3.919 \(\int x (a+b x)^n (c+d x) \, dx\)

Optimal. Leaf size=74 \[ -\frac{a (b c-a d) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{(b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]

[Out]

-((a*(b*c - a*d)*(a + b*x)^(1 + n))/(b^3*(1 + n))) + ((b*c - 2*a*d)*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d*(a +
b*x)^(3 + n))/(b^3*(3 + n))

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Rubi [A]  time = 0.0331179, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {77} \[ -\frac{a (b c-a d) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{(b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^n*(c + d*x),x]

[Out]

-((a*(b*c - a*d)*(a + b*x)^(1 + n))/(b^3*(1 + n))) + ((b*c - 2*a*d)*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d*(a +
b*x)^(3 + n))/(b^3*(3 + n))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x (a+b x)^n (c+d x) \, dx &=\int \left (\frac{a (-b c+a d) (a+b x)^n}{b^2}+\frac{(b c-2 a d) (a+b x)^{1+n}}{b^2}+\frac{d (a+b x)^{2+n}}{b^2}\right ) \, dx\\ &=-\frac{a (b c-a d) (a+b x)^{1+n}}{b^3 (1+n)}+\frac{(b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac{d (a+b x)^{3+n}}{b^3 (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.051196, size = 74, normalized size = 1. \[ -\frac{a (b c-a d) (a+b x)^{n+1}}{b^3 (n+1)}+\frac{(b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^n*(c + d*x),x]

[Out]

-((a*(b*c - a*d)*(a + b*x)^(1 + n))/(b^3*(1 + n))) + ((b*c - 2*a*d)*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d*(a +
b*x)^(3 + n))/(b^3*(3 + n))

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Maple [A]  time = 0.002, size = 114, normalized size = 1.5 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}d{n}^{2}{x}^{2}+{b}^{2}c{n}^{2}x+3\,{b}^{2}dn{x}^{2}-2\,abdnx+4\,{b}^{2}cnx+2\,d{x}^{2}{b}^{2}-abcn-2\,abdx+3\,{b}^{2}cx+2\,{a}^{2}d-3\,abc \right ) }{{b}^{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(x*(b*x+a)^n*(d*x+c),x)

[Out]

(b*x+a)^(1+n)*(b^2*d*n^2*x^2+b^2*c*n^2*x+3*b^2*d*n*x^2-2*a*b*d*n*x+4*b^2*c*n*x+2*b^2*d*x^2-a*b*c*n-2*a*b*d*x+3
*b^2*c*x+2*a^2*d-3*a*b*c)/b^3/(n^3+6*n^2+11*n+6)

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Maxima [A]  time = 1.09155, size = 153, normalized size = 2.07 \begin{align*} \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x+c),x, algorithm="maxima")

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c/((n^2 + 3*n + 2)*b^2) + ((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a
*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*d/((n^3 + 6*n^2 + 11*n + 6)*b^3)

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Fricas [B]  time = 1.61124, size = 323, normalized size = 4.36 \begin{align*} -\frac{{\left (a^{2} b c n + 3 \, a^{2} b c - 2 \, a^{3} d -{\left (b^{3} d n^{2} + 3 \, b^{3} d n + 2 \, b^{3} d\right )} x^{3} -{\left (3 \, b^{3} c +{\left (b^{3} c + a b^{2} d\right )} n^{2} +{\left (4 \, b^{3} c + a b^{2} d\right )} n\right )} x^{2} -{\left (a b^{2} c n^{2} +{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x+c),x, algorithm="fricas")

[Out]

-(a^2*b*c*n + 3*a^2*b*c - 2*a^3*d - (b^3*d*n^2 + 3*b^3*d*n + 2*b^3*d)*x^3 - (3*b^3*c + (b^3*c + a*b^2*d)*n^2 +
(4*b^3*c + a*b^2*d)*n)*x^2 - (a*b^2*c*n^2 + (3*a*b^2*c - 2*a^2*b*d)*n)*x)*(b*x + a)^n/(b^3*n^3 + 6*b^3*n^2 +
11*b^3*n + 6*b^3)

