3.808 \(\int (f+g x)^n (a+2 c d x+c e x^2) \, dx\)
Optimal. Leaf size=84 \[ \frac{(f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^3 (n+1)}-\frac{2 c (e f-d g) (f+g x)^{n+2}}{g^3 (n+2)}+\frac{c e (f+g x)^{n+3}}{g^3 (n+3)} \]
[Out]
((a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^3*(1 + n)) - (2*c*(e*f - d*g)*(f + g*x)^(2 + n))/(g^3*(2 +
n)) + (c*e*(f + g*x)^(3 + n))/(g^3*(3 + n))
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Rubi [A] time = 0.0634219, antiderivative size = 84, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used =
{698} \[ \frac{(f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^3 (n+1)}-\frac{2 c (e f-d g) (f+g x)^{n+2}}{g^3 (n+2)}+\frac{c e (f+g x)^{n+3}}{g^3 (n+3)} \]
Antiderivative was successfully verified.
[In]
Int[(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
((a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^3*(1 + n)) - (2*c*(e*f - d*g)*(f + g*x)^(2 + n))/(g^3*(2 +
n)) + (c*e*(f + g*x)^(3 + n))/(g^3*(3 + n))
Rule 698
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin{align*} \int (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\int \left (\frac{\left (a g^2+c f (e f-2 d g)\right ) (f+g x)^n}{g^2}+\frac{2 c (-e f+d g) (f+g x)^{1+n}}{g^2}+\frac{c e (f+g x)^{2+n}}{g^2}\right ) \, dx\\ &=\frac{\left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^3 (1+n)}-\frac{2 c (e f-d g) (f+g x)^{2+n}}{g^3 (2+n)}+\frac{c e (f+g x)^{3+n}}{g^3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.105957, size = 73, normalized size = 0.87 \[ \frac{(f+g x)^{n+1} \left (\frac{a g^2+c f (e f-2 d g)}{n+1}-\frac{2 c (f+g x) (e f-d g)}{n+2}+\frac{c e (f+g x)^2}{n+3}\right )}{g^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
((f + g*x)^(1 + n)*((a*g^2 + c*f*(e*f - 2*d*g))/(1 + n) - (2*c*(e*f - d*g)*(f + g*x))/(2 + n) + (c*e*(f + g*x)
^2)/(3 + n)))/g^3
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Maple [A] time = 0.048, size = 147, normalized size = 1.8 \begin{align*}{\frac{ \left ( gx+f \right ) ^{1+n} \left ( ce{g}^{2}{n}^{2}{x}^{2}+2\,cd{g}^{2}{n}^{2}x+3\,ce{g}^{2}n{x}^{2}+8\,cd{g}^{2}nx-2\,cefgnx+2\,ce{x}^{2}{g}^{2}+a{g}^{2}{n}^{2}-2\,cdfgn+6\,cd{g}^{2}x-2\,cefgx+5\,a{g}^{2}n-6\,cdfg+2\,ce{f}^{2}+6\,a{g}^{2} \right ) }{{g}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)^n*(c*e*x^2+2*c*d*x+a),x)
[Out]
(g*x+f)^(1+n)*(c*e*g^2*n^2*x^2+2*c*d*g^2*n^2*x+3*c*e*g^2*n*x^2+8*c*d*g^2*n*x-2*c*e*f*g*n*x+2*c*e*g^2*x^2+a*g^2
