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*(a^2*d+b^2*c)*(b*x+a)^(1+n)/b^4/(1+n)+(3*a^2*d+b^2*c)*(b*x+a)^(2+n)/b^4/(2+n)-3*a*d*(b*x+a)^(3+n)/b^4/(3+n)
+d*(b*x+a)^(4+n)/b^4/(4+n)
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00,
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 = 0.10, size = 109, normalized size = 1.07 \[ \frac {(a+b x)^{n+1} \left (-6 a^3 d+6 a^2 b d (n+1) x-a b^2 \left (c \left (n^2+7 n+12\right )+3 d \left (n^2+3 n+2\right ) x^2\right )+b^3 \left (n^2+4 n+3\right ) x \left (c (n+4)+d (n+2) x^2\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]
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))
________________________________________________________________________________________
fricas [B] time = 0.89, size = 250, normalized size = 2.45 \[ -\frac {{\left (a^{2} b^{2} c n^{2} + 7 \, a^{2} b^{2} c n + 12 \, a^{2} b^{2} c + 6 \, a^{4} d - {\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} - {\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} - {\left (b^{4} c n^{3} + 12 \, b^{4} c + {\left (8 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n^{2} + {\left (19 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} - {\left (a b^{3} c n^{3} + 7 \, a b^{3} c n^{2} + 6 \, {\left (2 \, a b^{3} c + a^{3} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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)
________________________________________________________________________________________
giac [B] time = 0.19, size = 410, normalized size = 4.02 \[ \frac {{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} + {\left (b x + a\right )}^{n} b^{4} c n^{3} x^{2} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d n x^{4} + {\left (b x + a\right )}^{n} a b^{3} c n^{3} x + 8 \, {\left (b x + a\right )}^{n} b^{4} c n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d x^{4} + 7 \, {\left (b x + a\right )}^{n} a b^{3} c n^{2} x + 19 \, {\left (b x + a\right )}^{n} b^{4} c n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} - {\left (b x + a\right )}^{n} a^{2} b^{2} c n^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d n x + 12 \, {\left (b x + a\right )}^{n} b^{4} c x^{2} - 7 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c n - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c - 6 \, {\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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)
________________________________________________________________________________________
maple [A] time = 0.00, size = 195, normalized size = 1.91 \[ -\frac {\left (-b^{3} d \,n^{3} x^{3}-6 b^{3} d \,n^{2} x^{3}+3 a \,b^{2} d \,n^{2} x^{2}-b^{3} c \,n^{3} x -11 b^{3} d n \,x^{3}+9 a \,b^{2} d n \,x^{2}-8 b^{3} c \,n^{2} x -6 d \,x^{3} b^{3}-6 a^{2} b d n x +a \,b^{2} c \,n^{2}+6 a d \,x^{2} b^{2}-19 b^{3} c n x -6 a^{2} b d x +7 a \,b^{2} c n -12 b^{3} c x +6 a^{3} d +12 a \,b^{2} c \right ) \left (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4}} \]
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)^(n+1)*(-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)
________________________________________________________________________________________
maxima [A] time = 0.47, size = 146, normalized size = 1.43 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
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)
________________________________________________________________________________________
mupad [B] time = 2.70, size = 255, normalized size = 2.50 \[ {\left (a+b\,x\right )}^n\,\left (\frac {d\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {a^2\,\left (6\,d\,a^2+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^2\,\left (n+1\right )\,\left (-3\,d\,a^2\,n+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x\,\left (6\,d\,a^2+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(c + d*x^2)*(a + b*x)^n,x)
[Out]
(a + b*x)^n*((d*x^4*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) - (a^2*(6*a^2*d + 12*b^2*c +
b^2*c*n^2 + 7*b^2*c*n))/(b^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (x^2*(n + 1)*(12*b^2*c + b^2*c*n^2 - 3*a^
2*d*n + 7*b^2*c*n))/(b^2*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (a*n*x*(6*a^2*d + 12*b^2*c + b^2*c*n^2 + 7*b^2
*c*n))/(b^3*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (a*d*n*x^3*(3*n + n^2 + 2))/(b*(50*n + 35*n^2 + 10*n^3 + n^
4 + 24)))
________________________________________________________________________________________
sympy [A] time = 3.53, size = 2181, normalized size = 21.38 \[ \text {result too large to display} \]
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) + 11*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) + 27*a**2*b*d*x/(6*a**3*b**4 +
18*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) + 18*
a*b**2*d*x**2/(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), 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
) - 12*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*log(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 + 3
5*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 + 1
0*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))
________________________________________________________________________________________