3.1.99 \(\int x^7 (a+b x)^7 \, dx\)

Optimal. Leaf size=95 \[ \frac {a^7 x^8}{8}+\frac {7}{9} a^6 b x^9+\frac {21}{10} a^5 b^2 x^{10}+\frac {35}{11} a^4 b^3 x^{11}+\frac {35}{12} a^3 b^4 x^{12}+\frac {21}{13} a^2 b^5 x^{13}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15} \]

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Rubi [A]  time = 0.04, antiderivative size = 95, 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 {21}{13} a^2 b^5 x^{13}+\frac {35}{12} a^3 b^4 x^{12}+\frac {35}{11} a^4 b^3 x^{11}+\frac {21}{10} a^5 b^2 x^{10}+\frac {7}{9} a^6 b x^9+\frac {a^7 x^8}{8}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x)^7,x]

[Out]

(a^7*x^8)/8 + (7*a^6*b*x^9)/9 + (21*a^5*b^2*x^10)/10 + (35*a^4*b^3*x^11)/11 + (35*a^3*b^4*x^12)/12 + (21*a^2*b
^5*x^13)/13 + (a*b^6*x^14)/2 + (b^7*x^15)/15

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^7 (a+b x)^7 \, dx &=\int \left (a^7 x^7+7 a^6 b x^8+21 a^5 b^2 x^9+35 a^4 b^3 x^{10}+35 a^3 b^4 x^{11}+21 a^2 b^5 x^{12}+7 a b^6 x^{13}+b^7 x^{14}\right ) \, dx\\ &=\frac {a^7 x^8}{8}+\frac {7}{9} a^6 b x^9+\frac {21}{10} a^5 b^2 x^{10}+\frac {35}{11} a^4 b^3 x^{11}+\frac {35}{12} a^3 b^4 x^{12}+\frac {21}{13} a^2 b^5 x^{13}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 95, normalized size = 1.00 \begin {gather*} \frac {a^7 x^8}{8}+\frac {7}{9} a^6 b x^9+\frac {21}{10} a^5 b^2 x^{10}+\frac {35}{11} a^4 b^3 x^{11}+\frac {35}{12} a^3 b^4 x^{12}+\frac {21}{13} a^2 b^5 x^{13}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x)^7,x]

[Out]

(a^7*x^8)/8 + (7*a^6*b*x^9)/9 + (21*a^5*b^2*x^10)/10 + (35*a^4*b^3*x^11)/11 + (35*a^3*b^4*x^12)/12 + (21*a^2*b
^5*x^13)/13 + (a*b^6*x^14)/2 + (b^7*x^15)/15

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^7 (a+b x)^7 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7*(a + b*x)^7,x]

[Out]

IntegrateAlgebraic[x^7*(a + b*x)^7, x]

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fricas [A]  time = 1.34, size = 79, normalized size = 0.83 \begin {gather*} \frac {1}{15} x^{15} b^{7} + \frac {1}{2} x^{14} b^{6} a + \frac {21}{13} x^{13} b^{5} a^{2} + \frac {35}{12} x^{12} b^{4} a^{3} + \frac {35}{11} x^{11} b^{3} a^{4} + \frac {21}{10} x^{10} b^{2} a^{5} + \frac {7}{9} x^{9} b a^{6} + \frac {1}{8} x^{8} a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/15*x^15*b^7 + 1/2*x^14*b^6*a + 21/13*x^13*b^5*a^2 + 35/12*x^12*b^4*a^3 + 35/11*x^11*b^3*a^4 + 21/10*x^10*b^2
*a^5 + 7/9*x^9*b*a^6 + 1/8*x^8*a^7

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giac [A]  time = 0.99, size = 79, normalized size = 0.83 \begin {gather*} \frac {1}{15} \, b^{7} x^{15} + \frac {1}{2} \, a b^{6} x^{14} + \frac {21}{13} \, a^{2} b^{5} x^{13} + \frac {35}{12} \, a^{3} b^{4} x^{12} + \frac {35}{11} \, a^{4} b^{3} x^{11} + \frac {21}{10} \, a^{5} b^{2} x^{10} + \frac {7}{9} \, a^{6} b x^{9} + \frac {1}{8} \, a^{7} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^7,x, algorithm="giac")

