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]

((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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Rubi [A]  time = 0.0460304, antiderivative size = 94, 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 = {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*}

Mathematica [A]  time = 0.0698985, size = 94, normalized size = 1. \[ \frac{\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

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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Maple [A]  time = 0.004, size = 167, normalized size = 1.8 \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}-11\,{b}^{3}dn{x}^{3}+9\,a{b}^{2}dn{x}^{2}-{b}^{3}c{n}^{3}-6\,d{x}^{3}{b}^{3}-6\,{a}^{2}bdnx+6\,ad{x}^{2}{b}^{2}-9\,{b}^{3}c{n}^{2}-6\,d{a}^{2}xb-26\,{b}^{3}cn+6\,{a}^{3}d-24\,{b}^{3}c \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((b*x+a)^n*(d*x^3+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^2-11*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 [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^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.20149, size = 473, normalized size = 5.03 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 3.0571, size = 1906, normalized size = 20.28 \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**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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Giac [B]  time = 1.18777, size = 487, normalized size = 5.18 \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} + 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}} \end{align*}

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)