Integrand size = 33, antiderivative size = 277 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^4 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (1+m) (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (2+m) (a+b x)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (3+m) (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (4+m) (a+b x)}+\frac {b^4 (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (5+m) (a+b x)} \] Output:
(-a*e+b*d)^4*(e*x+d)^(1+m)*((b*x+a)^2)^(1/2)/e^5/(1+m)/(b*x+a)-4*b*(-a*e+b *d)^3*(e*x+d)^(2+m)*((b*x+a)^2)^(1/2)/e^5/(2+m)/(b*x+a)+6*b^2*(-a*e+b*d)^2 *(e*x+d)^(3+m)*((b*x+a)^2)^(1/2)/e^5/(3+m)/(b*x+a)-4*b^3*(-a*e+b*d)*(e*x+d )^(4+m)*((b*x+a)^2)^(1/2)/e^5/(4+m)/(b*x+a)+b^4*(e*x+d)^(5+m)*((b*x+a)^2)^ (1/2)/e^5/(5+m)/(b*x+a)
Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.50 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {\sqrt {(a+b x)^2} (d+e x)^{1+m} \left (\frac {(b d-a e)^4}{1+m}-\frac {4 b (b d-a e)^3 (d+e x)}{2+m}+\frac {6 b^2 (b d-a e)^2 (d+e x)^2}{3+m}-\frac {4 b^3 (b d-a e) (d+e x)^3}{4+m}+\frac {b^4 (d+e x)^4}{5+m}\right )}{e^5 (a+b x)} \] Input:
Integrate[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*((b*d - a*e)^4/(1 + m) - (4*b*(b*d - a*e)^3*(d + e*x))/(2 + m) + (6*b^2*(b*d - a*e)^2*(d + e*x)^2)/(3 + m) - (4 *b^3*(b*d - a*e)*(d + e*x)^3)/(4 + m) + (b^4*(d + e*x)^4)/(5 + m)))/(e^5*( a + b*x))
Time = 0.57 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 53, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^mdx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^mdx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(a e-b d)^4 (d+e x)^m}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{m+1}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{m+2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{m+3}}{e^4}+\frac {b^4 (d+e x)^{m+4}}{e^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac {(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac {b^4 (d+e x)^{m+5}}{e^5 (m+5)}\right )}{a+b x}\) |
Input:
Int[(a + b*x)*(d + e*x)^m*(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*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*b*(b*d - a*e)^3*(d + e*x)^(2 + m))/(e^5*(2 + m)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (4*b^3*(b*d - a*e)*(d + e*x)^( 4 + m))/(e^5*(4 + m)) + (b^4*(d + e*x)^(5 + m))/(e^5*(5 + m))))/(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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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. \(783\) vs. \(2(222)=444\).
