Integrand size = 60, antiderivative size = 64 \[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=-\frac {(d+e x)^{-3-2 p} \left (d (e f+d g (1+p))+e (e f+d g (3+2 p)) x+e^2 g (2+p) x^2\right )^{1+p}}{e^2 (2+p)} \]
Time = 0.36 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.75 \[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=-\frac {(d+e x)^{-3-2 p} ((d+e x) (d g (1+p)+e (f+g (2+p) x)))^{1+p}}{e^2 (2+p)} \]
Integrate[(d + e*x)^(-3 - 2*p)*(f + g*x)*(d*(e*f + d*g + d*g*p) + e*(e*f + 3*d*g + 2*d*g*p)*x + e^2*g*(2 + p)*x^2)^p,x]
-(((d + e*x)^(-3 - 2*p)*((d + e*x)*(d*g*(1 + p) + e*(f + g*(2 + p)*x)))^(1 + p))/(e^2*(2 + p)))
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {1217}
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.
\(\displaystyle \int (f+g x) (d+e x)^{-2 p-3} \left (e x (2 d g p+3 d g+e f)+d (d g p+d g+e f)+e^2 g (p+2) x^2\right )^p \, dx\) |
\(\Big \downarrow \) 1217 |
\(\displaystyle -\frac {(d+e x)^{-2 p-3} \left (e x (d g (2 p+3)+e f)+d (d g (p+1)+e f)+e^2 g (p+2) x^2\right )^{p+1}}{e^2 (p+2)}\) |
Int[(d + e*x)^(-3 - 2*p)*(f + g*x)*(d*(e*f + d*g + d*g*p) + e*(e*f + 3*d*g + 2*d*g*p)*x + e^2*g*(2 + p)*x^2)^p,x]
-(((d + e*x)^(-3 - 2*p)*(d*(e*f + d*g*(1 + p)) + e*(e*f + d*g*(3 + 2*p))*x + e^2*g*(2 + p)*x^2)^(1 + p))/(e^2*(2 + p)))
3.23.92.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c* d^2 - b*d*e + a*e^2, 0] && EqQ[c*e*f*(m + 2*p + 2) + g*(c*d*m - b*e*(m + p + 1)), 0]
Time = 3.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.53
method | result | size |
gosper | \(-\frac {\left (e x +d \right )^{-2 p -2} \left (e g x p +d g p +2 e g x +d g +e f \right ) \left (e^{2} g \,x^{2} p +2 d e g p x +2 e^{2} g \,x^{2}+d^{2} g p +3 d e g x +e^{2} f x +d^{2} g +d e f \right )^{p}}{e^{2} \left (2+p \right )}\) | \(98\) |
risch | \(-\frac {\left (e x +d \right )^{-3-2 p} \left (e^{2} g \,x^{2} p +2 d e g p x +2 e^{2} g \,x^{2}+d^{2} g p +3 d e g x +e^{2} f x +d^{2} g +d e f \right ) \left (d g \left (1+p \right )+\left (f +\left (x p +2 x \right ) g \right ) e \right )^{p} \left (e x +d \right )^{p} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i \left (d g \left (1+p \right )+\left (f +\left (x p +2 x \right ) g \right ) e \right ) \left (e x +d \right )\right ) p \left (-\operatorname {csgn}\left (i \left (d g \left (1+p \right )+\left (f +\left (x p +2 x \right ) g \right ) e \right ) \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (e x +d \right )\right )\right ) \left (-\operatorname {csgn}\left (i \left (d g \left (1+p \right )+\left (f +\left (x p +2 x \right ) g \right ) e \right ) \left (e x +d \right )\right )+\operatorname {csgn}\left (i \left (d g \left (1+p \right )+\left (f +\left (x p +2 x \right ) g \right ) e \right )\right )\right )}{2}}}{e^{2} \left (2+p \right )}\) | \(236\) |
parallelrisch | \(-\frac {x^{2} \left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} e^{2} g^{2} p +2 x^{2} \left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} e^{2} g^{2}+2 x \left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} d e \,g^{2} p +3 x \left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} d e \,g^{2}+x \left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} e^{2} f g +\left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} d^{2} g^{2} p +\left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} d^{2} g^{2}+\left (e x +d \right )^{-3-2 p} \left (d \left (d g p +d g +e f \right )+e \left (2 d g p +3 d g +e f \right ) x +e^{2} g \left (2+p \right ) x^{2}\right )^{p} d e f g}{e^{2} g \left (2+p \right )}\) | \(513\) |
int((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f)*x+e^ 2*g*(2+p)*x^2)^p,x,method=_RETURNVERBOSE)
-1/e^2/(2+p)*(e*x+d)^(-2*p-2)*(e*g*p*x+d*g*p+2*e*g*x+d*g+e*f)*(e^2*g*p*x^2 +2*d*e*g*p*x+2*e^2*g*x^2+d^2*g*p+3*d*e*g*x+e^2*f*x+d^2*g+d*e*f)^p
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (64) = 128\).
Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.06 \[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=-\frac {{\left (d^{2} g p + d e f + d^{2} g + {\left (e^{2} g p + 2 \, e^{2} g\right )} x^{2} + {\left (2 \, d e g p + e^{2} f + 3 \, d e g\right )} x\right )} {\left (d^{2} g p + d e f + d^{2} g + {\left (e^{2} g p + 2 \, e^{2} g\right )} x^{2} + {\left (2 \, d e g p + e^{2} f + 3 \, d e g\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{e^{2} p + 2 \, e^{2}} \]
integrate((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f )*x+e^2*g*(2+p)*x^2)^p,x, algorithm="fricas")
-(d^2*g*p + d*e*f + d^2*g + (e^2*g*p + 2*e^2*g)*x^2 + (2*d*e*g*p + e^2*f + 3*d*e*g)*x)*(d^2*g*p + d*e*f + d^2*g + (e^2*g*p + 2*e^2*g)*x^2 + (2*d*e*g *p + e^2*f + 3*d*e*g)*x)^p*(e*x + d)^(-2*p - 3)/(e^2*p + 2*e^2)
Timed out. \[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=\text {Timed out} \]
integrate((e*x+d)**(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e* f)*x+e**2*g*(2+p)*x**2)**p,x)
\[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=\int { {\left (g x + f\right )} {\left (e^{2} g {\left (p + 2\right )} x^{2} + {\left (2 \, d g p + e f + 3 \, d g\right )} e x + {\left (d g p + e f + d g\right )} d\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \]
integrate((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f )*x+e^2*g*(2+p)*x^2)^p,x, algorithm="maxima")
integrate((g*x + f)*(e^2*g*(p + 2)*x^2 + (2*d*g*p + e*f + 3*d*g)*e*x + (d* g*p + e*f + d*g)*d)^p*(e*x + d)^(-2*p - 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (64) = 128\).
Time = 0.36 (sec) , antiderivative size = 412, normalized size of antiderivative = 6.44 \[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=-\frac {e^{2} g p x^{2} e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + 2 \, d e g p x e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + 2 \, e^{2} g x^{2} e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + d^{2} g p e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + e^{2} f x e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + 3 \, d e g x e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + d e f e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )} + d^{2} g e^{\left (p \log \left (e g p x + d g p + 2 \, e g x + e f + d g\right ) - p \log \left (e x + d\right ) - 3 \, \log \left (e x + d\right )\right )}}{e^{2} p + 2 \, e^{2}} \]
integrate((e*x+d)^(-3-2*p)*(g*x+f)*(d*(d*g*p+d*g+e*f)+e*(2*d*g*p+3*d*g+e*f )*x+e^2*g*(2+p)*x^2)^p,x, algorithm="giac")
-(e^2*g*p*x^2*e^(p*log(e*g*p*x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3*log(e*x + d)) + 2*d*e*g*p*x*e^(p*log(e*g*p*x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3*log(e*x + d)) + 2*e^2*g*x^2*e^(p*log(e*g*p *x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3*log(e*x + d)) + d^2 *g*p*e^(p*log(e*g*p*x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3* log(e*x + d)) + e^2*f*x*e^(p*log(e*g*p*x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3*log(e*x + d)) + 3*d*e*g*x*e^(p*log(e*g*p*x + d*g*p + 2* e*g*x + e*f + d*g) - p*log(e*x + d) - 3*log(e*x + d)) + d*e*f*e^(p*log(e*g *p*x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3*log(e*x + d)) + d ^2*g*e^(p*log(e*g*p*x + d*g*p + 2*e*g*x + e*f + d*g) - p*log(e*x + d) - 3* log(e*x + d)))/(e^2*p + 2*e^2)
Time = 11.54 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.16 \[ \int (d+e x)^{-3-2 p} (f+g x) \left (d (e f+d g+d g p)+e (e f+3 d g+2 d g p) x+e^2 g (2+p) x^2\right )^p \, dx=-{\left (d\,\left (d\,g+e\,f+d\,g\,p\right )+e\,x\,\left (3\,d\,g+e\,f+2\,d\,g\,p\right )+e^2\,g\,x^2\,\left (p+2\right )\right )}^p\,\left (\frac {g\,x^2}{{\left (d+e\,x\right )}^{2\,p+3}}+\frac {d^2\,g+d\,e\,f+d^2\,g\,p}{e^2\,\left (p+2\right )\,{\left (d+e\,x\right )}^{2\,p+3}}+\frac {x\,\left (3\,d\,g+e\,f+2\,d\,g\,p\right )}{e\,\left (p+2\right )\,{\left (d+e\,x\right )}^{2\,p+3}}\right ) \]
int(((f + g*x)*(d*(d*g + e*f + d*g*p) + e*x*(3*d*g + e*f + 2*d*g*p) + e^2* g*x^2*(p + 2))^p)/(d + e*x)^(2*p + 3),x)