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Sympy [A]  time = 1.80534, size = 1095, normalized size = 14.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**n*(d*x+c),x)

[Out]

Piecewise((a**n*(c*x**2/2 + d*x**3/3), Eq(b, 0)), (2*a**2*d*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x*
*2) + 3*a**2*d/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - a*b*c/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a
*b*d*x*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*d*x/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x*
*2) - 2*b**2*c*x/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 2*b**2*d*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4
*x + 2*b**5*x**2), Eq(n, -3)), (-2*a**2*d*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2*d/(a*b**3 + b**4*x) + a*b*c*
log(a/b + x)/(a*b**3 + b**4*x) + a*b*c/(a*b**3 + b**4*x) - 2*a*b*d*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*c*x
*log(a/b + x)/(a*b**3 + b**4*x) + b**2*d*x**2/(a*b**3 + b**4*x), Eq(n, -2)), (a**2*d*log(a/b + x)/b**3 - a*c*l
og(a/b + x)/b**2 - a*d*x/b**2 + c*x/b + d*x**2/(2*b), Eq(n, -1)), (2*a**3*d*(a + b*x)**n/(b**3*n**3 + 6*b**3*n
**2 + 11*b**3*n + 6*b**3) - a**2*b*c*n*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 3*a**2*b*
c*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*d*n*x*(a + b*x)**n/(b**3*n**3 + 6*b**
3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*c*n**2*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3
*a*b**2*c*n*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*d*n**2*x**2*(a + b*x)**n/(b
**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*d*n*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3
*n + 6*b**3) + b**3*c*n**2*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 4*b**3*c*n*x**2*
(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b**3*c*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n
**2 + 11*b**3*n + 6*b**3) + b**3*d*n**2*x**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b
**3*d*n*x**3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 2*b**3*d*x**3*(a + b*x)**n/(b**3*n*
*3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3), True))

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Giac [B]  time = 2.87774, size = 351, normalized size = 4.74 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{3} d n^{2} x^{3} +{\left (b x + a\right )}^{n} b^{3} c n^{2} x^{2} +{\left (b x + a\right )}^{n} a b^{2} d n^{2} x^{2} + 3 \,{\left (b x + a\right )}^{n} b^{3} d n x^{3} +{\left (b x + a\right )}^{n} a b^{2} c n^{2} x + 4 \,{\left (b x + a\right )}^{n} b^{3} c n x^{2} +{\left (b x + a\right )}^{n} a b^{2} d n x^{2} + 2 \,{\left (b x + a\right )}^{n} b^{3} d x^{3} + 3 \,{\left (b x + a\right )}^{n} a b^{2} c n x - 2 \,{\left (b x + a\right )}^{n} a^{2} b d n x + 3 \,{\left (b x + a\right )}^{n} b^{3} c x^{2} -{\left (b x + a\right )}^{n} a^{2} b c n - 3 \,{\left (b x + a\right )}^{n} a^{2} b c + 2 \,{\left (b x + a\right )}^{n} a^{3} d}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x+c),x, algorithm="giac")

[Out]

((b*x + a)^n*b^3*d*n^2*x^3 + (b*x + a)^n*b^3*c*n^2*x^2 + (b*x + a)^n*a*b^2*d*n^2*x^2 + 3*(b*x + a)^n*b^3*d*n*x
^3 + (b*x + a)^n*a*b^2*c*n^2*x + 4*(b*x + a)^n*b^3*c*n*x^2 + (b*x + a)^n*a*b^2*d*n*x^2 + 2*(b*x + a)^n*b^3*d*x
^3 + 3*(b*x + a)^n*a*b^2*c*n*x - 2*(b*x + a)^n*a^2*b*d*n*x + 3*(b*x + a)^n*b^3*c*x^2 - (b*x + a)^n*a^2*b*c*n -
3*(b*x + a)^n*a^2*b*c + 2*(b*x + a)^n*a^3*d)/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3)