Time = 1.29 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.83
| method | result | size |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{4} e^{4} m^{4} x^{4}+4 a \,b^{3} e^{4} m^{4} x^{3}+10 b^{4} e^{4} m^{3} x^{4}+6 a^{2} b^{2} e^{4} m^{4} x^{2}+44 a \,b^{3} e^{4} m^{3} x^{3}-4 b^{4} d \,e^{3} m^{3} x^{3}+35 b^{4} e^{4} m^{2} x^{4}+4 a^{3} b \,e^{4} m^{4} x +72 a^{2} b^{2} e^{4} m^{3} x^{2}-12 a \,b^{3} d \,e^{3} m^{3} x^{2}+164 a \,b^{3} e^{4} m^{2} x^{3}-24 b^{4} d \,e^{3} m^{2} x^{3}+50 b^{4} e^{4} m \,x^{4}+a^{4} e^{4} m^{4}+52 a^{3} b \,e^{4} m^{3} x -12 a^{2} b^{2} d \,e^{3} m^{3} x +294 a^{2} b^{2} e^{4} m^{2} x^{2}-96 a \,b^{3} d \,e^{3} m^{2} x^{2}+244 a \,b^{3} e^{4} m \,x^{3}+12 b^{4} d^{2} e^{2} m^{2} x^{2}-44 b^{4} d \,e^{3} m \,x^{3}+24 b^{4} x^{4} e^{4}+14 a^{4} e^{4} m^{3}-4 a^{3} b d \,e^{3} m^{3}+236 a^{3} b \,e^{4} m^{2} x -120 a^{2} b^{2} d \,e^{3} m^{2} x +468 a^{2} b^{2} e^{4} m \,x^{2}+24 a \,b^{3} d^{2} e^{2} m^{2} x -204 a \,b^{3} d \,e^{3} m \,x^{2}+120 x^{3} a \,b^{3} e^{4}+36 b^{4} d^{2} e^{2} m \,x^{2}-24 x^{3} b^{4} d \,e^{3}+71 a^{4} e^{4} m^{2}-48 a^{3} b d \,e^{3} m^{2}+428 a^{3} b \,e^{4} m x +12 a^{2} b^{2} d^{2} e^{2} m^{2}-348 a^{2} b^{2} d \,e^{3} m x +240 x^{2} a^{2} b^{2} e^{4}+144 a \,b^{3} d^{2} e^{2} m x -120 x^{2} a \,b^{3} d \,e^{3}-24 b^{4} d^{3} e m x +24 x^{2} b^{4} d^{2} e^{2}+154 a^{4} e^{4} m -188 a^{3} b d \,e^{3} m +240 x \,a^{3} b \,e^{4}+108 a^{2} b^{2} d^{2} e^{2} m -240 x \,a^{2} b^{2} d \,e^{3}-24 a \,b^{3} d^{3} e m +120 x a \,b^{3} d^{2} e^{2}-24 x \,b^{4} d^{3} e +120 a^{4} e^{4}-240 a^{3} b d \,e^{3}+240 a^{2} b^{2} d^{2} e^{2}-120 a \,b^{3} d^{3} e +24 b^{4} d^{4}\right )}{e^{5} \left (b x +a \right )^{3} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(784\) |
| orering | \(\frac {\left (b^{4} e^{4} m^{4} x^{4}+4 a \,b^{3} e^{4} m^{4} x^{3}+10 b^{4} e^{4} m^{3} x^{4}+6 a^{2} b^{2} e^{4} m^{4} x^{2}+44 a \,b^{3} e^{4} m^{3} x^{3}-4 b^{4} d \,e^{3} m^{3} x^{3}+35 b^{4} e^{4} m^{2} x^{4}+4 a^{3} b \,e^{4} m^{4} x +72 a^{2} b^{2} e^{4} m^{3} x^{2}-12 a \,b^{3} d \,e^{3} m^{3} x^{2}+164 a \,b^{3} e^{4} m^{2} x^{3}-24 b^{4} d \,e^{3} m^{2} x^{3}+50 b^{4} e^{4} m \,x^{4}+a^{4} e^{4} m^{4}+52 a^{3} b \,e^{4} m^{3} x -12 a^{2} b^{2} d \,e^{3} m^{3} x +294 a^{2} b^{2} e^{4} m^{2} x^{2}-96 a \,b^{3} d \,e^{3} m^{2} x^{2}+244 a \,b^{3} e^{4} m \,x^{3}+12 b^{4} d^{2} e^{2} m^{2} x^{2}-44 b^{4} d \,e^{3} m \,x^{3}+24 b^{4} x^{4} e^{4}+14 a^{4} e^{4} m^{3}-4 a^{3} b d \,e^{3} m^{3}+236 a^{3} b \,e^{4} m^{2} x -120 a^{2} b^{2} d \,e^{3} m^{2} x +468 a^{2} b^{2} e^{4} m \,x^{2}+24 a \,b^{3} d^{2} e^{2} m^{2} x -204 a \,b^{3} d \,e^{3} m \,x^{2}+120 x^{3} a \,b^{3} e^{4}+36 b^{4} d^{2} e^{2} m \,x^{2}-24 x^{3} b^{4} d \,e^{3}+71 a^{4} e^{4} m^{2}-48 a^{3} b d \,e^{3} m^{2}+428 a^{3} b \,e^{4} m x +12 a^{2} b^{2} d^{2} e^{2} m^{2}-348 a^{2} b^{2} d \,e^{3} m x +240 x^{2} a^{2} b^{2} e^{4}+144 a \,b^{3} d^{2} e^{2} m x -120 x^{2} a \,b^{3} d \,e^{3}-24 b^{4} d^{3} e m x +24 x^{2} b^{4} d^{2} e^{2}+154 a^{4} e^{4} m -188 a^{3} b d \,e^{3} m +240 x \,a^{3} b \,e^{4}+108 a^{2} b^{2} d^{2} e^{2} m -240 x \,a^{2} b^{2} d \,e^{3}-24 a \,b^{3} d^{3} e m +120 x a \,b^{3} d^{2} e^{2}-24 x \,b^{4} d^{3} e +120 a^{4} e^{4}-240 a^{3} b d \,e^{3}+240 a^{2} b^{2} d^{2} e^{2}-120 a \,b^{3} d^{3} e +24 b^{4} d^{4}\right ) \left (e x +d \right ) \left (e x +d \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) \left (b x +a \right )^{3}}\) | \(796\) |
| risch | \(\text {Expression too large to display}\) | \(1045\) |
Input:
int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/e^5*(e*x+d)^(1+m)/(b*x+a)^3*((b*x+a)^2)^(3/2)/(m^5+15*m^4+85*m^3+225*m^2 +274*m+120)*(b^4*e^4*m^4*x^4+4*a*b^3*e^4*m^4*x^3+10*b^4*e^4*m^3*x^4+6*a^2* b^2*e^4*m^4*x^2+44*a*b^3*e^4*m^3*x^3-4*b^4*d*e^3*m^3*x^3+35*b^4*e^4*m^2*x^ 4+4*a^3*b*e^4*m^4*x+72*a^2*b^2*e^4*m^3*x^2-12*a*b^3*d*e^3*m^3*x^2+164*a*b^ 3*e^4*m^2*x^3-24*b^4*d*e^3*m^2*x^3+50*b^4*e^4*m*x^4+a^4*e^4*m^4+52*a^3*b*e ^4*m^3*x-12*a^2*b^2*d*e^3*m^3*x+294*a^2*b^2*e^4*m^2*x^2-96*a*b^3*d*e^3*m^2 *x^2+244*a*b^3*e^4*m*x^3+12*b^4*d^2*e^2*m^2*x^2-44*b^4*d*e^3*m*x^3+24*b^4* e^4*x^4+14*a^4*e^4*m^3-4*a^3*b*d*e^3*m^3+236*a^3*b*e^4*m^2*x-120*a^2*b^2*d *e^3*m^2*x+468*a^2*b^2*e^4*m*x^2+24*a*b^3*d^2*e^2*m^2*x-204*a*b^3*d*e^3*m* x^2+120*a*b^3*e^4*x^3+36*b^4*d^2*e^2*m*x^2-24*b^4*d*e^3*x^3+71*a^4*e^4*m^2 -48*a^3*b*d*e^3*m^2+428*a^3*b*e^4*m*x+12*a^2*b^2*d^2*e^2*m^2-348*a^2*b^2*d *e^3*m*x+240*a^2*b^2*e^4*x^2+144*a*b^3*d^2*e^2*m*x-120*a*b^3*d*e^3*x^2-24* b^4*d^3*e*m*x+24*b^4*d^2*e^2*x^2+154*a^4*e^4*m-188*a^3*b*d*e^3*m+240*a^3*b *e^4*x+108*a^2*b^2*d^2*e^2*m-240*a^2*b^2*d*e^3*x-24*a*b^3*d^3*e*m+120*a*b^ 3*d^2*e^2*x-24*b^4*d^3*e*x+120*a^4*e^4-240*a^3*b*d*e^3+240*a^2*b^2*d^2*e^2 -120*a*b^3*d^3*e+24*b^4*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (222) = 444\).
