Integrand size = 29, antiderivative size = 83 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(b d-a e) (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^2 (1+p)}+\frac {e (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (3+2 p)} \]
1/2*(-a*e+b*d)*(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2)^p/b^2/(p+1)+e*(b*x+a)^3*(b^ 2*x^2+2*a*b*x+a^2)^p/b^2/(3+2*p)
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.61 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {\left ((a+b x)^2\right )^{1+p} (-a e+b d (3+2 p)+2 b e (1+p) x)}{2 b^2 (1+p) (3+2 p)} \]
(((a + b*x)^2)^(1 + p)*(-(a*e) + b*d*(3 + 2*p) + 2*b*e*(1 + p)*x))/(2*b^2* (1 + p)*(3 + 2*p))
Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1187, 35, 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.
| \(\displaystyle \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \int (a+b x) \left (x b^2+a b\right )^{2 p} (d+e x)dx\) |
| \(\Big \downarrow \) 35 |
| \(\displaystyle \frac {\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \int \left (x b^2+a b\right )^{2 p+1} (d+e x)dx}{b}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \int \left (\frac {(b d-a e) \left (x b^2+a b\right )^{2 p+1}}{b}+\frac {e \left (x b^2+a b\right )^{2 p+2}}{b^2}\right )dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \left (\frac {e \left (a b+b^2 x\right )^{2 p+3}}{b^4 (2 p+3)}+\frac {(b d-a e) \left (a b+b^2 x\right )^{2 (p+1)}}{2 b^3 (p+1)}\right )}{b}\) |
((a^2 + 2*a*b*x + b^2*x^2)^p*(((b*d - a*e)*(a*b + b^2*x)^(2*(1 + p)))/(2*b ^3*(1 + p)) + (e*(a*b + b^2*x)^(3 + 2*p))/(b^4*(3 + 2*p))))/(b*(a*b + b^2* x)^(2*p))
3.22.60.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*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]
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81
| method | result | size |
| gosper | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} \left (-2 b e p x -2 b d p -2 b e x +a e -3 b d \right ) \left (b x +a \right )^{2}}{2 b^{2} \left (2 p^{2}+5 p +3\right )}\) | \(67\) |
| risch | \(-\frac {\left (-2 b^{3} e p \,x^{3}-4 a \,b^{2} e p \,x^{2}-2 b^{3} d p \,x^{2}-2 b^{3} e \,x^{3}-2 a^{2} b e p x -4 a \,b^{2} d p x -3 a \,b^{2} e \,x^{2}-3 b^{3} d \,x^{2}-2 a^{2} b d p -6 a \,b^{2} d x +a^{3} e -3 a^{2} b d \right ) \left (\left (b x +a \right )^{2}\right )^{p}}{2 \left (1+p \right ) b^{2} \left (3+2 p \right )}\) | \(133\) |
| norman | \(\frac {b e \,x^{3} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{3+2 p}+\frac {a \left (a e p +2 b d p +3 b d \right ) x \,{\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b \left (2 p^{2}+5 p +3\right )}+\frac {\left (4 a e p +2 b d p +3 a e +3 b d \right ) x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{4 p^{2}+10 p +6}-\frac {a^{2} \left (-2 b d p +a e -3 b d \right ) {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{2} \left (2 p^{2}+5 p +3\right )}\) | \(196\) |
| parallelrisch | \(\frac {2 x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} e p +2 x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} e +4 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} e p +2 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d p +3 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} e +3 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d +2 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{3} b e p +4 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d p +6 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d +2 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{3} b d p -\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{4} e +3 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{3} b d}{2 a \left (1+p \right ) b^{2} \left (3+2 p \right )}\) | \(356\) |
-1/2*(b^2*x^2+2*a*b*x+a^2)^p*(-2*b*e*p*x-2*b*d*p-2*b*e*x+a*e-3*b*d)*(b*x+a )^2/b^2/(2*p^2+5*p+3)
Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.71 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (2 \, a^{2} b d p + 3 \, a^{2} b d - a^{3} e + 2 \, {\left (b^{3} e p + b^{3} e\right )} x^{3} + {\left (3 \, b^{3} d + 3 \, a b^{2} e + 2 \, {\left (b^{3} d + 2 \, a b^{2} e\right )} p\right )} x^{2} + 2 \, {\left (3 \, a b^{2} d + {\left (2 \, a b^{2} d + a^{2} b e\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, {\left (2 \, b^{2} p^{2} + 5 \, b^{2} p + 3 \, b^{2}\right )}} \]
1/2*(2*a^2*b*d*p + 3*a^2*b*d - a^3*e + 2*(b^3*e*p + b^3*e)*x^3 + (3*b^3*d + 3*a*b^2*e + 2*(b^3*d + 2*a*b^2*e)*p)*x^2 + 2*(3*a*b^2*d + (2*a*b^2*d + a ^2*b*e)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p/(2*b^2*p^2 + 5*b^2*p + 3*b^2)
