3.8.32 \(\int x^3 (a+b x)^n \, dx\) [732]

Optimal. Leaf size=83 \[ -\frac {a^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 a^2 (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a (a+b x)^{3+n}}{b^4 (3+n)}+\frac {(a+b x)^{4+n}}{b^4 (4+n)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} -\frac {a^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(a+b x)^{n+4}}{b^4 (n+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^n,x]

[Out]

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

Rule 45

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

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Mathematica [A]
time = 0.06, size = 67, normalized size = 0.81 \begin {gather*} \frac {(a+b x)^{1+n} \left (-\frac {a^3}{1+n}+\frac {3 a^2 (a+b x)}{2+n}-\frac {3 a (a+b x)^2}{3+n}+\frac {(a+b x)^3}{4+n}\right )}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^n,x]

[Out]

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

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Maple [A]
time = 0.11, size = 126, normalized size = 1.52

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} n^{3} x^{3}-6 b^{3} n^{2} x^{3}+3 a \,b^{2} n^{2} x^{2}-11 b^{3} n \,x^{3}+9 a \,b^{2} n \,x^{2}-6 b^{3} x^{3}-6 a^{2} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(126\)
risch	\(-\frac {\left (-b^{4} n^{3} x^{4}-a \,b^{3} n^{3} x^{3}-6 b^{4} n^{2} x^{4}-3 a \,b^{3} n^{2} x^{3}-11 b^{4} n \,x^{4}+3 a^{2} b^{2} n^{2} x^{2}-2 x^{3} a n \,b^{3}-6 b^{4} x^{4}+3 a^{2} n \,x^{2} b^{2}-6 a^{3} b n x +6 a^{4}\right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\)	\(146\)
norman	\(\frac {x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {a n \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {6 a^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {3 a^{2} n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}+\frac {6 n \,a^{3} x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^n,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 101, normalized size = 1.22 \begin {gather*} \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}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)

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Fricas [A]
time = 0.90, size = 143, normalized size = 1.72 \begin {gather*} \frac {{\left (6 \, a^{3} b n x + {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} + {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*a^3*b*n*x + (b^4*n^3 + 6*b^4*n^2 + 11*b^4*n + 6*b^4)*x^4 - 6*a^4 + (a*b^3*n^3 + 3*a*b^3*n^2 + 2*a*b^3*n)*x^
3 - 3*(a^2*b^2*n^2 + a^2*b^2*n)*x^2)*(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1318 vs. \(2 (71) = 142\).
time = 0.58, size = 1318, normalized size = 15.88 \begin {gather*} \begin {cases} \frac {a^{n} x^{4}}{4} & \text {for}\: b = 0 \\\frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {11 a^{3}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {27 a^{2} b x}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {18 a b^{2} x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {6 b^{3} x^{3} \log {\left (\frac {a}{b} + x \right )}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} & \text {for}\: n = -4 \\- \frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {9 a^{3}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {12 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {12 a^{2} b x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac {6 a b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {2 b^{3} x^{3}}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} & \text {for}\: n = -3 \\\frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{2 a b^{4} + 2 b^{5} x} + \frac {6 a^{3}}{2 a b^{4} + 2 b^{5} x} + \frac {6 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{2 a b^{4} + 2 b^{5} x} - \frac {3 a b^{2} x^{2}}{2 a b^{4} + 2 b^{5} x} + \frac {b^{3} x^{3}}{2 a b^{4} + 2 b^{5} x} & \text {for}\: n = -2 \\- \frac {a^{3} \log {\left (\frac {a}{b} + x \right )}}{b^{4}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b} & \text {for}\: n = -1 \\- \frac {6 a^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 a^{3} b n x \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} - \frac {3 a^{2} b^{2} n^{2} x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} - \frac {3 a^{2} b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {a b^{3} n^{3} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {3 a b^{3} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {2 a b^{3} n x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {b^{4} n^{3} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 b^{4} n^{2} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {11 b^{4} n x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} + \frac {6 b^{4} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} + 10 b^{4} n^{3} + 35 b^{4} n^{2} + 50 b^{4} n + 24 b^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*x**4/4, Eq(b, 0)), (6*a**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) + 11*a**3/(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*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*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*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*x**2/(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*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*log(a/b + x)/(2*a**2*b*
*4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*x*log(a/b + x)/(2
*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*x**2*
log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2),
Eq(n, -3)), (6*a**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*x*log(a/b + x
)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*x**2/(2*a*b**4 + 2*b**5*x) + b**3*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (
-a**3*log(a/b + x)/b**4 + a**2*x/b**3 - a*x**2/(2*b**2) + x**3/(3*b), Eq(n, -1)), (-6*a**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*a**3*b*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*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*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*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*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*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*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*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*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*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] Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (83) = 166\).
time = 1.73, size = 226, normalized size = 2.72 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{4} n^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} n^{2} x^{4} + 3 \, {\left (b x + a\right )}^{n} a b^{3} n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} n x^{4} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} n^{2} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} x^{4} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} n x^{2} + 6 \, {\left (b x + a\right )}^{n} a^{3} b n x - 6 \, {\left (b x + a\right )}^{n} a^{4}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^n,x, algorithm="giac")

[Out]

((b*x + a)^n*b^4*n^3*x^4 + (b*x + a)^n*a*b^3*n^3*x^3 + 6*(b*x + a)^n*b^4*n^2*x^4 + 3*(b*x + a)^n*a*b^3*n^2*x^3
+ 11*(b*x + a)^n*b^4*n*x^4 - 3*(b*x + a)^n*a^2*b^2*n^2*x^2 + 2*(b*x + a)^n*a*b^3*n*x^3 + 6*(b*x + a)^n*b^4*x^
4 - 3*(b*x + a)^n*a^2*b^2*n*x^2 + 6*(b*x + a)^n*a^3*b*n*x - 6*(b*x + a)^n*a^4)/(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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Mupad [B]
time = 0.53, size = 176, normalized size = 2.12 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,n\,x}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,n\,x^2\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^n,x)

[Out]

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

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