3.176 \(\int (a+b x)^n (c+d x^3) \, dx\)

Optimal. Leaf size=94 \[ \frac {\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {1850} \[ \frac {\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x^3),x]

[Out]

((b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 94, normalized size = 1.00 \[ \frac {\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n*(c + d*x^3),x]

[Out]

((b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (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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fricas [B]  time = 0.68, size = 222, normalized size = 2.36 \[ \frac {{\left (a b^{3} c n^{3} + 9 \, a b^{3} c n^{2} + 26 \, a b^{3} c n + 24 \, a b^{3} c - 6 \, a^{4} d + {\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} + {\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} d n^{2} + a^{2} b^{2} d n\right )} x^{2} + {\left (b^{4} c n^{3} + 9 \, b^{4} c n^{2} + 24 \, b^{4} c + 2 \, {\left (13 \, b^{4} c + 3 \, a^{3} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x^3+c),x, algorithm="fricas")

[Out]

(a*b^3*c*n^3 + 9*a*b^3*c*n^2 + 26*a*b^3*c*n + 24*a*b^3*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 - 3*(a^2*b^2*d*n^2 + a^2*b^2*d*n)*x^2 + (b^4*c*n
^3 + 9*b^4*c*n^2 + 24*b^4*c + 2*(13*b^4*c + 3*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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giac [B]  time = 0.33, size = 361, normalized size = 3.84 \[ \frac {{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d n x^{4} + {\left (b x + a\right )}^{n} b^{4} c n^{3} x - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d x^{4} + {\left (b x + a\right )}^{n} a b^{3} c n^{3} + 9 \, {\left (b x + a\right )}^{n} b^{4} c n^{2} x - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} + 9 \, {\left (b x + a\right )}^{n} a b^{3} c n^{2} + 26 \, {\left (b x + a\right )}^{n} b^{4} c n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d n x + 26 \, {\left (b x + a\right )}^{n} a b^{3} c n + 24 \, {\left (b x + a\right )}^{n} b^{4} c x + 24 \, {\left (b x + a\right )}^{n} a b^{3} c - 6 \, {\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x^3+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 + 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*b^4*c*n^3*x - 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 + (b*x + a)^n*a*b^3*c*n^3 + 9*(b*x + a)^n*b^4*c*n^2*x - 3*(b*x + a
)^n*a^2*b^2*d*n*x^2 + 9*(b*x + a)^n*a*b^3*c*n^2 + 26*(b*x + a)^n*b^4*c*n*x + 6*(b*x + a)^n*a^3*b*d*n*x + 26*(b
*x + a)^n*a*b^3*c*n + 24*(b*x + a)^n*b^4*c*x + 24*(b*x + a)^n*a*b^3*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)

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maple [A]  time = 0.00, size = 167, normalized size = 1.78 \[ -\frac {\left (-b^{3} d \,n^{3} x^{3}-6 b^{3} d \,n^{2} x^{3}+3 a \,b^{2} d \,n^{2} x^{2}-11 b^{3} d n \,x^{3}+9 a \,b^{2} d n \,x^{2}-b^{3} c \,n^{3}-6 d \,x^{3} b^{3}-6 a^{2} b d n x +6 a d \,x^{2} b^{2}-9 b^{3} c \,n^{2}-6 d \,a^{2} x b -26 b^{3} c n +6 a^{3} d -24 b^{3} c \right ) \left (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n*(d*x^3+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^2-11*b^3*d*n*x^3+9*a*b^2*d*n*x^2-b^3*c*n^3-6*b^
3*d*x^3-6*a^2*b*d*n*x+6*a*b^2*d*x^2-9*b^3*c*n^2-6*a^2*b*d*x-26*b^3*c*n+6*a^3*d-24*b^3*c)/b^4/(n^4+10*n^3+35*n^
2+50*n+24)

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maxima [A]  time = 0.96, size = 122, normalized size = 1.30 \[ \frac {{\left (b x + a\right )}^{n + 1} c}{b {\left (n + 1\right )}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x^3+c),x, algorithm="maxima")

[Out]

(b*x + a)^(n + 1)*c/(b*(n + 1)) + ((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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mupad [B]  time = 2.95, size = 247, normalized size = 2.63 \[ {\left (a+b\,x\right )}^n\,\left (\frac {x\,\left (6\,d\,a^3\,b\,n+c\,b^4\,n^3+9\,c\,b^4\,n^2+26\,c\,b^4\,n+24\,c\,b^4\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,\left (-6\,d\,a^3+c\,b^3\,n^3+9\,c\,b^3\,n^2+26\,c\,b^3\,n+24\,c\,b^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {3\,a^2\,d\,n\,x^2\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)*(a + b*x)^n,x)

[Out]

(a + b*x)^n*((x*(24*b^4*c + 9*b^4*c*n^2 + b^4*c*n^3 + 26*b^4*c*n + 6*a^3*b*d*n))/(b^4*(50*n + 35*n^2 + 10*n^3
+ n^4 + 24)) + (a*(24*b^3*c - 6*a^3*d + 9*b^3*c*n^2 + b^3*c*n^3 + 26*b^3*c*n))/(b^4*(50*n + 35*n^2 + 10*n^3 +
n^4 + 24)) + (d*x^4*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) - (3*a^2*d*n*x^2*(n + 1))/(b
^2*(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)))

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sympy [A]  time = 2.79, size = 1906, normalized size = 20.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n*(d*x**3+c),x)

[Out]

Piecewise((a**n*(c*x + 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) + 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) -
2*b**3*c/(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) - 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) - b**3*c/(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) - 3*a*b**2*d*x**2/(2*a*b**4 + 2
*b**5*x) - 2*b**3*c/(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*d*x**2/(2*b**2) + c*log(a/b + 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) - 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*(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) + 9*a*b**3*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) + 26*a*b**3*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) + 24*a*b**3*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) + 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*(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) + 9*b**4*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) + 26*b**4*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) + 24*b
**4*c*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) + 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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