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

Optimal. Leaf size=102 \[ -\frac{a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

[Out]

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

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Rubi [A]  time = 0.052293, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {772} \[ -\frac{a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^n*(c + d*x^2),x]

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x (a+b x)^n \left (c+d x^2\right ) \, dx &=\int \left (\frac{a \left (-b^2 c-a^2 d\right ) (a+b x)^n}{b^3}+\frac{\left (b^2 c+3 a^2 d\right ) (a+b x)^{1+n}}{b^3}-\frac{3 a d (a+b x)^{2+n}}{b^3}+\frac{d (a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=-\frac{a \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac{\left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac{3 a d (a+b x)^{3+n}}{b^4 (3+n)}+\frac{d (a+b x)^{4+n}}{b^4 (4+n)}\\ \end{align*}

Mathematica [A]  time = 0.118785, size = 109, normalized size = 1.07 \[ \frac{(a+b x)^{n+1} \left (6 a^2 b d (n+1) x-6 a^3 d-a b^2 \left (c \left (n^2+7 n+12\right )+3 d \left (n^2+3 n+2\right ) x^2\right )+b^3 \left (n^2+4 n+3\right ) x \left (c (n+4)+d (n+2) x^2\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^n*(c + d*x^2),x]

[Out]

((a + b*x)^(1 + n)*(-6*a^3*d + 6*a^2*b*d*(1 + n)*x + b^3*(3 + 4*n + n^2)*x*(c*(4 + n) + d*(2 + n)*x^2) - a*b^2
*(c*(12 + 7*n + n^2) + 3*d*(2 + 3*n + n^2)*x^2)))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))

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Maple [A]  time = 0.046, size = 195, normalized size = 1.9 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}d{n}^{3}{x}^{3}-6\,{b}^{3}d{n}^{2}{x}^{3}+3\,a{b}^{2}d{n}^{2}{x}^{2}-{b}^{3}c{n}^{3}x-11\,{b}^{3}dn{x}^{3}+9\,a{b}^{2}dn{x}^{2}-8\,{b}^{3}c{n}^{2}x-6\,d{x}^{3}{b}^{3}-6\,{a}^{2}bdnx+a{b}^{2}c{n}^{2}+6\,ad{x}^{2}{b}^{2}-19\,{b}^{3}cnx-6\,{a}^{2}bdx+7\,a{b}^{2}cn-12\,{b}^{3}cx+6\,{a}^{3}d+12\,ac{b}^{2} \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(1+n)*(-b^3*d*n^3*x^3-6*b^3*d*n^2*x^3+3*a*b^2*d*n^2*x^2-b^3*c*n^3*x-11*b^3*d*n*x^3+9*a*b^2*d*n*x^2-8*
b^3*c*n^2*x-6*b^3*d*x^3-6*a^2*b*d*n*x+a*b^2*c*n^2+6*a*b^2*d*x^2-19*b^3*c*n*x-6*a^2*b*d*x+7*a*b^2*c*n-12*b^3*c*
x+6*a^3*d+12*a*b^2*c)/b^4/(n^4+10*n^3+35*n^2+50*n+24)

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Maxima [A]  time = 1.02795, size = 197, normalized size = 1.93 \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^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^2+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^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n
^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*d/((n^4 + 10*n^3 + 35
*n^2 + 50*n + 24)*b^4)

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Fricas [B]  time = 1.88853, size = 516, normalized size = 5.06 \begin{align*} -\frac{{\left (a^{2} b^{2} c n^{2} + 7 \, a^{2} b^{2} c n + 12 \, a^{2} b^{2} c + 6 \, a^{4} d -{\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} -{\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} -{\left (b^{4} c n^{3} + 12 \, b^{4} c +{\left (8 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n^{2} +{\left (19 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} -{\left (a b^{3} c n^{3} + 7 \, a b^{3} c n^{2} + 6 \,{\left (2 \, a b^{3} c + a^{3} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^2+c),x, algorithm="fricas")

[Out]

-(a^2*b^2*c*n^2 + 7*a^2*b^2*c*n + 12*a^2*b^2*c + 6*a^4*d - (b^4*d*n^3 + 6*b^4*d*n^2 + 11*b^4*d*n + 6*b^4*d)*x^
4 - (a*b^3*d*n^3 + 3*a*b^3*d*n^2 + 2*a*b^3*d*n)*x^3 - (b^4*c*n^3 + 12*b^4*c + (8*b^4*c - 3*a^2*b^2*d)*n^2 + (1
9*b^4*c - 3*a^2*b^2*d)*n)*x^2 - (a*b^3*c*n^3 + 7*a*b^3*c*n^2 + 6*(2*a*b^3*c + a^3*b*d)*n)*x)*(b*x + a)^n/(b^4*
n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)

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Sympy [A]  time = 4.73363, size = 2179, normalized size = 21.36 \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**2+c),x)

[Out]