\[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\begin {cases} a \left (d x + \frac {e x^{2}}{2}\right ) \left (a^{2}\right )^{p} & \text {for}\: b = 0 \\\int \frac {\left (a + b x\right ) \left (d + e x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx & \text {for}\: p = - \frac {3}{2} \\- \frac {a e \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {d \log {\left (\frac {a}{b} + x \right )}}{b} + \frac {e x}{b} & \text {for}\: p = -1 \\- \frac {a^{3} e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {2 a^{2} b d p \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {3 a^{2} b d \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {2 a^{2} b e p x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {4 a b^{2} d p x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {6 a b^{2} d x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {4 a b^{2} e p x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {3 a b^{2} e x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {2 b^{3} d p x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {3 b^{3} d x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {2 b^{3} e p x^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} + \frac {2 b^{3} e x^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 10 b^{2} p + 6 b^{2}} & \text {otherwise} \end {cases} \]
Piecewise((a*(d*x + e*x**2/2)*(a**2)**p, Eq(b, 0)), (Integral((a + b*x)*(d + e*x)/((a + b*x)**2)**(3/2), x), Eq(p, -3/2)), (-a*e*log(a/b + x)/b**2 + d*log(a/b + x)/b + e*x/b, Eq(p, -1)), (-a**3*e*(a**2 + 2*a*b*x + b**2*x** 2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 2*a**2*b*d*p*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 3*a**2*b*d*(a**2 + 2*a* b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 2*a**2*b*e*p*x*(a **2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 4*a*b** 2*d*p*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 6*a*b**2*d*x*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 4*a*b**2*e*p*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 3*a*b**2*e*x**2*(a**2 + 2*a*b*x + b**2*x**2)**p/(4 *b**2*p**2 + 10*b**2*p + 6*b**2) + 2*b**3*d*p*x**2*(a**2 + 2*a*b*x + b**2* x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 3*b**3*d*x**2*(a**2 + 2*a*b* x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 2*b**3*e*p*x**3*(a* *2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2) + 2*b**3*e *x**3*(a**2 + 2*a*b*x + b**2*x**2)**p/(4*b**2*p**2 + 10*b**2*p + 6*b**2), True))
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (81) = 162\).
Time = 0.22 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.48 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} a d}{b {\left (2 \, p + 1\right )}} + \frac {{\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} {\left (b x + a\right )}^{2 \, p} d}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b} + \frac {{\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} {\left (b x + a\right )}^{2 \, p} a e}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac {{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} + {\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )} {\left (b x + a\right )}^{2 \, p} e}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{2}} \]
(b*x + a)*(b*x + a)^(2*p)*a*d/(b*(2*p + 1)) + 1/2*(b^2*(2*p + 1)*x^2 + 2*a *b*p*x - a^2)*(b*x + a)^(2*p)*d/((2*p^2 + 3*p + 1)*b) + 1/2*(b^2*(2*p + 1) *x^2 + 2*a*b*p*x - a^2)*(b*x + a)^(2*p)*a*e/((2*p^2 + 3*p + 1)*b^2) + ((2* p^2 + 3*p + 1)*b^3*x^3 + (2*p^2 + p)*a*b^2*x^2 - 2*a^2*b*p*x + a^3)*(b*x + a)^(2*p)*e/((4*p^3 + 12*p^2 + 11*p + 3)*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (81) = 162\).
Time = 0.27 (sec) , antiderivative size = 347, normalized size of antiderivative = 4.18 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} e p x^{3} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d p x^{2} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} e p x^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} e x^{3} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d p x + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b e p x + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d x^{2} + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} e x^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b d p + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d x + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b d - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} e}{2 \, {\left (2 \, b^{2} p^{2} + 5 \, b^{2} p + 3 \, b^{2}\right )}} \]
1/2*(2*(b^2*x^2 + 2*a*b*x + a^2)^p*b^3*e*p*x^3 + 2*(b^2*x^2 + 2*a*b*x + a^ 2)^p*b^3*d*p*x^2 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^2*e*p*x^2 + 2*(b^2*x^ 2 + 2*a*b*x + a^2)^p*b^3*e*x^3 + 4*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^2*d*p*x + 2*(b^2*x^2 + 2*a*b*x + a^2)^p*a^2*b*e*p*x + 3*(b^2*x^2 + 2*a*b*x + a^2) ^p*b^3*d*x^2 + 3*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^2*e*x^2 + 2*(b^2*x^2 + 2* a*b*x + a^2)^p*a^2*b*d*p + 6*(b^2*x^2 + 2*a*b*x + a^2)^p*a*b^2*d*x + 3*(b^ 2*x^2 + 2*a*b*x + a^2)^p*a^2*b*d - (b^2*x^2 + 2*a*b*x + a^2)^p*a^3*e)/(2*b ^2*p^2 + 5*b^2*p + 3*b^2)
Time = 11.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.71 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx={\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {x^2\,\left (3\,a\,e+3\,b\,d+4\,a\,e\,p+2\,b\,d\,p\right )}{2\,\left (2\,p^2+5\,p+3\right )}+\frac {a^2\,\left (3\,b\,d-a\,e+2\,b\,d\,p\right )}{2\,b^2\,\left (2\,p^2+5\,p+3\right )}+\frac {a\,x\,\left (3\,b\,d+a\,e\,p+2\,b\,d\,p\right )}{b\,\left (2\,p^2+5\,p+3\right )}+\frac {b\,e\,x^3\,\left (p+1\right )}{2\,p^2+5\,p+3}\right ) \]