\(\int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx\) [857]

Optimal result

Integrand size = 27, antiderivative size = 202 \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\frac {8 (b e+a f)^4}{35 a^4 b^5 \sqrt {a-b x} \sqrt {a+b x}}-\frac {8 e (b e+a f)^3 \sqrt {a-b x}}{35 a^5 b^4 \sqrt {a+b x}}-\frac {4 (b e+a f)^2 (e+f x)^2}{35 a^3 b^3 \sqrt {a-b x} (a+b x)^{3/2}}-\frac {4 (b e+a f) (e+f x)^3}{35 a^2 b^2 \sqrt {a-b x} (a+b x)^{5/2}}-\frac {(e+f x)^4}{7 a b \sqrt {a-b x} (a+b x)^{7/2}} \] Output:

8/35*(a*f+b*e)^4/a^4/b^5/(-b*x+a)^(1/2)/(b*x+a)^(1/2)-8/35*e*(a*f+b*e)^3*( 
-b*x+a)^(1/2)/a^5/b^4/(b*x+a)^(1/2)-4/35*(a*f+b*e)^2*(f*x+e)^2/a^3/b^3/(-b 
*x+a)^(1/2)/(b*x+a)^(3/2)-4/35*(a*f+b*e)*(f*x+e)^3/a^2/b^2/(-b*x+a)^(1/2)/ 
(b*x+a)^(5/2)-1/7*(f*x+e)^4/a/b/(-b*x+a)^(1/2)/(b*x+a)^(7/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.24 \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\frac {8 a^8 f^4+8 b^8 e^4 x^4+24 a^7 b f^3 (e+f x)+24 a b^7 e^3 x^3 (e+f x)+4 a^6 b^2 f^2 \left (5 e^2+18 e f x+5 f^2 x^2\right )+4 a^2 b^6 e^2 x^2 \left (5 e^2+18 e f x+5 f^2 x^2\right )-a^4 b^4 (e-f x)^2 \left (13 e^2+38 e f x+13 f^2 x^2\right )-4 a^5 b^3 f \left (e^3-15 e^2 f x-15 e f^2 x^2+f^3 x^3\right )-4 a^3 b^5 e x \left (e^3-15 e^2 f x-15 e f^2 x^2+f^3 x^3\right )}{35 a^5 b^5 \sqrt {a-b x} (a+b x)^{7/2}} \] Input:

Integrate[(e + f*x)^4/((a - b*x)^(3/2)*(a + b*x)^(9/2)),x]
 

Output:

(8*a^8*f^4 + 8*b^8*e^4*x^4 + 24*a^7*b*f^3*(e + f*x) + 24*a*b^7*e^3*x^3*(e 
+ f*x) + 4*a^6*b^2*f^2*(5*e^2 + 18*e*f*x + 5*f^2*x^2) + 4*a^2*b^6*e^2*x^2* 
(5*e^2 + 18*e*f*x + 5*f^2*x^2) - a^4*b^4*(e - f*x)^2*(13*e^2 + 38*e*f*x + 
13*f^2*x^2) - 4*a^5*b^3*f*(e^3 - 15*e^2*f*x - 15*e*f^2*x^2 + f^3*x^3) - 4* 
a^3*b^5*e*x*(e^3 - 15*e^2*f*x - 15*e*f^2*x^2 + f^3*x^3))/(35*a^5*b^5*Sqrt[ 
a - b*x]*(a + b*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {105, 105, 100, 27, 87, 48}

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.