Piecewise((a**n*(c*x**2/2 + d*x**4/4), Eq(b, 0)), (6*a**3*d*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*
b**6*x**2 + 6*b**7*x**3) + 5*a**3*d/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a**2*b*
d*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 9*a**2*b*d*x/(6*a**3*b**4 + 1
8*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - a*b**2*c/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b*
*7*x**3) + 18*a*b**2*d*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*b**
3*c*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*d*x**3*log(a/b + x)/(6*a**3*b**4
+ 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*b**3*d*x**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**
2 + 6*b**7*x**3), Eq(n, -4)), (-6*a**3*d*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d/(2*a
**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 1
2*a**2*b*d*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - a*b**2*c/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*
a*b**2*d*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*b**3*c*x/(2*a**2*b**4 + 4*a*b**5*x + 2
*b**6*x**2) + 2*b**3*d*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*d*log(a/b + x)/(2*a*
b**4 + 2*b**5*x) + 6*a**3*d/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2
*c*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*c/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d*x**2/(2*a*b**4 + 2*b**5*
x) + 2*b**3*c*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + b**3*d*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d*l
og(a/b + x)/b**4 + a**2*d*x/b**3 - a*c*log(a/b + x)/b**2 - a*d*x**2/(2*b**2) + c*x/b + d*x**3/(3*b), Eq(n, -1)
), (-6*a**4*d*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*d*n*x*(a
+ b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - a**2*b**2*c*n**2*(a + b*x)**n/(b*
*4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 7*a**2*b**2*c*n*(a + b*x)**n/(b**4*n**4 + 10*b*
*4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 12*a**2*b**2*c*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**
4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2
+ 50*b**4*n + 24*b**4) - 3*a**2*b**2*d*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*
n + 24*b**4) + a*b**3*c*n**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) +
7*a*b**3*c*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 12*a*b**3*c*n
*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*d*n**3*x**3*(a + b*x)
**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*d*n**2*x**3*(a + b*x)**n/(b**4*
n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*d*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**
4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*c*n**3*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b*
*4*n**2 + 50*b**4*n + 24*b**4) + 8*b**4*c*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50
*b**4*n + 24*b**4) + 19*b**4*c*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b
**4) + 12*b**4*c*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*d*n*
*3*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d*n**2*x**4*(a +
b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 11*b**4*d*n*x**4*(a + b*x)**n/(b**4
*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n
**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4), True))

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Giac [B]  time = 1.13034, size = 554, normalized size = 5.43 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} +{\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \,{\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} +{\left (b x + a\right )}^{n} b^{4} c n^{3} x^{2} + 3 \,{\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \,{\left (b x + a\right )}^{n} b^{4} d n x^{4} +{\left (b x + a\right )}^{n} a b^{3} c n^{3} x + 8 \,{\left (b x + a\right )}^{n} b^{4} c n^{2} x^{2} - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 2 \,{\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \,{\left (b x + a\right )}^{n} b^{4} d x^{4} + 7 \,{\left (b x + a\right )}^{n} a b^{3} c n^{2} x + 19 \,{\left (b x + a\right )}^{n} b^{4} c n x^{2} - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} -{\left (b x + a\right )}^{n} a^{2} b^{2} c n^{2} + 12 \,{\left (b x + a\right )}^{n} a b^{3} c n x + 6 \,{\left (b x + a\right )}^{n} a^{3} b d n x + 12 \,{\left (b x + a\right )}^{n} b^{4} c x^{2} - 7 \,{\left (b x + a\right )}^{n} a^{2} b^{2} c n - 12 \,{\left (b x + a\right )}^{n} a^{2} b^{2} c - 6 \,{\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^2+c),x, algorithm="giac")

[Out]

((b*x + a)^n*b^4*d*n^3*x^4 + (b*x + a)^n*a*b^3*d*n^3*x^3 + 6*(b*x + a)^n*b^4*d*n^2*x^4 + (b*x + a)^n*b^4*c*n^3
*x^2 + 3*(b*x + a)^n*a*b^3*d*n^2*x^3 + 11*(b*x + a)^n*b^4*d*n*x^4 + (b*x + a)^n*a*b^3*c*n^3*x + 8*(b*x + a)^n*
b^4*c*n^2*x^2 - 3*(b*x + a)^n*a^2*b^2*d*n^2*x^2 + 2*(b*x + a)^n*a*b^3*d*n*x^3 + 6*(b*x + a)^n*b^4*d*x^4 + 7*(b
*x + a)^n*a*b^3*c*n^2*x + 19*(b*x + a)^n*b^4*c*n*x^2 - 3*(b*x + a)^n*a^2*b^2*d*n*x^2 - (b*x + a)^n*a^2*b^2*c*n
^2 + 12*(b*x + a)^n*a*b^3*c*n*x + 6*(b*x + a)^n*a^3*b*d*n*x + 12*(b*x + a)^n*b^4*c*x^2 - 7*(b*x + a)^n*a^2*b^2
*c*n - 12*(b*x + a)^n*a^2*b^2*c - 6*(b*x + a)^n*a^4*d)/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)