[Out]

1/15*b^7*x^15 + 1/2*a*b^6*x^14 + 21/13*a^2*b^5*x^13 + 35/12*a^3*b^4*x^12 + 35/11*a^4*b^3*x^11 + 21/10*a^5*b^2*
x^10 + 7/9*a^6*b*x^9 + 1/8*a^7*x^8

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maple [A]  time = 0.00, size = 80, normalized size = 0.84 \begin {gather*} \frac {1}{15} b^{7} x^{15}+\frac {1}{2} a \,b^{6} x^{14}+\frac {21}{13} a^{2} b^{5} x^{13}+\frac {35}{12} a^{3} b^{4} x^{12}+\frac {35}{11} a^{4} b^{3} x^{11}+\frac {21}{10} a^{5} b^{2} x^{10}+\frac {7}{9} a^{6} b \,x^{9}+\frac {1}{8} a^{7} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x+a)^7,x)

[Out]

1/8*a^7*x^8+7/9*a^6*b*x^9+21/10*a^5*b^2*x^10+35/11*a^4*b^3*x^11+35/12*a^3*b^4*x^12+21/13*a^2*b^5*x^13+1/2*a*b^
6*x^14+1/15*b^7*x^15

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maxima [A]  time = 1.30, size = 79, normalized size = 0.83 \begin {gather*} \frac {1}{15} \, b^{7} x^{15} + \frac {1}{2} \, a b^{6} x^{14} + \frac {21}{13} \, a^{2} b^{5} x^{13} + \frac {35}{12} \, a^{3} b^{4} x^{12} + \frac {35}{11} \, a^{4} b^{3} x^{11} + \frac {21}{10} \, a^{5} b^{2} x^{10} + \frac {7}{9} \, a^{6} b x^{9} + \frac {1}{8} \, a^{7} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/15*b^7*x^15 + 1/2*a*b^6*x^14 + 21/13*a^2*b^5*x^13 + 35/12*a^3*b^4*x^12 + 35/11*a^4*b^3*x^11 + 21/10*a^5*b^2*
x^10 + 7/9*a^6*b*x^9 + 1/8*a^7*x^8

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mupad [B]  time = 0.07, size = 79, normalized size = 0.83 \begin {gather*} \frac {a^7\,x^8}{8}+\frac {7\,a^6\,b\,x^9}{9}+\frac {21\,a^5\,b^2\,x^{10}}{10}+\frac {35\,a^4\,b^3\,x^{11}}{11}+\frac {35\,a^3\,b^4\,x^{12}}{12}+\frac {21\,a^2\,b^5\,x^{13}}{13}+\frac {a\,b^6\,x^{14}}{2}+\frac {b^7\,x^{15}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x)^7,x)

[Out]

(a^7*x^8)/8 + (b^7*x^15)/15 + (7*a^6*b*x^9)/9 + (a*b^6*x^14)/2 + (21*a^5*b^2*x^10)/10 + (35*a^4*b^3*x^11)/11 +
(35*a^3*b^4*x^12)/12 + (21*a^2*b^5*x^13)/13

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sympy [A]  time = 0.09, size = 92, normalized size = 0.97 \begin {gather*} \frac {a^{7} x^{8}}{8} + \frac {7 a^{6} b x^{9}}{9} + \frac {21 a^{5} b^{2} x^{10}}{10} + \frac {35 a^{4} b^{3} x^{11}}{11} + \frac {35 a^{3} b^{4} x^{12}}{12} + \frac {21 a^{2} b^{5} x^{13}}{13} + \frac {a b^{6} x^{14}}{2} + \frac {b^{7} x^{15}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x+a)**7,x)

[Out]

a**7*x**8/8 + 7*a**6*b*x**9/9 + 21*a**5*b**2*x**10/10 + 35*a**4*b**3*x**11/11 + 35*a**3*b**4*x**12/12 + 21*a**
2*b**5*x**13/13 + a*b**6*x**14/2 + b**7*x**15/15

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