*B*e)*(-1/3*(d + e*x)^(9/2)/(b*(a + b*x)^3) 
+ (3*e*(-1/2*(d + e*x)^(7/2)/(b*(a + b*x)^2) + (7*e*(-((d + e*x)^(5/2)/(b* 
(a + b*x))) + (5*e*((2*(d + e*x)^(3/2))/(3*b) + ((b*d - a*e)*((2*Sqrt[d + 
e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a* 
e]])/b^(3/2)))/b))/(2*b)))/(4*b)))/(2*b)))/(8*b*(b*d - a*e))))/Sqrt[a^2 + 
2*a*b*x + b^2*x^2]
 

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 51
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] - Simp[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 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 60
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 221
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 1187
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(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)^n*(b/2 
+ c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 
 - 4*a*c, 0] &&  !IntegerQ[p]
 
Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.04

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

Input:

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

Output:

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*a*b^ 
3-55/16*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-1171 
*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*a^3*b-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+187/12 
8*A*b^4*d^3*e+515/128*B*e^4*a^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*b^4*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)/(b*(a*e-b*d))^(1/2)*arctan(b 
*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (311) = 622\).

Time = 0.16 (sec) , antiderivative size = 1414, normalized size of antiderivative = 3.33 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[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*sq...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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. \(2 (311) = 622\).

Time = 0.21 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.94 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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 - 3560 
*(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 - 2295*(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 - 975*(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 - 57 
87*(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^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {315 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{4} e^{4}-315 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{3} b d \,e^{3}+945 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{3} b \,e^{4} x -945 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{2} b^{2} d \,e^{3} x +945 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{2} b^{2} e^{4} x^{2}-945 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a \,b^{3} d \,e^{3} x^{2}+315 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a \,b^{3} e^{4} x^{3}-315 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b^{4} d \,e^{3} x^{3}-315 \sqrt {e x +d}\, a^{4} b \,e^{4}+420 \sqrt {e x +d}\, a^{3} b^{2} d \,e^{3}-840 \sqrt {e x +d}\, a^{3} b^{2} e^{4} x -63 \sqrt {e x +d}\, a^{2} b^{3} d^{2} e^{2}+1134 \sqrt {e x +d}\, a^{2} b^{3} d \,e^{3} x -693 \sqrt {e x +d}\, a^{2} b^{3} e^{4} x^{2}-18 \sqrt {e x +d}\, a \,b^{4} d^{3} e -180 \sqrt {e x +d}\, a \,b^{4} d^{2} e^{2} x +954 \sqrt {e x +d}\, a \,b^{4} d \,e^{3} x^{2}-144 \sqrt {e x +d}\, a \,b^{4} e^{4} x^{3}-8 \sqrt {e x +d}\, b^{5} d^{4}-50 \sqrt {e x +d}\, b^{5} d^{3} e x -165 \sqrt {e x +d}\, b^{5} d^{2} e^{2} x^{2}+208 \sqrt {e x +d}\, b^{5} d \,e^{3} x^{3}+16 \sqrt {e x +d}\, b^{5} e^{4} x^{4}}{24 b^{6} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:

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

Output:

(315*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b* 
d)))*a**4*e**4 - 315*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt( 
b)*sqrt(a*e - b*d)))*a**3*b*d*e**3 + 945*sqrt(b)*sqrt(a*e - b*d)*atan((sqr 
t(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b*e**4*x - 945*sqrt(b)*sqrt( 
a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**2*d*e 
**3*x + 945*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a 
*e - b*d)))*a**2*b**2*e**4*x**2 - 945*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d 
 + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**3*d*e**3*x**2 + 315*sqrt(b)*sqr 
t(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**3*e**4 
*x**3 - 315*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a 
*e - b*d)))*b**4*d*e**3*x**3 - 315*sqrt(d + e*x)*a**4*b*e**4 + 420*sqrt(d 
+ e*x)*a**3*b**2*d*e**3 - 840*sqrt(d + e*x)*a**3*b**2*e**4*x - 63*sqrt(d + 
 e*x)*a**2*b**3*d**2*e**2 + 1134*sqrt(d + e*x)*a**2*b**3*d*e**3*x - 693*sq 
rt(d + e*x)*a**2*b**3*e**4*x**2 - 18*sqrt(d + e*x)*a*b**4*d**3*e - 180*sqr 
t(d + e*x)*a*b**4*d**2*e**2*x + 954*sqrt(d + e*x)*a*b**4*d*e**3*x**2 - 144 
*sqrt(d + e*x)*a*b**4*e**4*x**3 - 8*sqrt(d + e*x)*b**5*d**4 - 50*sqrt(d + 
e*x)*b**5*d**3*e*x - 165*sqrt(d + e*x)*b**5*d**2*e**2*x**2 + 208*sqrt(d + 
e*x)*b**5*d*e**3*x**3 + 16*sqrt(d + e*x)*b**5*e**4*x**4)/(24*b**6*(a**3 + 
3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))