\(\displaystyle \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {4 (b e-a f) \int \frac {(e+f x)^3}{\sqrt {a-b x} (a+b x)^{9/2}}dx}{a b}+\frac {(e+f x)^4}{a b \sqrt {a-b x} (a+b x)^{7/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {4 (b e-a f) \left (\frac {3 (a f+b e) \int \frac {(e+f x)^2}{\sqrt {a-b x} (a+b x)^{7/2}}dx}{7 a b}-\frac {\sqrt {a-b x} (e+f x)^3}{7 a b (a+b x)^{7/2}}\right )}{a b}+\frac {(e+f x)^4}{a b \sqrt {a-b x} (a+b x)^{7/2}}\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {4 (b e-a f) \left (\frac {3 (a f+b e) \left (\frac {\int \frac {b \left (2 b^2 e^2+6 a b f e-3 a^2 f^2+5 a b f^2 x\right )}{\sqrt {a-b x} (a+b x)^{5/2}}dx}{5 a b^3}-\frac {\sqrt {a-b x} (b e-a f)^2}{5 a b^3 (a+b x)^{5/2}}\right )}{7 a b}-\frac {\sqrt {a-b x} (e+f x)^3}{7 a b (a+b x)^{7/2}}\right )}{a b}+\frac {(e+f x)^4}{a b \sqrt {a-b x} (a+b x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 (b e-a f) \left (\frac {3 (a f+b e) \left (\frac {\int \frac {2 b^2 e^2+6 a b f e-3 a^2 f^2+5 a b f^2 x}{\sqrt {a-b x} (a+b x)^{5/2}}dx}{5 a b^2}-\frac {\sqrt {a-b x} (b e-a f)^2}{5 a b^3 (a+b x)^{5/2}}\right )}{7 a b}-\frac {\sqrt {a-b x} (e+f x)^3}{7 a b (a+b x)^{7/2}}\right )}{a b}+\frac {(e+f x)^4}{a b \sqrt {a-b x} (a+b x)^{7/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {4 (b e-a f) \left (\frac {3 (a f+b e) \left (\frac {\frac {\left (7 a^2 f^2+6 a b e f+2 b^2 e^2\right ) \int \frac {1}{\sqrt {a-b x} (a+b x)^{3/2}}dx}{3 a}-\frac {2 \sqrt {a-b x} \left (\frac {b e^2}{a}-\frac {4 a f^2}{b}+3 e f\right )}{3 (a+b x)^{3/2}}}{5 a b^2}-\frac {\sqrt {a-b x} (b e-a f)^2}{5 a b^3 (a+b x)^{5/2}}\right )}{7 a b}-\frac {\sqrt {a-b x} (e+f x)^3}{7 a b (a+b x)^{7/2}}\right )}{a b}+\frac {(e+f x)^4}{a b \sqrt {a-b x} (a+b x)^{7/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {4 (b e-a f) \left (\frac {3 (a f+b e) \left (\frac {-\frac {\sqrt {a-b x} \left (7 a^2 f^2+6 a b e f+2 b^2 e^2\right )}{3 a^2 b \sqrt {a+b x}}-\frac {2 \sqrt {a-b x} \left (\frac {b e^2}{a}-\frac {4 a f^2}{b}+3 e f\right )}{3 (a+b x)^{3/2}}}{5 a b^2}-\frac {\sqrt {a-b x} (b e-a f)^2}{5 a b^3 (a+b x)^{5/2}}\right )}{7 a b}-\frac {\sqrt {a-b x} (e+f x)^3}{7 a b (a+b x)^{7/2}}\right )}{a b}+\frac {(e+f x)^4}{a b \sqrt {a-b x} (a+b x)^{7/2}}\)

Input:

Int[(e + f*x)^4/((a - b*x)^(3/2)*(a + b*x)^(9/2)),x]
 

Output:

(e + f*x)^4/(a*b*Sqrt[a - b*x]*(a + b*x)^(7/2)) + (4*(b*e - a*f)*(-1/7*(Sq 
rt[a - b*x]*(e + f*x)^3)/(a*b*(a + b*x)^(7/2)) + (3*(b*e + a*f)*(-1/5*((b* 
e - a*f)^2*Sqrt[a - b*x])/(a*b^3*(a + b*x)^(5/2)) + ((-2*((b*e^2)/a + 3*e* 
f - (4*a*f^2)/b)*Sqrt[a - b*x])/(3*(a + b*x)^(3/2)) - ((2*b^2*e^2 + 6*a*b* 
e*f + 7*a^2*f^2)*Sqrt[a - b*x])/(3*a^2*b*Sqrt[a + b*x]))/(5*a*b^2)))/(7*a* 
b)))/(a*b)
 

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 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

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 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(172)=344\).