*n^2-2*c*d*f*g*n+6*c*d*g^2*x-2*c*e*f*g*x+5*a*g^2*n-6*c*d*f*g+2*c*e*f^2+6*a*g^2)/g^3/(n^3+6*n^2+11*n+6)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.57113, size = 462, normalized size = 5.5 \begin{align*} \frac{{\left (a f g^{2} n^{2} + 2 \, c e f^{3} - 6 \, c d f^{2} g + 6 \, a f g^{2} +{\left (c e g^{3} n^{2} + 3 \, c e g^{3} n + 2 \, c e g^{3}\right )} x^{3} +{\left (6 \, c d g^{3} +{\left (c e f g^{2} + 2 \, c d g^{3}\right )} n^{2} +{\left (c e f g^{2} + 8 \, c d g^{3}\right )} n\right )} x^{2} -{\left (2 \, c d f^{2} g - 5 \, a f g^{2}\right )} n +{\left (6 \, a g^{3} +{\left (2 \, c d f g^{2} + a g^{3}\right )} n^{2} -{\left (2 \, c e f^{2} g - 6 \, c d f g^{2} - 5 \, a g^{3}\right )} n\right )} x\right )}{\left (g x + f\right )}^{n}}{g^{3} n^{3} + 6 \, g^{3} n^{2} + 11 \, g^{3} n + 6 \, g^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="fricas")
[Out]
(a*f*g^2*n^2 + 2*c*e*f^3 - 6*c*d*f^2*g + 6*a*f*g^2 + (c*e*g^3*n^2 + 3*c*e*g^3*n + 2*c*e*g^3)*x^3 + (6*c*d*g^3
+ (c*e*f*g^2 + 2*c*d*g^3)*n^2 + (c*e*f*g^2 + 8*c*d*g^3)*n)*x^2 - (2*c*d*f^2*g - 5*a*f*g^2)*n + (6*a*g^3 + (2*c
*d*f*g^2 + a*g^3)*n^2 - (2*c*e*f^2*g - 6*c*d*f*g^2 - 5*a*g^3)*n)*x)*(g*x + f)^n/(g^3*n^3 + 6*g^3*n^2 + 11*g^3*
n + 6*g^3)
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Sympy [A] time = 3.98552, size = 1532, normalized size = 18.24 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)
[Out]
Piecewise((f**n*(a*x + c*d*x**2 + c*e*x**3/3), Eq(g, 0)), (-a*g**2/(2*f**2*g**3 + 4*f*g**4*x + 2*g**5*x**2) -
2*c*d*f*g/(2*f**2*g**3 + 4*f*g**4*x + 2*g**5*x**2) - 4*c*d*g**2*x/(2*f**2*g**3 + 4*f*g**4*x + 2*g**5*x**2) + 2
*c*e*f**2*log(f/g + x)/(2*f**2*g**3 + 4*f*g**4*x + 2*g**5*x**2) + 3*c*e*f**2/(2*f**2*g**3 + 4*f*g**4*x + 2*g**
5*x**2) + 4*c*e*f*g*x*log(f/g + x)/(2*f**2*g**3 + 4*f*g**4*x + 2*g**5*x**2) + 4*c*e*f*g*x/(2*f**2*g**3 + 4*f*g
**4*x + 2*g**5*x**2) + 2*c*e*g**2*x**2*log(f/g + x)/(2*f**2*g**3 + 4*f*g**4*x + 2*g**5*x**2), Eq(n, -3)), (a*g
**3*x/(f**2*g**3 + f*g**4*x) + 2*c*d*f**2*g*log(f/g + x)/(f**2*g**3 + f*g**4*x) + 2*c*d*f*g**2*x*log(f/g + x)/
(f**2*g**3 + f*g**4*x) - 2*c*d*f*g**2*x/(f**2*g**3 + f*g**4*x) - 2*c*e*f**3*log(f/g + x)/(f**2*g**3 + f*g**4*x
) - 2*c*e*f**2*g*x*log(f/g + x)/(f**2*g**3 + f*g**4*x) + 2*c*e*f**2*g*x/(f**2*g**3 + f*g**4*x) + c*e*f*g**2*x*
*2/(f**2*g**3 + f*g**4*x), Eq(n, -2)), (a*log(f/g + x)/g - 2*c*d*f*log(f/g + x)/g**2 + 2*c*d*x/g + c*e*f**2*lo
g(f/g + x)/g**3 - c*e*f*x/g**2 + c*e*x**2/(2*g), Eq(n, -1)), (a*f*g**2*n**2*(f + g*x)**n/(g**3*n**3 + 6*g**3*n
**2 + 11*g**3*n + 6*g**3) + 5*a*f*g**2*n*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + 6*a*f*g
**2*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + a*g**3*n**2*x*(f + g*x)**n/(g**3*n**3 + 6*g*
*3*n**2 + 11*g**3*n + 6*g**3) + 5*a*g**3*n*x*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + 6*a
*g**3*x*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) - 2*c*d*f**2*g*n*(f + g*x)**n/(g**3*n**3 +
6*g**3*n**2 + 11*g**3*n + 6*g**3) - 6*c*d*f**2*g*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3)
+ 2*c*d*f*g**2*n**2*x*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + 6*c*d*f*g**2*n*x*(f + g*x)