Time = 0.09 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.25 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fric as")
Output:
(a^4*d*e^4*m^4 + 24*b^4*d^5 - 120*a*b^3*d^4*e + 240*a^2*b^2*d^3*e^2 - 240* a^3*b*d^2*e^3 + 120*a^4*d*e^4 + (b^4*e^5*m^4 + 10*b^4*e^5*m^3 + 35*b^4*e^5 *m^2 + 50*b^4*e^5*m + 24*b^4*e^5)*x^5 + (120*a*b^3*e^5 + (b^4*d*e^4 + 4*a* b^3*e^5)*m^4 + 2*(3*b^4*d*e^4 + 22*a*b^3*e^5)*m^3 + (11*b^4*d*e^4 + 164*a* b^3*e^5)*m^2 + 2*(3*b^4*d*e^4 + 122*a*b^3*e^5)*m)*x^4 - 2*(2*a^3*b*d^2*e^3 - 7*a^4*d*e^4)*m^3 + 2*(120*a^2*b^2*e^5 + (2*a*b^3*d*e^4 + 3*a^2*b^2*e^5) *m^4 - 2*(b^4*d^2*e^3 - 8*a*b^3*d*e^4 - 18*a^2*b^2*e^5)*m^3 - (6*b^4*d^2*e ^3 - 34*a*b^3*d*e^4 - 147*a^2*b^2*e^5)*m^2 - 2*(2*b^4*d^2*e^3 - 10*a*b^3*d *e^4 - 117*a^2*b^2*e^5)*m)*x^3 + (12*a^2*b^2*d^3*e^2 - 48*a^3*b*d^2*e^3 + 71*a^4*d*e^4)*m^2 + 2*(120*a^3*b*e^5 + (3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*m^4 - 2*(3*a*b^3*d^2*e^3 - 15*a^2*b^2*d*e^4 - 13*a^3*b*e^5)*m^3 + (6*b^4*d^3* e^2 - 36*a*b^3*d^2*e^3 + 87*a^2*b^2*d*e^4 + 118*a^3*b*e^5)*m^2 + 2*(3*b^4* d^3*e^2 - 15*a*b^3*d^2*e^3 + 30*a^2*b^2*d*e^4 + 107*a^3*b*e^5)*m)*x^2 - 2* (12*a*b^3*d^4*e - 54*a^2*b^2*d^3*e^2 + 94*a^3*b*d^2*e^3 - 77*a^4*d*e^4)*m + (120*a^4*e^5 + (4*a^3*b*d*e^4 + a^4*e^5)*m^4 - 2*(6*a^2*b^2*d^2*e^3 - 24 *a^3*b*d*e^4 - 7*a^4*e^5)*m^3 + (24*a*b^3*d^3*e^2 - 108*a^2*b^2*d^2*e^3 + 188*a^3*b*d*e^4 + 71*a^4*e^5)*m^2 - 2*(12*b^4*d^4*e - 60*a*b^3*d^3*e^2 + 1 20*a^2*b^2*d^2*e^3 - 120*a^3*b*d*e^4 - 77*a^4*e^5)*m)*x)*(e*x + d)^m/(e^5* m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)
Exception generated. \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Leaf count of result is larger than twice the leaf count of optimal. 756 vs. \(2 (222) = 444\).