Time = 0.37 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.79

method	result	size
gosper	\(\frac {-13 a^{4} b^{4} f^{4} x^{4}-4 a^{3} b^{5} e \,f^{3} x^{4}+20 a^{2} b^{6} e^{2} f^{2} x^{4}+24 a \,b^{7} e^{3} f \,x^{4}+8 b^{8} e^{4} x^{4}-4 a^{5} b^{3} f^{4} x^{3}-12 a^{4} b^{4} e \,f^{3} x^{3}+60 a^{3} b^{5} e^{2} f^{2} x^{3}+72 a^{2} b^{6} e^{3} f \,x^{3}+24 a \,b^{7} e^{4} x^{3}+20 a^{6} b^{2} f^{4} x^{2}+60 a^{5} b^{3} e \,f^{3} x^{2}+50 a^{4} b^{4} e^{2} f^{2} x^{2}+60 a^{3} b^{5} e^{3} f \,x^{2}+20 a^{2} b^{6} e^{4} x^{2}+24 a^{7} b \,f^{4} x +72 a^{6} b^{2} e \,f^{3} x +60 a^{5} b^{3} e^{2} f^{2} x -12 a^{4} b^{4} e^{3} f x -4 a^{3} b^{5} e^{4} x +8 a^{8} f^{4}+24 a^{7} b e \,f^{3}+20 a^{6} b^{2} e^{2} f^{2}-4 a^{5} b^{3} e^{3} f -13 a^{4} b^{4} e^{4}}{35 \sqrt {-b x +a}\, \left (b x +a \right )^{\frac {7}{2}} a^{5} b^{5}}\)	\(361\)
default	\(\frac {-13 a^{4} b^{4} f^{4} x^{4}-4 a^{3} b^{5} e \,f^{3} x^{4}+20 a^{2} b^{6} e^{2} f^{2} x^{4}+24 a \,b^{7} e^{3} f \,x^{4}+8 b^{8} e^{4} x^{4}-4 a^{5} b^{3} f^{4} x^{3}-12 a^{4} b^{4} e \,f^{3} x^{3}+60 a^{3} b^{5} e^{2} f^{2} x^{3}+72 a^{2} b^{6} e^{3} f \,x^{3}+24 a \,b^{7} e^{4} x^{3}+20 a^{6} b^{2} f^{4} x^{2}+60 a^{5} b^{3} e \,f^{3} x^{2}+50 a^{4} b^{4} e^{2} f^{2} x^{2}+60 a^{3} b^{5} e^{3} f \,x^{2}+20 a^{2} b^{6} e^{4} x^{2}+24 a^{7} b \,f^{4} x +72 a^{6} b^{2} e \,f^{3} x +60 a^{5} b^{3} e^{2} f^{2} x -12 a^{4} b^{4} e^{3} f x -4 a^{3} b^{5} e^{4} x +8 a^{8} f^{4}+24 a^{7} b e \,f^{3}+20 a^{6} b^{2} e^{2} f^{2}-4 a^{5} b^{3} e^{3} f -13 a^{4} b^{4} e^{4}}{35 \sqrt {-b x +a}\, \left (b x +a \right )^{\frac {7}{2}} a^{5} b^{5}}\)	\(361\)
orering	\(\frac {-13 a^{4} b^{4} f^{4} x^{4}-4 a^{3} b^{5} e \,f^{3} x^{4}+20 a^{2} b^{6} e^{2} f^{2} x^{4}+24 a \,b^{7} e^{3} f \,x^{4}+8 b^{8} e^{4} x^{4}-4 a^{5} b^{3} f^{4} x^{3}-12 a^{4} b^{4} e \,f^{3} x^{3}+60 a^{3} b^{5} e^{2} f^{2} x^{3}+72 a^{2} b^{6} e^{3} f \,x^{3}+24 a \,b^{7} e^{4} x^{3}+20 a^{6} b^{2} f^{4} x^{2}+60 a^{5} b^{3} e \,f^{3} x^{2}+50 a^{4} b^{4} e^{2} f^{2} x^{2}+60 a^{3} b^{5} e^{3} f \,x^{2}+20 a^{2} b^{6} e^{4} x^{2}+24 a^{7} b \,f^{4} x +72 a^{6} b^{2} e \,f^{3} x +60 a^{5} b^{3} e^{2} f^{2} x -12 a^{4} b^{4} e^{3} f x -4 a^{3} b^{5} e^{4} x +8 a^{8} f^{4}+24 a^{7} b e \,f^{3}+20 a^{6} b^{2} e^{2} f^{2}-4 a^{5} b^{3} e^{3} f -13 a^{4} b^{4} e^{4}}{35 \sqrt {-b x +a}\, \left (b x +a \right )^{\frac {7}{2}} a^{5} b^{5}}\)	\(361\)

