3.1.80 \(\int x^3 (a+b x)^5 \, dx\) [80]

Optimal. Leaf size=64 \[ -\frac {a^3 (a+b x)^6}{6 b^4}+\frac {3 a^2 (a+b x)^7}{7 b^4}-\frac {3 a (a+b x)^8}{8 b^4}+\frac {(a+b x)^9}{9 b^4} \]

[Out]

-1/6*a^3*(b*x+a)^6/b^4+3/7*a^2*(b*x+a)^7/b^4-3/8*a*(b*x+a)^8/b^4+1/9*(b*x+a)^9/b^4

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Rubi [A]
time = 0.02, antiderivative size = 64, 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)^6}{6 b^4}+\frac {3 a^2 (a+b x)^7}{7 b^4}+\frac {(a+b x)^9}{9 b^4}-\frac {3 a (a+b x)^8}{8 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^5,x]

[Out]

-1/6*(a^3*(a + b*x)^6)/b^4 + (3*a^2*(a + b*x)^7)/(7*b^4) - (3*a*(a + b*x)^8)/(8*b^4) + (a + b*x)^9/(9*b^4)

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)^5 \, dx &=\int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx\\ &=-\frac {a^3 (a+b x)^6}{6 b^4}+\frac {3 a^2 (a+b x)^7}{7 b^4}-\frac {3 a (a+b x)^8}{8 b^4}+\frac {(a+b x)^9}{9 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 66, normalized size = 1.03 \begin {gather*} \frac {a^5 x^4}{4}+a^4 b x^5+\frac {5}{3} a^3 b^2 x^6+\frac {10}{7} a^2 b^3 x^7+\frac {5}{8} a b^4 x^8+\frac {b^5 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^5,x]

[Out]

(a^5*x^4)/4 + a^4*b*x^5 + (5*a^3*b^2*x^6)/3 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^8)/8 + (b^5*x^9)/9

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Mathics [A]
time = 1.84, size = 57, normalized size = 0.89 \begin {gather*} \frac {x^4 \left (126 a^5+504 a^4 b x+840 a^3 b^2 x^2+720 a^2 b^3 x^3+315 a b^4 x^4+56 b^5 x^5\right )}{504} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^3*(a + b*x)^5,x]')

[Out]

x ^ 4 (126 a ^ 5 + 504 a ^ 4 b x + 840 a ^ 3 b ^ 2 x ^ 2 + 720 a ^ 2 b ^ 3 x ^ 3 + 315 a b ^ 4 x ^ 4 + 56 b ^
5 x ^ 5) / 504

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Maple [A]
time = 0.09, size = 57, normalized size = 0.89

method	result	size
gosper	\(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\)	\(57\)
default	\(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\)	\(57\)
norman	\(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\)	\(57\)
risch	\(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/9*b^5*x^9+5/8*a*b^4*x^8+10/7*a^2*b^3*x^7+5/3*a^3*b^2*x^6+a^4*b*x^5+1/4*a^5*x^4

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Maxima [A]
time = 0.25, size = 56, normalized size = 0.88 \begin {gather*} \frac {1}{9} \, b^{5} x^{9} + \frac {5}{8} \, a b^{4} x^{8} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^5,x, algorithm="maxima")

[Out]

1/9*b^5*x^9 + 5/8*a*b^4*x^8 + 10/7*a^2*b^3*x^7 + 5/3*a^3*b^2*x^6 + a^4*b*x^5 + 1/4*a^5*x^4

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Fricas [A]
time = 0.30, size = 56, normalized size = 0.88 \begin {gather*} \frac {1}{9} \, b^{5} x^{9} + \frac {5}{8} \, a b^{4} x^{8} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^5,x, algorithm="fricas")

[Out]

1/9*b^5*x^9 + 5/8*a*b^4*x^8 + 10/7*a^2*b^3*x^7 + 5/3*a^3*b^2*x^6 + a^4*b*x^5 + 1/4*a^5*x^4

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Sympy [A]
time = 0.04, size = 63, normalized size = 0.98 \begin {gather*} \frac {a^{5} x^{4}}{4} + a^{4} b x^{5} + \frac {5 a^{3} b^{2} x^{6}}{3} + \frac {10 a^{2} b^{3} x^{7}}{7} + \frac {5 a b^{4} x^{8}}{8} + \frac {b^{5} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**5,x)

[Out]

a**5*x**4/4 + a**4*b*x**5 + 5*a**3*b**2*x**6/3 + 10*a**2*b**3*x**7/7 + 5*a*b**4*x**8/8 + b**5*x**9/9

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Giac [A]
time = 0.00, size = 66, normalized size = 1.03 \begin {gather*} \frac {1}{9} x^{9} b^{5}+\frac {5}{8} x^{8} b^{4} a+\frac {10}{7} x^{7} b^{3} a^{2}+\frac {5}{3} x^{6} b^{2} a^{3}+x^{5} b a^{4}+\frac {1}{4} x^{4} a^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^5,x)

[Out]

1/9*b^5*x^9 + 5/8*a*b^4*x^8 + 10/7*a^2*b^3*x^7 + 5/3*a^3*b^2*x^6 + a^4*b*x^5 + 1/4*a^5*x^4

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Mupad [B]
time = 0.02, size = 56, normalized size = 0.88 \begin {gather*} \frac {a^5\,x^4}{4}+a^4\,b\,x^5+\frac {5\,a^3\,b^2\,x^6}{3}+\frac {10\,a^2\,b^3\,x^7}{7}+\frac {5\,a\,b^4\,x^8}{8}+\frac {b^5\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^5,x)

[Out]

(a^5*x^4)/4 + (b^5*x^9)/9 + a^4*b*x^5 + (5*a*b^4*x^8)/8 + (5*a^3*b^2*x^6)/3 + (10*a^2*b^3*x^7)/7

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