Integrand size = 33, antiderivative size = 219 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^4 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac {2 e (b d-a e)^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^5}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {e^3 (b d-a e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac {e^4 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5} \] Output:
1/5*(-a*e+b*d)^4*(b*x+a)^4*((b*x+a)^2)^(1/2)/b^5+2/3*e*(-a*e+b*d)^3*(b*x+a )^5*((b*x+a)^2)^(1/2)/b^5+6/7*e^2*(-a*e+b*d)^2*(b*x+a)^6*((b*x+a)^2)^(1/2) /b^5+1/2*e^3*(-a*e+b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^5+1/9*e^4*(b*x+a)^8* ((b*x+a)^2)^(1/2)/b^5
Time = 1.10 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.22 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (126 a^4 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+84 a^3 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+36 a^2 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+9 a b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )\right )}{630 (a+b x)} \] Input:
Integrate[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
(x*Sqrt[(a + b*x)^2]*(126*a^4*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e ^3*x^3 + e^4*x^4) + 84*a^3*b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24* d*e^3*x^3 + 5*e^4*x^4) + 36*a^2*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^ 2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 9*a*b^3*x^3*(70*d^4 + 224*d^3*e*x + 2 80*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + b^4*x^4*(126*d^4 + 420*d^3* e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4)))/(630*(a + b*x))
Time = 0.67 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 2009}
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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^4dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^4dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^4 (a+b x)^8}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^7}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^5}{b^4}+\frac {(b d-a e)^4 (a+b x)^4}{b^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac {6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac {2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac {(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac {e^4 (a+b x)^9}{9 b^5}\right )}{a+b x}\) |
Input:
Int[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^4*(a + b*x)^5)/(5*b^5) + (2*e *(b*d - a*e)^3*(a + b*x)^6)/(3*b^5) + (6*e^2*(b*d - a*e)^2*(a + b*x)^7)/(7 *b^5) + (e^3*(b*d - a*e)*(a + b*x)^8)/(2*b^5) + (e^4*(a + b*x)^9)/(9*b^5)) )/(a + b*x)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(154)=308\).
Time = 1.49 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55
| method | result | size |
| gosper | \(\frac {x \left (70 b^{4} e^{4} x^{8}+315 x^{7} a \,b^{3} e^{4}+315 x^{7} d \,e^{3} b^{4}+540 x^{6} a^{2} b^{2} e^{4}+1440 x^{6} a \,b^{3} d \,e^{3}+540 x^{6} b^{4} d^{2} e^{2}+420 x^{5} a^{3} b \,e^{4}+2520 x^{5} a^{2} b^{2} d \,e^{3}+2520 x^{5} a \,b^{3} d^{2} e^{2}+420 x^{5} b^{4} d^{3} e +126 a^{4} e^{4} x^{4}+2016 a^{3} b d \,e^{3} x^{4}+4536 a^{2} b^{2} d^{2} e^{2} x^{4}+2016 a \,b^{3} d^{3} e \,x^{4}+126 b^{4} d^{4} x^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 a^{4} d^{2} e^{2} x^{2}+3360 a^{3} b \,d^{3} e \,x^{2}+1260 a^{2} b^{2} d^{4} x^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 a^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{630 \left (b x +a \right )^{3}}\) | \(339\) |