Input:

int((f*x+e)^4/(-b*x+a)^(3/2)/(b*x+a)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

1/35*(-13*a^4*b^4*f^4*x^4-4*a^3*b^5*e*f^3*x^4+20*a^2*b^6*e^2*f^2*x^4+24*a* 
b^7*e^3*f*x^4+8*b^8*e^4*x^4-4*a^5*b^3*f^4*x^3-12*a^4*b^4*e*f^3*x^3+60*a^3* 
b^5*e^2*f^2*x^3+72*a^2*b^6*e^3*f*x^3+24*a*b^7*e^4*x^3+20*a^6*b^2*f^4*x^2+6 
0*a^5*b^3*e*f^3*x^2+50*a^4*b^4*e^2*f^2*x^2+60*a^3*b^5*e^3*f*x^2+20*a^2*b^6 
*e^4*x^2+24*a^7*b*f^4*x+72*a^6*b^2*e*f^3*x+60*a^5*b^3*e^2*f^2*x-12*a^4*b^4 
*e^3*f*x-4*a^3*b^5*e^4*x+8*a^8*f^4+24*a^7*b*e*f^3+20*a^6*b^2*e^2*f^2-4*a^5 
*b^3*e^3*f-13*a^4*b^4*e^4)/(-b*x+a)^(1/2)/(b*x+a)^(7/2)/a^5/b^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (172) = 344\).

