3.1.78 \(\int x^5 (a+b x)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac {a^5 x^6}{6}+\frac {5}{7} a^4 b x^7+\frac {5}{4} a^3 b^2 x^8+\frac {10}{9} a^2 b^3 x^9+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{11}}{11} \]

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Rubi [A]  time = 0.03, antiderivative size = 69, 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 = {43} \begin {gather*} \frac {10}{9} a^2 b^3 x^9+\frac {5}{4} a^3 b^2 x^8+\frac {5}{7} a^4 b x^7+\frac {a^5 x^6}{6}+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^5,x]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^7)/7 + (5*a^3*b^2*x^8)/4 + (10*a^2*b^3*x^9)/9 + (a*b^4*x^10)/2 + (b^5*x^11)/11

Rule 43

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^5 (a+b x)^5 \, dx &=\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx\\ &=\frac {a^5 x^6}{6}+\frac {5}{7} a^4 b x^7+\frac {5}{4} a^3 b^2 x^8+\frac {10}{9} a^2 b^3 x^9+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{11}}{11}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} \frac {a^5 x^6}{6}+\frac {5}{7} a^4 b x^7+\frac {5}{4} a^3 b^2 x^8+\frac {10}{9} a^2 b^3 x^9+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^5,x]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^7)/7 + (5*a^3*b^2*x^8)/4 + (10*a^2*b^3*x^9)/9 + (a*b^4*x^10)/2 + (b^5*x^11)/11

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^5 (a+b x)^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5*(a + b*x)^5,x]

[Out]

IntegrateAlgebraic[x^5*(a + b*x)^5, x]

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fricas [A]  time = 1.29, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{11} x^{11} b^{5} + \frac {1}{2} x^{10} b^{4} a + \frac {10}{9} x^{9} b^{3} a^{2} + \frac {5}{4} x^{8} b^{2} a^{3} + \frac {5}{7} x^{7} b a^{4} + \frac {1}{6} x^{6} a^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*b^5 + 1/2*x^10*b^4*a + 10/9*x^9*b^3*a^2 + 5/4*x^8*b^2*a^3 + 5/7*x^7*b*a^4 + 1/6*x^6*a^5

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giac [A]  time = 1.20, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} + \frac {1}{2} \, a b^{4} x^{10} + \frac {10}{9} \, a^{2} b^{3} x^{9} + \frac {5}{4} \, a^{3} b^{2} x^{8} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{6} \, a^{5} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^5*x^11 + 1/2*a*b^4*x^10 + 10/9*a^2*b^3*x^9 + 5/4*a^3*b^2*x^8 + 5/7*a^4*b*x^7 + 1/6*a^5*x^6

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maple [A]  time = 0.00, size = 58, normalized size = 0.84 \begin {gather*} \frac {1}{11} b^{5} x^{11}+\frac {1}{2} a \,b^{4} x^{10}+\frac {10}{9} a^{2} b^{3} x^{9}+\frac {5}{4} a^{3} b^{2} x^{8}+\frac {5}{7} a^{4} b \,x^{7}+\frac {1}{6} a^{5} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^5,x)

[Out]

1/6*a^5*x^6+5/7*a^4*b*x^7+5/4*a^3*b^2*x^8+10/9*a^2*b^3*x^9+1/2*a*b^4*x^10+1/11*b^5*x^11

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maxima [A]  time = 1.36, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} + \frac {1}{2} \, a b^{4} x^{10} + \frac {10}{9} \, a^{2} b^{3} x^{9} + \frac {5}{4} \, a^{3} b^{2} x^{8} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{6} \, a^{5} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^5*x^11 + 1/2*a*b^4*x^10 + 10/9*a^2*b^3*x^9 + 5/4*a^3*b^2*x^8 + 5/7*a^4*b*x^7 + 1/6*a^5*x^6

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mupad [B]  time = 0.02, size = 57, normalized size = 0.83 \begin {gather*} \frac {a^5\,x^6}{6}+\frac {5\,a^4\,b\,x^7}{7}+\frac {5\,a^3\,b^2\,x^8}{4}+\frac {10\,a^2\,b^3\,x^9}{9}+\frac {a\,b^4\,x^{10}}{2}+\frac {b^5\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x)^5,x)

[Out]

(a^5*x^6)/6 + (b^5*x^11)/11 + (5*a^4*b*x^7)/7 + (a*b^4*x^10)/2 + (5*a^3*b^2*x^8)/4 + (10*a^2*b^3*x^9)/9

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sympy [A]  time = 0.08, size = 65, normalized size = 0.94 \begin {gather*} \frac {a^{5} x^{6}}{6} + \frac {5 a^{4} b x^{7}}{7} + \frac {5 a^{3} b^{2} x^{8}}{4} + \frac {10 a^{2} b^{3} x^{9}}{9} + \frac {a b^{4} x^{10}}{2} + \frac {b^{5} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x**6/6 + 5*a**4*b*x**7/7 + 5*a**3*b**2*x**8/4 + 10*a**2*b**3*x**9/9 + a*b**4*x**10/2 + b**5*x**11/11

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