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

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Rubi [A]  time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, size = 74, normalized size = 1.00 \[ -\frac {a (b c-a d) (a+b x)^{n+1}}{b^3 (n+1)}+\frac {(b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d (a+b x)^{n+3}}{b^3 (n+3)} \]

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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fricas [B]  time = 0.72, size = 159, normalized size = 2.15 \[ -\frac {{\left (a^{2} b c n + 3 \, a^{2} b c - 2 \, a^{3} d - {\left (b^{3} d n^{2} + 3 \, b^{3} d n + 2 \, b^{3} d\right )} x^{3} - {\left (3 \, b^{3} c + {\left (b^{3} c + a b^{2} d\right )} n^{2} + {\left (4 \, b^{3} c + a b^{2} d\right )} n\right )} x^{2} - {\left (a b^{2} c n^{2} + {\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \]

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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giac [B]  time = 0.99, size = 260, normalized size = 3.51 \[ \frac {{\left (b x + a\right )}^{n} b^{3} d n^{2} x^{3} + {\left (b x + a\right )}^{n} b^{3} c n^{2} x^{2} + {\left (b x + a\right )}^{n} a b^{2} d n^{2} x^{2} + 3 \, {\left (b x + a\right )}^{n} b^{3} d n x^{3} + {\left (b x + a\right )}^{n} a b^{2} c n^{2} x + 4 \, {\left (b x + a\right )}^{n} b^{3} c n x^{2} + {\left (b x + a\right )}^{n} a b^{2} d n x^{2} + 2 \, {\left (b x + a\right )}^{n} b^{3} d x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{2} c n x - 2 \, {\left (b x + a\right )}^{n} a^{2} b d n x + 3 \, {\left (b x + a\right )}^{n} b^{3} c x^{2} - {\left (b x + a\right )}^{n} a^{2} b c n - 3 \, {\left (b x + a\right )}^{n} a^{2} b c + 2 \, {\left (b x + a\right )}^{n} a^{3} d}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \]

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)

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maple [A]  time = 0.00, size = 114, normalized size = 1.54 \[ \frac {\left (b^{2} d \,n^{2} x^{2}+b^{2} c \,n^{2} x +3 b^{2} d n \,x^{2}-2 a b d n x +4 b^{2} c n x +2 d \,x^{2} b^{2}-a b c n -2 a b d x +3 b^{2} c x +2 a^{2} d -3 a b c \right ) \left (b x +a \right )^{n +1}}{\left (n^{3}+6 n^{2}+11 n +6\right ) b^{3}} \]

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)^(n+1)*(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 = 0.57, size = 113, normalized size = 1.53 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \]

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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mupad [B]  time = 1.12, size = 146, normalized size = 1.97 \[ {\left (a+b\,x\right )}^n\,\left (\frac {d\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}-\frac {a^2\,\left (3\,b\,c-2\,a\,d+b\,c\,n\right )}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {x^2\,\left (n+1\right )\,\left (3\,b\,c+a\,d\,n+b\,c\,n\right )}{b\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,n\,x\,\left (3\,b\,c-2\,a\,d+b\,c\,n\right )}{b^2\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^n*(c + d*x),x)

[Out]

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

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sympy [A]  time = 1.70, size = 1095, normalized size = 14.80 \[ \begin {cases} a^{n} \left (\frac {c x^{2}}{2} + \frac {d x^{3}}{3}\right ) & \text {for}\: b = 0 \\\frac {2 a^{2} d \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {3 a^{2} d}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} - \frac {a b c}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b d x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b d x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} - \frac {2 b^{2} c x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {2 b^{2} d x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {2 a^{2} d \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {2 a^{2} d}{a b^{3} + b^{4} x} + \frac {a b c \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {a b c}{a b^{3} + b^{4} x} - \frac {2 a b d x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {b^{2} c x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {b^{2} d x^{2}}{a b^{3} + b^{4} x} & \text {for}\: n = -2 \\\frac {a^{2} d \log {\left (\frac {a}{b} + x \right )}}{b^{3}} - \frac {a c \log {\left (\frac {a}{b} + x \right )}}{b^{2}} - \frac {a d x}{b^{2}} + \frac {c x}{b} + \frac {d x^{2}}{2 b} & \text {for}\: n = -1 \\\frac {2 a^{3} d \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {a^{2} b c n \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {3 a^{2} b c \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {2 a^{2} b d n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} c n^{2} x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 a b^{2} c n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} d n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} d n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} c n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {4 b^{3} c n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 b^{3} c x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} d n^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 b^{3} d n x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 b^{3} d x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} & \text {otherwise} \end {cases} \]

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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