Time = 0.26 (sec) , antiderivative size = 764, normalized size of antiderivative = 3.78 \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\frac {13 \, a^{5} b^{4} e^{4} + 4 \, a^{6} b^{3} e^{3} f - 20 \, a^{7} b^{2} e^{2} f^{2} - 24 \, a^{8} b e f^{3} - 8 \, a^{9} f^{4} - {\left (13 \, b^{9} e^{4} + 4 \, a b^{8} e^{3} f - 20 \, a^{2} b^{7} e^{2} f^{2} - 24 \, a^{3} b^{6} e f^{3} - 8 \, a^{4} b^{5} f^{4}\right )} x^{5} - 3 \, {\left (13 \, a b^{8} e^{4} + 4 \, a^{2} b^{7} e^{3} f - 20 \, a^{3} b^{6} e^{2} f^{2} - 24 \, a^{4} b^{5} e f^{3} - 8 \, a^{5} b^{4} f^{4}\right )} x^{4} - 2 \, {\left (13 \, a^{2} b^{7} e^{4} + 4 \, a^{3} b^{6} e^{3} f - 20 \, a^{4} b^{5} e^{2} f^{2} - 24 \, a^{5} b^{4} e f^{3} - 8 \, a^{6} b^{3} f^{4}\right )} x^{3} + 2 \, {\left (13 \, a^{3} b^{6} e^{4} + 4 \, a^{4} b^{5} e^{3} f - 20 \, a^{5} b^{4} e^{2} f^{2} - 24 \, a^{6} b^{3} e f^{3} - 8 \, a^{7} b^{2} f^{4}\right )} x^{2} + {\left (13 \, a^{4} b^{4} e^{4} + 4 \, a^{5} b^{3} e^{3} f - 20 \, a^{6} b^{2} e^{2} f^{2} - 24 \, a^{7} b e f^{3} - 8 \, a^{8} f^{4} - {\left (8 \, b^{8} e^{4} + 24 \, a b^{7} e^{3} f + 20 \, a^{2} b^{6} e^{2} f^{2} - 4 \, a^{3} b^{5} e f^{3} - 13 \, a^{4} b^{4} f^{4}\right )} x^{4} - 4 \, {\left (6 \, a b^{7} e^{4} + 18 \, a^{2} b^{6} e^{3} f + 15 \, a^{3} b^{5} e^{2} f^{2} - 3 \, a^{4} b^{4} e f^{3} - a^{5} b^{3} f^{4}\right )} x^{3} - 10 \, {\left (2 \, a^{2} b^{6} e^{4} + 6 \, a^{3} b^{5} e^{3} f + 5 \, a^{4} b^{4} e^{2} f^{2} + 6 \, a^{5} b^{3} e f^{3} + 2 \, a^{6} b^{2} f^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} e^{4} + 3 \, a^{4} b^{4} e^{3} f - 15 \, a^{5} b^{3} e^{2} f^{2} - 18 \, a^{6} b^{2} e f^{3} - 6 \, a^{7} b f^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {-b x + a} + 3 \, {\left (13 \, a^{4} b^{5} e^{4} + 4 \, a^{5} b^{4} e^{3} f - 20 \, a^{6} b^{3} e^{2} f^{2} - 24 \, a^{7} b^{2} e f^{3} - 8 \, a^{8} b f^{4}\right )} x}{35 \, {\left (a^{5} b^{10} x^{5} + 3 \, a^{6} b^{9} x^{4} + 2 \, a^{7} b^{8} x^{3} - 2 \, a^{8} b^{7} x^{2} - 3 \, a^{9} b^{6} x - a^{10} b^{5}\right )}} \] Input:

integrate((f*x+e)^4/(-b*x+a)^(3/2)/(b*x+a)^(9/2),x, algorithm="fricas")
 

Output:

1/35*(13*a^5*b^4*e^4 + 4*a^6*b^3*e^3*f - 20*a^7*b^2*e^2*f^2 - 24*a^8*b*e*f 
^3 - 8*a^9*f^4 - (13*b^9*e^4 + 4*a*b^8*e^3*f - 20*a^2*b^7*e^2*f^2 - 24*a^3 
*b^6*e*f^3 - 8*a^4*b^5*f^4)*x^5 - 3*(13*a*b^8*e^4 + 4*a^2*b^7*e^3*f - 20*a 
^3*b^6*e^2*f^2 - 24*a^4*b^5*e*f^3 - 8*a^5*b^4*f^4)*x^4 - 2*(13*a^2*b^7*e^4 
 + 4*a^3*b^6*e^3*f - 20*a^4*b^5*e^2*f^2 - 24*a^5*b^4*e*f^3 - 8*a^6*b^3*f^4 
)*x^3 + 2*(13*a^3*b^6*e^4 + 4*a^4*b^5*e^3*f - 20*a^5*b^4*e^2*f^2 - 24*a^6* 
b^3*e*f^3 - 8*a^7*b^2*f^4)*x^2 + (13*a^4*b^4*e^4 + 4*a^5*b^3*e^3*f - 20*a^ 
6*b^2*e^2*f^2 - 24*a^7*b*e*f^3 - 8*a^8*f^4 - (8*b^8*e^4 + 24*a*b^7*e^3*f + 
 20*a^2*b^6*e^2*f^2 - 4*a^3*b^5*e*f^3 - 13*a^4*b^4*f^4)*x^4 - 4*(6*a*b^7*e 
^4 + 18*a^2*b^6*e^3*f + 15*a^3*b^5*e^2*f^2 - 3*a^4*b^4*e*f^3 - a^5*b^3*f^4 
)*x^3 - 10*(2*a^2*b^6*e^4 + 6*a^3*b^5*e^3*f + 5*a^4*b^4*e^2*f^2 + 6*a^5*b^ 
3*e*f^3 + 2*a^6*b^2*f^4)*x^2 + 4*(a^3*b^5*e^4 + 3*a^4*b^4*e^3*f - 15*a^5*b 
^3*e^2*f^2 - 18*a^6*b^2*e*f^3 - 6*a^7*b*f^4)*x)*sqrt(b*x + a)*sqrt(-b*x + 
a) + 3*(13*a^4*b^5*e^4 + 4*a^5*b^4*e^3*f - 20*a^6*b^3*e^2*f^2 - 24*a^7*b^2 
*e*f^3 - 8*a^8*b*f^4)*x)/(a^5*b^10*x^5 + 3*a^6*b^9*x^4 + 2*a^7*b^8*x^3 - 2 
*a^8*b^7*x^2 - 3*a^9*b^6*x - a^10*b^5)
 

