3.925 \(\int (a+b x)^n (c+d x)^2 \, dx\)

Optimal. Leaf size=78 \[ \frac{(b c-a d)^2 (a+b x)^{n+1}}{b^3 (n+1)}+\frac{2 d (b c-a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d^2 (a+b x)^{n+3}}{b^3 (n+3)} \]

[Out]

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

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Rubi [A]  time = 0.0331732, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{(b c-a d)^2 (a+b x)^{n+1}}{b^3 (n+1)}+\frac{2 d (b c-a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d^2 (a+b x)^{n+3}}{b^3 (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x)^2,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0704539, size = 67, normalized size = 0.86 \[ \frac{(a+b x)^{n+1} \left (\frac{2 d (a+b x) (b c-a d)}{n+2}+\frac{(b c-a d)^2}{n+1}+\frac{d^2 (a+b x)^2}{n+3}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n*(c + d*x)^2,x]

[Out]

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

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Maple [B]  time = 0.007, size = 159, normalized size = 2. \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}{d}^{2}{n}^{2}{x}^{2}+2\,{b}^{2}cd{n}^{2}x+3\,{b}^{2}{d}^{2}n{x}^{2}-2\,ab{d}^{2}nx+{b}^{2}{c}^{2}{n}^{2}+8\,{b}^{2}cdnx+2\,{b}^{2}{d}^{2}{x}^{2}-2\,abcdn-2\,ab{d}^{2}x+5\,{b}^{2}{c}^{2}n+6\,{b}^{2}cdx+2\,{a}^{2}{d}^{2}-6\,abcd+6\,{b}^{2}{c}^{2} \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((b*x+a)^n*(d*x+c)^2,x)

[Out]

(b*x+a)^(1+n)*(b^2*d^2*n^2*x^2+2*b^2*c*d*n^2*x+3*b^2*d^2*n*x^2-2*a*b*d^2*n*x+b^2*c^2*n^2+8*b^2*c*d*n*x+2*b^2*d
^2*x^2-2*a*b*c*d*n-2*a*b*d^2*x+5*b^2*c^2*n+6*b^2*c*d*x+2*a^2*d^2-6*a*b*c*d+6*b^2*c^2)/b^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((b*x+a)^n*(d*x+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.6764, size = 478, normalized size = 6.13 \begin{align*} \frac{{\left (a b^{2} c^{2} n^{2} + 6 \, a b^{2} c^{2} - 6 \, a^{2} b c d + 2 \, a^{3} d^{2} +{\left (b^{3} d^{2} n^{2} + 3 \, b^{3} d^{2} n + 2 \, b^{3} d^{2}\right )} x^{3} +{\left (6 \, b^{3} c d +{\left (2 \, b^{3} c d + a b^{2} d^{2}\right )} n^{2} +{\left (8 \, b^{3} c d + a b^{2} d^{2}\right )} n\right )} x^{2} +{\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d\right )} n +{\left (6 \, b^{3} c^{2} +{\left (b^{3} c^{2} + 2 \, a b^{2} c d\right )} n^{2} +{\left (5 \, b^{3} c^{2} + 6 \, a b^{2} c d - 2 \, a^{2} b d^{2}\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((b*x+a)^n*(d*x+c)^2,x, algorithm="fricas")

[Out]

(a*b^2*c^2*n^2 + 6*a*b^2*c^2 - 6*a^2*b*c*d + 2*a^3*d^2 + (b^3*d^2*n^2 + 3*b^3*d^2*n + 2*b^3*d^2)*x^3 + (6*b^3*
c*d + (2*b^3*c*d + a*b^2*d^2)*n^2 + (8*b^3*c*d + a*b^2*d^2)*n)*x^2 + (5*a*b^2*c^2 - 2*a^2*b*c*d)*n + (6*b^3*c^
2 + (b^3*c^2 + 2*a*b^2*c*d)*n^2 + (5*b^3*c^2 + 6*a*b^2*c*d - 2*a^2*b*d^2)*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 = 2.0591, size = 1506, normalized size = 19.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n*(d*x+c)**2,x)

[Out]