Time = 0.06 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.73 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxi ma")
Output:
((m^3 + 6*m^2 + 11*m + 6)*b^3*e^4*x^4 - 3*(m^2 + 7*m + 12)*a^2*b*d^2*e^2 + (m^3 + 9*m^2 + 26*m + 24)*a^3*d*e^3 + 6*a*b^2*d^3*e*(m + 4) - 6*b^3*d^4 + ((m^3 + 3*m^2 + 2*m)*b^3*d*e^3 + 3*(m^3 + 7*m^2 + 14*m + 8)*a*b^2*e^4)*x^ 3 - 3*((m^2 + m)*b^3*d^2*e^2 - (m^3 + 5*m^2 + 4*m)*a*b^2*d*e^3 - (m^3 + 8* m^2 + 19*m + 12)*a^2*b*e^4)*x^2 - (6*(m^2 + 4*m)*a*b^2*d^2*e^2 - 3*(m^3 + 7*m^2 + 12*m)*a^2*b*d*e^3 - (m^3 + 9*m^2 + 26*m + 24)*a^3*e^4 - 6*b^3*d^3* e*m)*x)*(e*x + d)^m*a/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^3*e^5*x^5 + 6*(m^2 + 9*m + 20)*a^2*b*d^3*e^ 2 - (m^3 + 12*m^2 + 47*m + 60)*a^3*d^2*e^3 - 18*a*b^2*d^4*e*(m + 5) + 24*b ^3*d^5 + ((m^4 + 6*m^3 + 11*m^2 + 6*m)*b^3*d*e^4 + 3*(m^4 + 11*m^3 + 41*m^ 2 + 61*m + 30)*a*b^2*e^5)*x^4 - (4*(m^3 + 3*m^2 + 2*m)*b^3*d^2*e^3 - 3*(m^ 4 + 8*m^3 + 17*m^2 + 10*m)*a*b^2*d*e^4 - 3*(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)*a^2*b*e^5)*x^3 + (12*(m^2 + m)*b^3*d^3*e^2 - 9*(m^3 + 6*m^2 + 5*m)*a* b^2*d^2*e^3 + 3*(m^4 + 10*m^3 + 29*m^2 + 20*m)*a^2*b*d*e^4 + (m^4 + 13*m^3 + 59*m^2 + 107*m + 60)*a^3*e^5)*x^2 + (18*(m^2 + 5*m)*a*b^2*d^3*e^2 - 6*( m^3 + 9*m^2 + 20*m)*a^2*b*d^2*e^3 + (m^4 + 12*m^3 + 47*m^2 + 60*m)*a^3*d*e ^4 - 24*b^3*d^4*e*m)*x)*(e*x + d)^m*b/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 1948 vs. \(2 (222) = 444\).
Time = 0.23 (sec) , antiderivative size = 1948, normalized size of antiderivative = 7.03 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac ")
Output:
((e*x + d)^m*b^4*e^5*m^4*x^5*sgn(b*x + a) + (e*x + d)^m*b^4*d*e^4*m^4*x^4* sgn(b*x + a) + 4*(e*x + d)^m*a*b^3*e^5*m^4*x^4*sgn(b*x + a) + 10*(e*x + d) ^m*b^4*e^5*m^3*x^5*sgn(b*x + a) + 4*(e*x + d)^m*a*b^3*d*e^4*m^4*x^3*sgn(b* x + a) + 6*(e*x + d)^m*a^2*b^2*e^5*m^4*x^3*sgn(b*x + a) + 6*(e*x + d)^m*b^ 4*d*e^4*m^3*x^4*sgn(b*x + a) + 44*(e*x + d)^m*a*b^3*e^5*m^3*x^4*sgn(b*x + a) + 35*(e*x + d)^m*b^4*e^5*m^2*x^5*sgn(b*x + a) + 6*(e*x + d)^m*a^2*b^2*d *e^4*m^4*x^2*sgn(b*x + a) + 4*(e*x + d)^m*a^3*b*e^5*m^4*x^2*sgn(b*x + a) - 4*(e*x + d)^m*b^4*d^2*e^3*m^3*x^3*sgn(b*x + a) + 32*(e*x + d)^m*a*b^3*d*e ^4*m^3*x^3*sgn(b*x + a) + 72*(e*x + d)^m*a^2*b^2*e^5*m^3*x^3*sgn(b*x + a) + 11*(e*x + d)^m*b^4*d*e^4*m^2*x^4*sgn(b*x + a) + 164*(e*x + d)^m*a*b^3*e^ 5*m^2*x^4*sgn(b*x + a) + 50*(e*x + d)^m*b^4*e^5*m*x^5*sgn(b*x + a) + 4*(e* x + d)^m*a^3*b*d*e^4*m^4*x*sgn(b*x + a) + (e*x + d)^m*a^4*e^5*m^4*x*sgn(b* x + a) - 12*(e*x + d)^m*a*b^3*d^2*e^3*m^3*x^2*sgn(b*x + a) + 60*(e*x + d)^ m*a^2*b^2*d*e^4*m^3*x^2*sgn(b*x + a) + 52*(e*x + d)^m*a^3*b*e^5*m^3*x^2*sg n(b*x + a) - 12*(e*x + d)^m*b^4*d^2*e^3*m^2*x^3*sgn(b*x + a) + 68*(e*x + d )^m*a*b^3*d*e^4*m^2*x^3*sgn(b*x + a) + 294*(e*x + d)^m*a^2*b^2*e^5*m^2*x^3 *sgn(b*x + a) + 6*(e*x + d)^m*b^4*d*e^4*m*x^4*sgn(b*x + a) + 244*(e*x + d) ^m*a*b^3*e^5*m*x^4*sgn(b*x + a) + 24*(e*x + d)^m*b^4*e^5*x^5*sgn(b*x + a) + (e*x + d)^m*a^4*d*e^4*m^4*sgn(b*x + a) - 12*(e*x + d)^m*a^2*b^2*d^2*e^3* m^3*x*sgn(b*x + a) + 48*(e*x + d)^m*a^3*b*d*e^4*m^3*x*sgn(b*x + a) + 14...
