3.100 \(\int x^6 (a+b x)^7 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0385288, antiderivative size = 95, normalized size of antiderivative = 1., 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} \[ \frac{7}{4} a^2 b^5 x^{12}+\frac{35}{11} a^3 b^4 x^{11}+\frac{7}{2} a^4 b^3 x^{10}+\frac{7}{3} a^5 b^2 x^9+\frac{7}{8} a^6 b x^8+\frac{a^7 x^7}{7}+\frac{7}{13} a b^6 x^{13}+\frac{b^7 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0035383, size = 95, normalized size = 1. \[ \frac{7}{4} a^2 b^5 x^{12}+\frac{35}{11} a^3 b^4 x^{11}+\frac{7}{2} a^4 b^3 x^{10}+\frac{7}{3} a^5 b^2 x^9+\frac{7}{8} a^6 b x^8+\frac{a^7 x^7}{7}+\frac{7}{13} a b^6 x^{13}+\frac{b^7 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^7,x]

[Out]

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

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Maple [A]  time = 0.002, size = 80, normalized size = 0.8 \begin{align*}{\frac{{a}^{7}{x}^{7}}{7}}+{\frac{7\,{a}^{6}b{x}^{8}}{8}}+{\frac{7\,{a}^{5}{b}^{2}{x}^{9}}{3}}+{\frac{7\,{a}^{4}{b}^{3}{x}^{10}}{2}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{11}}{11}}+{\frac{7\,{a}^{2}{b}^{5}{x}^{12}}{4}}+{\frac{7\,a{b}^{6}{x}^{13}}{13}}+{\frac{{b}^{7}{x}^{14}}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^7,x)

[Out]

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

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Maxima [A]  time = 1.0641, size = 107, normalized size = 1.13 \begin{align*} \frac{1}{14} \, b^{7} x^{14} + \frac{7}{13} \, a b^{6} x^{13} + \frac{7}{4} \, a^{2} b^{5} x^{12} + \frac{35}{11} \, a^{3} b^{4} x^{11} + \frac{7}{2} \, a^{4} b^{3} x^{10} + \frac{7}{3} \, a^{5} b^{2} x^{9} + \frac{7}{8} \, a^{6} b x^{8} + \frac{1}{7} \, a^{7} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.39136, size = 189, normalized size = 1.99 \begin{align*} \frac{1}{14} x^{14} b^{7} + \frac{7}{13} x^{13} b^{6} a + \frac{7}{4} x^{12} b^{5} a^{2} + \frac{35}{11} x^{11} b^{4} a^{3} + \frac{7}{2} x^{10} b^{3} a^{4} + \frac{7}{3} x^{9} b^{2} a^{5} + \frac{7}{8} x^{8} b a^{6} + \frac{1}{7} x^{7} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.09656, size = 94, normalized size = 0.99 \begin{align*} \frac{a^{7} x^{7}}{7} + \frac{7 a^{6} b x^{8}}{8} + \frac{7 a^{5} b^{2} x^{9}}{3} + \frac{7 a^{4} b^{3} x^{10}}{2} + \frac{35 a^{3} b^{4} x^{11}}{11} + \frac{7 a^{2} b^{5} x^{12}}{4} + \frac{7 a b^{6} x^{13}}{13} + \frac{b^{7} x^{14}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**7,x)

[Out]

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

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Giac [A]  time = 1.16793, size = 107, normalized size = 1.13 \begin{align*} \frac{1}{14} \, b^{7} x^{14} + \frac{7}{13} \, a b^{6} x^{13} + \frac{7}{4} \, a^{2} b^{5} x^{12} + \frac{35}{11} \, a^{3} b^{4} x^{11} + \frac{7}{2} \, a^{4} b^{3} x^{10} + \frac{7}{3} \, a^{5} b^{2} x^{9} + \frac{7}{8} \, a^{6} b x^{8} + \frac{1}{7} \, a^{7} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^7,x, algorithm="giac")

[Out]

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