Sympy [F]

\[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\int \frac {\left (e + f x\right )^{4}}{\left (a - b x\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((f*x+e)**4/(-b*x+a)**(3/2)/(b*x+a)**(9/2),x)
 

Output:

Integral((e + f*x)**4/((a - b*x)**(3/2)*(a + b*x)**(9/2)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2002 vs. \(2 (172) = 344\).

Time = 0.09 (sec) , antiderivative size = 2002, normalized size of antiderivative = 9.91 \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^4/(-b*x+a)^(3/2)/(b*x+a)^(9/2),x, algorithm="maxima")
 

Output:

-1/7*a^4*f^4/(sqrt(-b^2*x^2 + a^2)*a*b^8*x^3 + 3*sqrt(-b^2*x^2 + a^2)*a^2* 
b^7*x^2 + 3*sqrt(-b^2*x^2 + a^2)*a^3*b^6*x + sqrt(-b^2*x^2 + a^2)*a^4*b^5) 
 - 4/35*a^4*f^4/(sqrt(-b^2*x^2 + a^2)*a^2*b^7*x^2 + 2*sqrt(-b^2*x^2 + a^2) 
*a^3*b^6*x + sqrt(-b^2*x^2 + a^2)*a^4*b^5) - 4/35*a^4*f^4/(sqrt(-b^2*x^2 + 
 a^2)*a^3*b^6*x + sqrt(-b^2*x^2 + a^2)*a^4*b^5) + 4/7*a^3*e*f^3/(sqrt(-b^2 
*x^2 + a^2)*a*b^7*x^3 + 3*sqrt(-b^2*x^2 + a^2)*a^2*b^6*x^2 + 3*sqrt(-b^2*x 
^2 + a^2)*a^3*b^5*x + sqrt(-b^2*x^2 + a^2)*a^4*b^4) + 16/35*a^3*e*f^3/(sqr 
t(-b^2*x^2 + a^2)*a^2*b^6*x^2 + 2*sqrt(-b^2*x^2 + a^2)*a^3*b^5*x + sqrt(-b 
^2*x^2 + a^2)*a^4*b^4) + 16/35*a^3*e*f^3/(sqrt(-b^2*x^2 + a^2)*a^3*b^5*x + 
 sqrt(-b^2*x^2 + a^2)*a^4*b^4) + 4/5*a^3*f^4/(sqrt(-b^2*x^2 + a^2)*a*b^7*x 
^2 + 2*sqrt(-b^2*x^2 + a^2)*a^2*b^6*x + sqrt(-b^2*x^2 + a^2)*a^3*b^5) + 4/ 
5*a^3*f^4/(sqrt(-b^2*x^2 + a^2)*a^2*b^6*x + sqrt(-b^2*x^2 + a^2)*a^3*b^5) 
- 6/7*a^2*e^2*f^2/(sqrt(-b^2*x^2 + a^2)*a*b^6*x^3 + 3*sqrt(-b^2*x^2 + a^2) 
*a^2*b^5*x^2 + 3*sqrt(-b^2*x^2 + a^2)*a^3*b^4*x + sqrt(-b^2*x^2 + a^2)*a^4 
*b^3) - 24/35*a^2*e^2*f^2/(sqrt(-b^2*x^2 + a^2)*a^2*b^5*x^2 + 2*sqrt(-b^2* 
x^2 + a^2)*a^3*b^4*x + sqrt(-b^2*x^2 + a^2)*a^4*b^3) - 24/35*a^2*e^2*f^2/( 
sqrt(-b^2*x^2 + a^2)*a^3*b^4*x + sqrt(-b^2*x^2 + a^2)*a^4*b^3) - 12/5*a^2* 
e*f^3/(sqrt(-b^2*x^2 + a^2)*a*b^6*x^2 + 2*sqrt(-b^2*x^2 + a^2)*a^2*b^5*x + 
 sqrt(-b^2*x^2 + a^2)*a^3*b^4) - 12/5*a^2*e*f^3/(sqrt(-b^2*x^2 + a^2)*a^2* 
b^5*x + sqrt(-b^2*x^2 + a^2)*a^3*b^4) - 2*a^2*f^4/(sqrt(-b^2*x^2 + a^2)...
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((f*x+e)^4/(-b*x+a)^(3/2)/(b*x+a)^(9/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 1.90 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.04 \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=-\frac {\sqrt {a-b\,x}\,\left (\frac {8\,a^8\,f^4+24\,a^7\,b\,e\,f^3+20\,a^6\,b^2\,e^2\,f^2-4\,a^5\,b^3\,e^3\,f-13\,a^4\,b^4\,e^4}{35\,a^5\,b^9}+\frac {x\,\left (24\,a^7\,b\,f^4+72\,a^6\,b^2\,e\,f^3+60\,a^5\,b^3\,e^2\,f^2-12\,a^4\,b^4\,e^3\,f-4\,a^3\,b^5\,e^4\right )}{35\,a^5\,b^9}+\frac {x^3\,\left (-4\,a^5\,b^3\,f^4-12\,a^4\,b^4\,e\,f^3+60\,a^3\,b^5\,e^2\,f^2+72\,a^2\,b^6\,e^3\,f+24\,a\,b^7\,e^4\right )}{35\,a^5\,b^9}+\frac {x^4\,\left (-13\,a^4\,b^4\,f^4-4\,a^3\,b^5\,e\,f^3+20\,a^2\,b^6\,e^2\,f^2+24\,a\,b^7\,e^3\,f+8\,b^8\,e^4\right )}{35\,a^5\,b^9}+\frac {x^2\,\left (20\,a^6\,b^2\,f^4+60\,a^5\,b^3\,e\,f^3+50\,a^4\,b^4\,e^2\,f^2+60\,a^3\,b^5\,e^3\,f+20\,a^2\,b^6\,e^4\right )}{35\,a^5\,b^9}\right )}{x^4\,\sqrt {a+b\,x}-\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {2\,a\,x^3\,\sqrt {a+b\,x}}{b}-\frac {2\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \] Input:

int((e + f*x)^4/((a + b*x)^(9/2)*(a - b*x)^(3/2)),x)
 

Output:

-((a - b*x)^(1/2)*((8*a^8*f^4 - 13*a^4*b^4*e^4 - 4*a^5*b^3*e^3*f + 20*a^6* 
b^2*e^2*f^2 + 24*a^7*b*e*f^3)/(35*a^5*b^9) + (x*(24*a^7*b*f^4 - 4*a^3*b^5* 
e^4 - 12*a^4*b^4*e^3*f + 72*a^6*b^2*e*f^3 + 60*a^5*b^3*e^2*f^2))/(35*a^5*b 
^9) + (x^3*(24*a*b^7*e^4 - 4*a^5*b^3*f^4 + 72*a^2*b^6*e^3*f - 12*a^4*b^4*e 
*f^3 + 60*a^3*b^5*e^2*f^2))/(35*a^5*b^9) + (x^4*(8*b^8*e^4 - 13*a^4*b^4*f^ 
4 - 4*a^3*b^5*e*f^3 + 20*a^2*b^6*e^2*f^2 + 24*a*b^7*e^3*f))/(35*a^5*b^9) + 
 (x^2*(20*a^2*b^6*e^4 + 20*a^6*b^2*f^4 + 60*a^3*b^5*e^3*f + 60*a^5*b^3*e*f 
^3 + 50*a^4*b^4*e^2*f^2))/(35*a^5*b^9)))/(x^4*(a + b*x)^(1/2) - (a^4*(a + 
b*x)^(1/2))/b^4 + (2*a*x^3*(a + b*x)^(1/2))/b - (2*a^3*x*(a + b*x)^(1/2))/ 
b^3)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.94 \[ \int \frac {(e+f x)^4}{(a-b x)^{3/2} (a+b x)^{9/2}} \, dx=\frac {-13 a^{4} b^{4} f^{4} x^{4}-4 a^{3} b^{5} e \,f^{3} x^{4}+20 a^{2} b^{6} e^{2} f^{2} x^{4}+24 a \,b^{7} e^{3} f \,x^{4}+8 b^{8} e^{4} x^{4}-4 a^{5} b^{3} f^{4} x^{3}-12 a^{4} b^{4} e \,f^{3} x^{3}+60 a^{3} b^{5} e^{2} f^{2} x^{3}+72 a^{2} b^{6} e^{3} f \,x^{3}+24 a \,b^{7} e^{4} x^{3}+20 a^{6} b^{2} f^{4} x^{2}+60 a^{5} b^{3} e \,f^{3} x^{2}+50 a^{4} b^{4} e^{2} f^{2} x^{2}+60 a^{3} b^{5} e^{3} f \,x^{2}+20 a^{2} b^{6} e^{4} x^{2}+24 a^{7} b \,f^{4} x +72 a^{6} b^{2} e \,f^{3} x +60 a^{5} b^{3} e^{2} f^{2} x -12 a^{4} b^{4} e^{3} f x -4 a^{3} b^{5} e^{4} x +8 a^{8} f^{4}+24 a^{7} b e \,f^{3}+20 a^{6} b^{2} e^{2} f^{2}-4 a^{5} b^{3} e^{3} f -13 a^{4} b^{4} e^{4}}{35 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{5} b^{5} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:

int((f*x+e)^4/(-b*x+a)^(3/2)/(b*x+a)^(9/2),x)
 

Output:

(8*a**8*f**4 + 24*a**7*b*e*f**3 + 24*a**7*b*f**4*x + 20*a**6*b**2*e**2*f** 
2 + 72*a**6*b**2*e*f**3*x + 20*a**6*b**2*f**4*x**2 - 4*a**5*b**3*e**3*f + 
60*a**5*b**3*e**2*f**2*x + 60*a**5*b**3*e*f**3*x**2 - 4*a**5*b**3*f**4*x** 
3 - 13*a**4*b**4*e**4 - 12*a**4*b**4*e**3*f*x + 50*a**4*b**4*e**2*f**2*x** 
2 - 12*a**4*b**4*e*f**3*x**3 - 13*a**4*b**4*f**4*x**4 - 4*a**3*b**5*e**4*x 
 + 60*a**3*b**5*e**3*f*x**2 + 60*a**3*b**5*e**2*f**2*x**3 - 4*a**3*b**5*e* 
f**3*x**4 + 20*a**2*b**6*e**4*x**2 + 72*a**2*b**6*e**3*f*x**3 + 20*a**2*b* 
*6*e**2*f**2*x**4 + 24*a*b**7*e**4*x**3 + 24*a*b**7*e**3*f*x**4 + 8*b**8*e 
**4*x**4)/(35*sqrt(a + b*x)*sqrt(a - b*x)*a**5*b**5*(a**3 + 3*a**2*b*x + 3 
*a*b**2*x**2 + b**3*x**3))