**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + 2*c*d*g**3*n**2*x**2*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**
2 + 11*g**3*n + 6*g**3) + 8*c*d*g**3*n*x**2*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + 6*c*
d*g**3*x**2*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + 2*c*e*f**3*(f + g*x)**n/(g**3*n**3 +
6*g**3*n**2 + 11*g**3*n + 6*g**3) - 2*c*e*f**2*g*n*x*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g*
*3) + c*e*f*g**2*n**2*x**2*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + c*e*f*g**2*n*x**2*(f
+ g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) + c*e*g**3*n**2*x**3*(f + g*x)**n/(g**3*n**3 + 6*g**3
*n**2 + 11*g**3*n + 6*g**3) + 3*c*e*g**3*n*x**3*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3) +
2*c*e*g**3*x**3*(f + g*x)**n/(g**3*n**3 + 6*g**3*n**2 + 11*g**3*n + 6*g**3), True))
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Giac [B] time = 1.10491, size = 504, normalized size = 6. \begin{align*} \frac{{\left (g x + f\right )}^{n} c g^{3} n^{2} x^{3} e + 2 \,{\left (g x + f\right )}^{n} c d g^{3} n^{2} x^{2} +{\left (g x + f\right )}^{n} c f g^{2} n^{2} x^{2} e + 3 \,{\left (g x + f\right )}^{n} c g^{3} n x^{3} e + 2 \,{\left (g x + f\right )}^{n} c d f g^{2} n^{2} x + 8 \,{\left (g x + f\right )}^{n} c d g^{3} n x^{2} +{\left (g x + f\right )}^{n} c f g^{2} n x^{2} e + 2 \,{\left (g x + f\right )}^{n} c g^{3} x^{3} e + 6 \,{\left (g x + f\right )}^{n} c d f g^{2} n x +{\left (g x + f\right )}^{n} a g^{3} n^{2} x + 6 \,{\left (g x + f\right )}^{n} c d g^{3} x^{2} - 2 \,{\left (g x + f\right )}^{n} c f^{2} g n x e - 2 \,{\left (g x + f\right )}^{n} c d f^{2} g n +{\left (g x + f\right )}^{n} a f g^{2} n^{2} + 5 \,{\left (g x + f\right )}^{n} a g^{3} n x - 6 \,{\left (g x + f\right )}^{n} c d f^{2} g + 5 \,{\left (g x + f\right )}^{n} a f g^{2} n + 6 \,{\left (g x + f\right )}^{n} a g^{3} x + 2 \,{\left (g x + f\right )}^{n} c f^{3} e + 6 \,{\left (g x + f\right )}^{n} a f g^{2}}{g^{3} n^{3} + 6 \, g^{3} n^{2} + 11 \, g^{3} n + 6 \, g^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="giac")
[Out]
((g*x + f)^n*c*g^3*n^2*x^3*e + 2*(g*x + f)^n*c*d*g^3*n^2*x^2 + (g*x + f)^n*c*f*g^2*n^2*x^2*e + 3*(g*x + f)^n*c
*g^3*n*x^3*e + 2*(g*x + f)^n*c*d*f*g^2*n^2*x + 8*(g*x + f)^n*c*d*g^3*n*x^2 + (g*x + f)^n*c*f*g^2*n*x^2*e + 2*(
g*x + f)^n*c*g^3*x^3*e + 6*(g*x + f)^n*c*d*f*g^2*n*x + (g*x + f)^n*a*g^3*n^2*x + 6*(g*x + f)^n*c*d*g^3*x^2 - 2
*(g*x + f)^n*c*f^2*g*n*x*e - 2*(g*x + f)^n*c*d*f^2*g*n + (g*x + f)^n*a*f*g^2*n^2 + 5*(g*x + f)^n*a*g^3*n*x - 6
*(g*x + f)^n*c*d*f^2*g + 5*(g*x + f)^n*a*f*g^2*n + 6*(g*x + f)^n*a*g^3*x + 2*(g*x + f)^n*c*f^3*e + 6*(g*x + f)
^n*a*f*g^2)/(g^3*n^3 + 6*g^3*n^2 + 11*g^3*n + 6*g^3)