| default | \(\frac {x \left (70 b^{4} e^{4} x^{8}+315 x^{7} a \,b^{3} e^{4}+315 x^{7} d \,e^{3} b^{4}+540 x^{6} a^{2} b^{2} e^{4}+1440 x^{6} a \,b^{3} d \,e^{3}+540 x^{6} b^{4} d^{2} e^{2}+420 x^{5} a^{3} b \,e^{4}+2520 x^{5} a^{2} b^{2} d \,e^{3}+2520 x^{5} a \,b^{3} d^{2} e^{2}+420 x^{5} b^{4} d^{3} e +126 a^{4} e^{4} x^{4}+2016 a^{3} b d \,e^{3} x^{4}+4536 a^{2} b^{2} d^{2} e^{2} x^{4}+2016 a \,b^{3} d^{3} e \,x^{4}+126 b^{4} d^{4} x^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 a^{4} d^{2} e^{2} x^{2}+3360 a^{3} b \,d^{3} e \,x^{2}+1260 a^{2} b^{2} d^{4} x^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 a^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{630 \left (b x +a \right )^{3}}\) | \(339\) |
| orering | \(\frac {x \left (70 b^{4} e^{4} x^{8}+315 x^{7} a \,b^{3} e^{4}+315 x^{7} d \,e^{3} b^{4}+540 x^{6} a^{2} b^{2} e^{4}+1440 x^{6} a \,b^{3} d \,e^{3}+540 x^{6} b^{4} d^{2} e^{2}+420 x^{5} a^{3} b \,e^{4}+2520 x^{5} a^{2} b^{2} d \,e^{3}+2520 x^{5} a \,b^{3} d^{2} e^{2}+420 x^{5} b^{4} d^{3} e +126 a^{4} e^{4} x^{4}+2016 a^{3} b d \,e^{3} x^{4}+4536 a^{2} b^{2} d^{2} e^{2} x^{4}+2016 a \,b^{3} d^{3} e \,x^{4}+126 b^{4} d^{4} x^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 a^{4} d^{2} e^{2} x^{2}+3360 a^{3} b \,d^{3} e \,x^{2}+1260 a^{2} b^{2} d^{4} x^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 a^{4} d^{4}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{630 \left (b x +a \right )^{3}}\) | \(348\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{4} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a \,b^{3} e^{4}+4 d \,e^{3} b^{4}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{2} b^{2} e^{4}+16 a \,b^{3} d \,e^{3}+6 b^{4} d^{2} e^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} b \,e^{4}+24 a^{2} b^{2} d \,e^{3}+24 a \,b^{3} d^{2} e^{2}+4 b^{4} d^{3} e \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{4}+16 a^{3} b d \,e^{3}+36 a^{2} b^{2} d^{2} e^{2}+16 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 d \,e^{3} a^{4}+24 d^{2} e^{2} a^{3} b +24 a^{2} b^{2} d^{3} e +4 d^{4} a \,b^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 d^{2} e^{2} a^{4}+16 d^{3} e \,a^{3} b +6 a^{2} b^{2} d^{4}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{4} d^{3} e +4 a^{3} b \,d^{4}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} d^{4} x}{b x +a}\) | \(439\) |
Input:
int((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/630*x*(70*b^4*e^4*x^8+315*a*b^3*e^4*x^7+315*b^4*d*e^3*x^7+540*a^2*b^2*e^ 4*x^6+1440*a*b^3*d*e^3*x^6+540*b^4*d^2*e^2*x^6+420*a^3*b*e^4*x^5+2520*a^2* b^2*d*e^3*x^5+2520*a*b^3*d^2*e^2*x^5+420*b^4*d^3*e*x^5+126*a^4*e^4*x^4+201 6*a^3*b*d*e^3*x^4+4536*a^2*b^2*d^2*e^2*x^4+2016*a*b^3*d^3*e*x^4+126*b^4*d^ 4*x^4+630*a^4*d*e^3*x^3+3780*a^3*b*d^2*e^2*x^3+3780*a^2*b^2*d^3*e*x^3+630* a*b^3*d^4*x^3+1260*a^4*d^2*e^2*x^2+3360*a^3*b*d^3*e*x^2+1260*a^2*b^2*d^4*x ^2+1260*a^4*d^3*e*x+1260*a^3*b*d^4*x+630*a^4*d^4)*((b*x+a)^2)^(3/2)/(b*x+a )^3
Time = 0.07 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.30 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{9} \, b^{4} e^{4} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (3 \, b^{4} d^{2} e^{2} + 8 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{4} + 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} + 16 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} x^{5} + {\left (a b^{3} d^{4} + 6 \, a^{2} b^{2} d^{3} e + 6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{4} + 8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fric as")
Output:
1/9*b^4*e^4*x^9 + a^4*d^4*x + 1/2*(b^4*d*e^3 + a*b^3*e^4)*x^8 + 2/7*(3*b^4 *d^2*e^2 + 8*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^7 + 2/3*(b^4*d^3*e + 6*a*b^3*d ^2*e^2 + 6*a^2*b^2*d*e^3 + a^3*b*e^4)*x^6 + 1/5*(b^4*d^4 + 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 + 16*a^3*b*d*e^3 + a^4*e^4)*x^5 + (a*b^3*d^4 + 6*a^2* b^2*d^3*e + 6*a^3*b*d^2*e^2 + a^4*d*e^3)*x^4 + 2/3*(3*a^2*b^2*d^4 + 8*a^3* b*d^3*e + 3*a^4*d^2*e^2)*x^3 + 2*(a^3*b*d^4 + a^4*d^3*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 10220 vs. \(2 (156) = 312\).