Timed out. \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \] Input:
int((a + b*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
Output:
int((a + b*x)*(d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
Time = 0.20 (sec) , antiderivative size = 1028, normalized size of antiderivative = 3.71 \[ \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
Output:
((d + e*x)**m*(a**4*d*e**4*m**4 + 14*a**4*d*e**4*m**3 + 71*a**4*d*e**4*m** 2 + 154*a**4*d*e**4*m + 120*a**4*d*e**4 + a**4*e**5*m**4*x + 14*a**4*e**5* m**3*x + 71*a**4*e**5*m**2*x + 154*a**4*e**5*m*x + 120*a**4*e**5*x - 4*a** 3*b*d**2*e**3*m**3 - 48*a**3*b*d**2*e**3*m**2 - 188*a**3*b*d**2*e**3*m - 2 40*a**3*b*d**2*e**3 + 4*a**3*b*d*e**4*m**4*x + 48*a**3*b*d*e**4*m**3*x + 1 88*a**3*b*d*e**4*m**2*x + 240*a**3*b*d*e**4*m*x + 4*a**3*b*e**5*m**4*x**2 + 52*a**3*b*e**5*m**3*x**2 + 236*a**3*b*e**5*m**2*x**2 + 428*a**3*b*e**5*m *x**2 + 240*a**3*b*e**5*x**2 + 12*a**2*b**2*d**3*e**2*m**2 + 108*a**2*b**2 *d**3*e**2*m + 240*a**2*b**2*d**3*e**2 - 12*a**2*b**2*d**2*e**3*m**3*x - 1 08*a**2*b**2*d**2*e**3*m**2*x - 240*a**2*b**2*d**2*e**3*m*x + 6*a**2*b**2* d*e**4*m**4*x**2 + 60*a**2*b**2*d*e**4*m**3*x**2 + 174*a**2*b**2*d*e**4*m* *2*x**2 + 120*a**2*b**2*d*e**4*m*x**2 + 6*a**2*b**2*e**5*m**4*x**3 + 72*a* *2*b**2*e**5*m**3*x**3 + 294*a**2*b**2*e**5*m**2*x**3 + 468*a**2*b**2*e**5 *m*x**3 + 240*a**2*b**2*e**5*x**3 - 24*a*b**3*d**4*e*m - 120*a*b**3*d**4*e + 24*a*b**3*d**3*e**2*m**2*x + 120*a*b**3*d**3*e**2*m*x - 12*a*b**3*d**2* e**3*m**3*x**2 - 72*a*b**3*d**2*e**3*m**2*x**2 - 60*a*b**3*d**2*e**3*m*x** 2 + 4*a*b**3*d*e**4*m**4*x**3 + 32*a*b**3*d*e**4*m**3*x**3 + 68*a*b**3*d*e **4*m**2*x**3 + 40*a*b**3*d*e**4*m*x**3 + 4*a*b**3*e**5*m**4*x**4 + 44*a*b **3*e**5*m**3*x**4 + 164*a*b**3*e**5*m**2*x**4 + 244*a*b**3*e**5*m*x**4 + 120*a*b**3*e**5*x**4 + 24*b**4*d**5 - 24*b**4*d**4*e*m*x + 12*b**4*d**3...