Piecewise((a**n*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(b, 0)), (2*a**2*d**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**
4*x + 2*b**5*x**2) + 3*a**2*d**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 2*a*b*c*d/(2*a**2*b**3 + 4*a*b**4*
x + 2*b**5*x**2) + 4*a*b*d**2*x*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*d**2*x/(2*a**2*b
**3 + 4*a*b**4*x + 2*b**5*x**2) - b**2*c**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) - 4*b**2*c*d*x/(2*a**2*b*
*3 + 4*a*b**4*x + 2*b**5*x**2) + 2*b**2*d**2*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**2*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2*d**2/(a*b**3 + b**4*x) + 2*a*b*c*d*log(a/b + x)/(
a*b**3 + b**4*x) + 2*a*b*c*d/(a*b**3 + b**4*x) - 2*a*b*d**2*x*log(a/b + x)/(a*b**3 + b**4*x) - b**2*c**2/(a*b*
*3 + b**4*x) + 2*b**2*c*d*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*d**2*x**2/(a*b**3 + b**4*x), Eq(n, -2)), (a*
*2*d**2*log(a/b + x)/b**3 - 2*a*c*d*log(a/b + x)/b**2 - a*d**2*x/b**2 + c**2*log(a/b + x)/b + 2*c*d*x/b + d**2
*x**2/(2*b), Eq(n, -1)), (2*a**3*d**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*c
*d*n*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 6*a**2*b*c*d*(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**2*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**2*n**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 5*a*b**2*c**2*n*(a + b*x)**n/
(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 6*a*b**2*c**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3
*n + 6*b**3) + 2*a*b**2*c*d*n**2*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 6*a*b**2*c*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*d**2*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**2*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**2*n**2*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 5*b**3*c**2*n*x*(a +
b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 6*b**3*c**2*x*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2
+ 11*b**3*n + 6*b**3) + 2*b**3*c*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) + 8*b
**3*c*d*n*x**2*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 6*b**3*c*d*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**2*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**2*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**2*
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 = 1.45536, size = 520, normalized size = 6.67 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{3} d^{2} n^{2} x^{3} + 2 \,{\left (b x + a\right )}^{n} b^{3} c d n^{2} x^{2} +{\left (b x + a\right )}^{n} a b^{2} d^{2} n^{2} x^{2} + 3 \,{\left (b x + a\right )}^{n} b^{3} d^{2} n x^{3} +{\left (b x + a\right )}^{n} b^{3} c^{2} n^{2} x + 2 \,{\left (b x + a\right )}^{n} a b^{2} c d n^{2} x + 8 \,{\left (b x + a\right )}^{n} b^{3} c d n x^{2} +{\left (b x + a\right )}^{n} a b^{2} d^{2} n x^{2} + 2 \,{\left (b x + a\right )}^{n} b^{3} d^{2} x^{3} +{\left (b x + a\right )}^{n} a b^{2} c^{2} n^{2} + 5 \,{\left (b x + a\right )}^{n} b^{3} c^{2} n x + 6 \,{\left (b x + a\right )}^{n} a b^{2} c d n x - 2 \,{\left (b x + a\right )}^{n} a^{2} b d^{2} n x + 6 \,{\left (b x + a\right )}^{n} b^{3} c d x^{2} + 5 \,{\left (b x + a\right )}^{n} a b^{2} c^{2} n - 2 \,{\left (b x + a\right )}^{n} a^{2} b c d n + 6 \,{\left (b x + a\right )}^{n} b^{3} c^{2} x + 6 \,{\left (b x + a\right )}^{n} a b^{2} c^{2} - 6 \,{\left (b x + a\right )}^{n} a^{2} b c d + 2 \,{\left (b x + a\right )}^{n} a^{3} d^{2}}{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((b*x+a)^n*(d*x+c)^2,x, algorithm="giac")

[Out]

((b*x + a)^n*b^3*d^2*n^2*x^3 + 2*(b*x + a)^n*b^3*c*d*n^2*x^2 + (b*x + a)^n*a*b^2*d^2*n^2*x^2 + 3*(b*x + a)^n*b
^3*d^2*n*x^3 + (b*x + a)^n*b^3*c^2*n^2*x + 2*(b*x + a)^n*a*b^2*c*d*n^2*x + 8*(b*x + a)^n*b^3*c*d*n*x^2 + (b*x
+ a)^n*a*b^2*d^2*n*x^2 + 2*(b*x + a)^n*b^3*d^2*x^3 + (b*x + a)^n*a*b^2*c^2*n^2 + 5*(b*x + a)^n*b^3*c^2*n*x + 6
*(b*x + a)^n*a*b^2*c*d*n*x - 2*(b*x + a)^n*a^2*b*d^2*n*x + 6*(b*x + a)^n*b^3*c*d*x^2 + 5*(b*x + a)^n*a*b^2*c^2
*n - 2*(b*x + a)^n*a^2*b*c*d*n + 6*(b*x + a)^n*b^3*c^2*x + 6*(b*x + a)^n*a*b^2*c^2 - 6*(b*x + a)^n*a^2*b*c*d +
2*(b*x + a)^n*a^3*d^2)/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3)