Time = 1.67 (sec) , antiderivative size = 10220, normalized size of antiderivative = 46.67 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
Output:
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**3*e**4*x**8/9 + x**7*(28*a *b**4*e**4/9 + 4*b**5*d*e**3)/(8*b**2) + x**6*(82*a**2*b**3*e**4/9 + 20*a* b**4*d*e**3 - 15*a*(28*a*b**4*e**4/9 + 4*b**5*d*e**3)/(8*b) + 6*b**5*d**2* e**2)/(7*b**2) + x**5*(10*a**3*b**2*e**4 + 40*a**2*b**3*d*e**3 - 7*a**2*(2 8*a*b**4*e**4/9 + 4*b**5*d*e**3)/(8*b**2) + 30*a*b**4*d**2*e**2 - 13*a*(82 *a**2*b**3*e**4/9 + 20*a*b**4*d*e**3 - 15*a*(28*a*b**4*e**4/9 + 4*b**5*d*e **3)/(8*b) + 6*b**5*d**2*e**2)/(7*b) + 4*b**5*d**3*e)/(6*b**2) + x**4*(5*a **4*b*e**4 + 40*a**3*b**2*d*e**3 + 60*a**2*b**3*d**2*e**2 - 6*a**2*(82*a** 2*b**3*e**4/9 + 20*a*b**4*d*e**3 - 15*a*(28*a*b**4*e**4/9 + 4*b**5*d*e**3) /(8*b) + 6*b**5*d**2*e**2)/(7*b**2) + 20*a*b**4*d**3*e - 11*a*(10*a**3*b** 2*e**4 + 40*a**2*b**3*d*e**3 - 7*a**2*(28*a*b**4*e**4/9 + 4*b**5*d*e**3)/( 8*b**2) + 30*a*b**4*d**2*e**2 - 13*a*(82*a**2*b**3*e**4/9 + 20*a*b**4*d*e* *3 - 15*a*(28*a*b**4*e**4/9 + 4*b**5*d*e**3)/(8*b) + 6*b**5*d**2*e**2)/(7* b) + 4*b**5*d**3*e)/(6*b) + b**5*d**4)/(5*b**2) + x**3*(a**5*e**4 + 20*a** 4*b*d*e**3 + 60*a**3*b**2*d**2*e**2 + 40*a**2*b**3*d**3*e - 5*a**2*(10*a** 3*b**2*e**4 + 40*a**2*b**3*d*e**3 - 7*a**2*(28*a*b**4*e**4/9 + 4*b**5*d*e* *3)/(8*b**2) + 30*a*b**4*d**2*e**2 - 13*a*(82*a**2*b**3*e**4/9 + 20*a*b**4 *d*e**3 - 15*a*(28*a*b**4*e**4/9 + 4*b**5*d*e**3)/(8*b) + 6*b**5*d**2*e**2 )/(7*b) + 4*b**5*d**3*e)/(6*b**2) + 5*a*b**4*d**4 - 9*a*(5*a**4*b*e**4 + 4 0*a**3*b**2*d*e**3 + 60*a**2*b**3*d**2*e**2 - 6*a**2*(82*a**2*b**3*e**4...
Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (154) = 308\).
Time = 0.05 (sec) , antiderivative size = 998, normalized size of antiderivative = 4.56 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxi ma")
Output:
1/9*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^4*x^4/b - 13/72*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^4*x^3/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^4*x - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5*e^4*x/b^4 + 37/168*(b^2*x^2 + 2*a* b*x + a^2)^(5/2)*a^2*e^4*x^2/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2 *d^4/b - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^6*e^4/b^5 - 121/504*(b^2*x^ 2 + 2*a*b*x + a^2)^(5/2)*a^3*e^4*x/b^4 + 125/504*(b^2*x^2 + 2*a*b*x + a^2) ^(5/2)*a^4*e^4/b^5 + 1/8*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/ 2)*x^3/b^2 + 1/4*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*x /b^4 - 1/2*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x /b^3 + 1/2*(2*b*d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x /b^2 - 1/4*(b*d^4 + 4*a*d^3*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b - 11/ 56*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^2/b^3 + 2/7*(3* b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^2/b^2 + 1/4*(4*b* d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5/b^5 - 1/2*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^4 + 1/2*(2*b*d^3*e + 3* a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^3 - 1/4*(b*d^4 + 4*a*d^3* e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 + 13/56*(4*b*d*e^3 + a*e^4)*(b^ 2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^4 - 3/7*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^ 2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b^3 + 1/3*(2*b*d^3*e + 3*a*d^2*e^2)*(b^2* x^2 + 2*a*b*x + a^2)^(5/2)*x/b^2 - 69/280*(4*b*d*e^3 + a*e^4)*(b^2*x^2 ...
Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (154) = 308\).
