Integrand size = 30, antiderivative size = 43 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {(d+e x)^{1+m} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (1+m+2 p)} \] Output:
(e*x+d)^(1+m)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p/e/(1+m+2*p)
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {(d+e x)^{1+m} \left (c (d+e x)^2\right )^p}{e (1+m+2 p)} \] Input:
Integrate[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]
Output:
((d + e*x)^(1 + m)*(c*(d + e*x)^2)^p)/(e*(1 + m + 2*p))
Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1102, 37, 17}
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.
\(\displaystyle \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle \left (c d e+c e^2 x\right )^{-2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^p \int (d+e x)^m \left (c x e^2+c d e\right )^{2 p}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle (d+e x)^{-2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^p \int (d+e x)^{m+2 p}dx\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {(d+e x)^{m+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (m+2 p+1)}\) |
Input:
Int[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]
Output:
((d + e*x)^(1 + m)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p)/(e*(1 + m + 2*p))
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 1.82 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (x^{2} c \,e^{2}+2 c d x e +c \,d^{2}\right )^{p}}{e \left (1+m +2 p \right )}\) | \(44\) |
orering | \(\frac {\left (e x +d \right ) \left (e x +d \right )^{m} \left (x^{2} c \,e^{2}+2 c d x e +c \,d^{2}\right )^{p}}{e \left (1+m +2 p \right )}\) | \(47\) |
parallelrisch | \(\frac {x \left (e x +d \right )^{m} {\left (c \left (e^{2} x^{2}+2 d e x +d^{2}\right )\right )}^{p} e +\left (e x +d \right )^{m} {\left (c \left (e^{2} x^{2}+2 d e x +d^{2}\right )\right )}^{p} d}{e \left (1+m +2 p \right )}\) | \(73\) |
norman | \(\frac {x \,{\mathrm e}^{m \ln \left (e x +d \right )} {\mathrm e}^{p \ln \left (x^{2} c \,e^{2}+2 c d x e +c \,d^{2}\right )}}{1+m +2 p}+\frac {d \,{\mathrm e}^{m \ln \left (e x +d \right )} {\mathrm e}^{p \ln \left (x^{2} c \,e^{2}+2 c d x e +c \,d^{2}\right )}}{e \left (1+m +2 p \right )}\) | \(91\) |
risch | \(\frac {\left (e x +d \right ) \left (e x +d \right )^{m} c^{p} \left (e x +d \right )^{2 p} {\mathrm e}^{-\frac {i \pi p \left (\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right )^{3}-2 \operatorname {csgn}\left (i \left (e x +d \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i \left (e x +d \right )\right )^{2}-\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2}+\operatorname {csgn}\left (i \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right ) \operatorname {csgn}\left (i c \right )+\operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{3}-\operatorname {csgn}\left (i c \left (e x +d \right )^{2}\right )^{2} \operatorname {csgn}\left (i c \right )\right )}{2}}}{e \left (1+m +2 p \right )}\) | \(195\) |
Input:
int((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x,method=_RETURNVERBOSE)
Output:
(e*x+d)^(1+m)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p/e/(1+m+2*p)
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} e^{\left (2 \, p \log \left (e x + d\right ) + p \log \left (c\right )\right )}}{e m + 2 \, e p + e} \] Input:
integrate((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x, algorithm="fricas")
Output:
(e*x + d)*(e*x + d)^m*e^(2*p*log(e*x + d) + p*log(c))/(e*m + 2*e*p + e)
\[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\begin {cases} d^{- 2 p - 1} x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \wedge m = - 2 p - 1 \\d^{m} x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \\\int \left (c \left (d + e x\right )^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1}\, dx & \text {for}\: m = - 2 p - 1 \\\frac {d \left (d + e x\right )^{m} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{e m + 2 e p + e} + \frac {e x \left (d + e x\right )^{m} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{e m + 2 e p + e} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**m*(c*e**2*x**2+2*c*d*e*x+c*d**2)**p,x)
Output:
Piecewise((d**(-2*p - 1)*x*(c*d**2)**p, Eq(e, 0) & Eq(m, -2*p - 1)), (d**m *x*(c*d**2)**p, Eq(e, 0)), (Integral((c*(d + e*x)**2)**p*(d + e*x)**(-2*p - 1), x), Eq(m, -2*p - 1)), (d*(d + e*x)**m*(c*d**2 + 2*c*d*e*x + c*e**2*x **2)**p/(e*m + 2*e*p + e) + e*x*(d + e*x)**m*(c*d**2 + 2*c*d*e*x + c*e**2* x**2)**p/(e*m + 2*e*p + e), True))
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {{\left (c^{p} e x + c^{p} d\right )} e^{\left (m \log \left (e x + d\right ) + 2 \, p \log \left (e x + d\right )\right )}}{e {\left (m + 2 \, p + 1\right )}} \] Input:
integrate((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x, algorithm="maxima")
Output:
(c^p*e*x + c^p*d)*e^(m*log(e*x + d) + 2*p*log(e*x + d))/(e*(m + 2*p + 1))
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.44 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {{\left (e x + d\right )}^{m} e x e^{\left (2 \, p \log \left (e x + d\right ) + p \log \left (c\right )\right )} + {\left (e x + d\right )}^{m} d e^{\left (2 \, p \log \left (e x + d\right ) + p \log \left (c\right )\right )}}{e m + 2 \, e p + e} \] Input:
integrate((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x, algorithm="giac")
Output:
((e*x + d)^m*e*x*e^(2*p*log(e*x + d) + p*log(c)) + (e*x + d)^m*d*e^(2*p*lo g(e*x + d) + p*log(c)))/(e*m + 2*e*p + e)
Time = 5.81 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {{\left (d+e\,x\right )}^{m+1}\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p}{e\,\left (m+2\,p+1\right )} \] Input:
int((d + e*x)^m*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^p,x)
Output:
((d + e*x)^(m + 1)*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^p)/(e*(m + 2*p + 1))
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx=\frac {\left (e x +d \right )^{m} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p} \left (e x +d \right )}{e \left (m +2 p +1\right )} \] Input:
int((e*x+d)^m*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x)
Output:
((d + e*x)**m*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p*(d + e*x))/(e*(m + 2*p + 1))