Time = 0.19 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.45 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{9} \, b^{4} e^{4} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{4} d e^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{3} e^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, b^{4} d^{2} e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{7} \, a b^{3} d e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, a^{2} b^{2} e^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b^{4} d^{3} e x^{6} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{2} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b^{2} d e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a^{3} b e^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{5} \, a b^{3} d^{3} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {36}{5} \, a^{2} b^{2} d^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{5} \, a^{3} b d e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{4} e^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{3} e x^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b d^{2} e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} d e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{3} \, a^{3} b d^{3} e x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{4} d^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{4} d^{3} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{4} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (126 \, a^{5} b^{4} d^{4} - 84 \, a^{6} b^{3} d^{3} e + 36 \, a^{7} b^{2} d^{2} e^{2} - 9 \, a^{8} b d e^{3} + a^{9} e^{4}\right )} \mathrm {sgn}\left (b x + a\right )}{630 \, b^{5}} \] Input:
integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac ")
Output:
1/9*b^4*e^4*x^9*sgn(b*x + a) + 1/2*b^4*d*e^3*x^8*sgn(b*x + a) + 1/2*a*b^3* e^4*x^8*sgn(b*x + a) + 6/7*b^4*d^2*e^2*x^7*sgn(b*x + a) + 16/7*a*b^3*d*e^3 *x^7*sgn(b*x + a) + 6/7*a^2*b^2*e^4*x^7*sgn(b*x + a) + 2/3*b^4*d^3*e*x^6*s gn(b*x + a) + 4*a*b^3*d^2*e^2*x^6*sgn(b*x + a) + 4*a^2*b^2*d*e^3*x^6*sgn(b *x + a) + 2/3*a^3*b*e^4*x^6*sgn(b*x + a) + 1/5*b^4*d^4*x^5*sgn(b*x + a) + 16/5*a*b^3*d^3*e*x^5*sgn(b*x + a) + 36/5*a^2*b^2*d^2*e^2*x^5*sgn(b*x + a) + 16/5*a^3*b*d*e^3*x^5*sgn(b*x + a) + 1/5*a^4*e^4*x^5*sgn(b*x + a) + a*b^3 *d^4*x^4*sgn(b*x + a) + 6*a^2*b^2*d^3*e*x^4*sgn(b*x + a) + 6*a^3*b*d^2*e^2 *x^4*sgn(b*x + a) + a^4*d*e^3*x^4*sgn(b*x + a) + 2*a^2*b^2*d^4*x^3*sgn(b*x + a) + 16/3*a^3*b*d^3*e*x^3*sgn(b*x + a) + 2*a^4*d^2*e^2*x^3*sgn(b*x + a) + 2*a^3*b*d^4*x^2*sgn(b*x + a) + 2*a^4*d^3*e*x^2*sgn(b*x + a) + a^4*d^4*x *sgn(b*x + a) + 1/630*(126*a^5*b^4*d^4 - 84*a^6*b^3*d^3*e + 36*a^7*b^2*d^2 *e^2 - 9*a^8*b*d*e^3 + a^9*e^4)*sgn(b*x + a)/b^5
Timed out. \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \] Input:
int((a + b*x)*(d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
Output:
int((a + b*x)*(d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
Time = 0.26 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.47 \[ \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \left (70 b^{4} e^{4} x^{8}+315 a \,b^{3} e^{4} x^{7}+315 b^{4} d \,e^{3} x^{7}+540 a^{2} b^{2} e^{4} x^{6}+1440 a \,b^{3} d \,e^{3} x^{6}+540 b^{4} d^{2} e^{2} x^{6}+420 a^{3} b \,e^{4} x^{5}+2520 a^{2} b^{2} d \,e^{3} x^{5}+2520 a \,b^{3} d^{2} e^{2} x^{5}+420 b^{4} d^{3} e \,x^{5}+126 a^{4} e^{4} x^{4}+2016 a^{3} b d \,e^{3} x^{4}+4536 a^{2} b^{2} d^{2} e^{2} x^{4}+2016 a \,b^{3} d^{3} e \,x^{4}+126 b^{4} d^{4} x^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 a^{4} d^{2} e^{2} x^{2}+3360 a^{3} b \,d^{3} e \,x^{2}+1260 a^{2} b^{2} d^{4} x^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 a^{4} d^{4}\right )}{630} \] Input:
int((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
Output:
(x*(630*a**4*d**4 + 1260*a**4*d**3*e*x + 1260*a**4*d**2*e**2*x**2 + 630*a* *4*d*e**3*x**3 + 126*a**4*e**4*x**4 + 1260*a**3*b*d**4*x + 3360*a**3*b*d** 3*e*x**2 + 3780*a**3*b*d**2*e**2*x**3 + 2016*a**3*b*d*e**3*x**4 + 420*a**3 *b*e**4*x**5 + 1260*a**2*b**2*d**4*x**2 + 3780*a**2*b**2*d**3*e*x**3 + 453 6*a**2*b**2*d**2*e**2*x**4 + 2520*a**2*b**2*d*e**3*x**5 + 540*a**2*b**2*e* *4*x**6 + 630*a*b**3*d**4*x**3 + 2016*a*b**3*d**3*e*x**4 + 2520*a*b**3*d** 2*e**2*x**5 + 1440*a*b**3*d*e**3*x**6 + 315*a*b**3*e**4*x**7 + 126*b**4*d* *4*x**4 + 420*b**4*d**3*e*x**5 + 540*b**4*d**2*e**2*x**6 + 315*b**4*d*e**3 *x**7 + 70*b**4*e**4*x**8))/630