3.101 \(\int x^5 (a+b x)^7 \, dx\)

Optimal. Leaf size=96 \[ \frac{10 a^2 (a+b x)^{11}}{11 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^5 (a+b x)^8}{8 b^6}+\frac{(a+b x)^{13}}{13 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6} \]

[Out]

-(a^5*(a + b*x)^8)/(8*b^6) + (5*a^4*(a + b*x)^9)/(9*b^6) - (a^3*(a + b*x)^10)/b^6 + (10*a^2*(a + b*x)^11)/(11*
b^6) - (5*a*(a + b*x)^12)/(12*b^6) + (a + b*x)^13/(13*b^6)

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Rubi [A]  time = 0.0402941, antiderivative size = 96, 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} \[ \frac{10 a^2 (a+b x)^{11}}{11 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^5 (a+b x)^8}{8 b^6}+\frac{(a+b x)^{13}}{13 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^7,x]

[Out]

-(a^5*(a + b*x)^8)/(8*b^6) + (5*a^4*(a + b*x)^9)/(9*b^6) - (a^3*(a + b*x)^10)/b^6 + (10*a^2*(a + b*x)^11)/(11*
b^6) - (5*a*(a + b*x)^12)/(12*b^6) + (a + b*x)^13/(13*b^6)

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

Mathematica [A]  time = 0.0024743, size = 92, normalized size = 0.96 \[ \frac{21}{11} a^2 b^5 x^{11}+\frac{7}{2} a^3 b^4 x^{10}+\frac{35}{9} a^4 b^3 x^9+\frac{21}{8} a^5 b^2 x^8+a^6 b x^7+\frac{a^7 x^6}{6}+\frac{7}{12} a b^6 x^{12}+\frac{b^7 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^7,x]

[Out]

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

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Maple [A]  time = 0., size = 79, normalized size = 0.8 \begin{align*}{\frac{{b}^{7}{x}^{13}}{13}}+{\frac{7\,a{b}^{6}{x}^{12}}{12}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{11}}{11}}+{\frac{7\,{a}^{3}{b}^{4}{x}^{10}}{2}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{9}}{9}}+{\frac{21\,{a}^{5}{b}^{2}{x}^{8}}{8}}+{a}^{6}b{x}^{7}+{\frac{{a}^{7}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^7,x)

[Out]

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

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Maxima [A]  time = 1.06887, size = 105, normalized size = 1.09 \begin{align*} \frac{1}{13} \, b^{7} x^{13} + \frac{7}{12} \, a b^{6} x^{12} + \frac{21}{11} \, a^{2} b^{5} x^{11} + \frac{7}{2} \, a^{3} b^{4} x^{10} + \frac{35}{9} \, a^{4} b^{3} x^{9} + \frac{21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac{1}{6} \, a^{7} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.35283, size = 185, normalized size = 1.93 \begin{align*} \frac{1}{13} x^{13} b^{7} + \frac{7}{12} x^{12} b^{6} a + \frac{21}{11} x^{11} b^{5} a^{2} + \frac{7}{2} x^{10} b^{4} a^{3} + \frac{35}{9} x^{9} b^{3} a^{4} + \frac{21}{8} x^{8} b^{2} a^{5} + x^{7} b a^{6} + \frac{1}{6} x^{6} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.08211, size = 90, normalized size = 0.94 \begin{align*} \frac{a^{7} x^{6}}{6} + a^{6} b x^{7} + \frac{21 a^{5} b^{2} x^{8}}{8} + \frac{35 a^{4} b^{3} x^{9}}{9} + \frac{7 a^{3} b^{4} x^{10}}{2} + \frac{21 a^{2} b^{5} x^{11}}{11} + \frac{7 a b^{6} x^{12}}{12} + \frac{b^{7} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x+a)**7,x)

[Out]

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

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Giac [A]  time = 1.22312, size = 105, normalized size = 1.09 \begin{align*} \frac{1}{13} \, b^{7} x^{13} + \frac{7}{12} \, a b^{6} x^{12} + \frac{21}{11} \, a^{2} b^{5} x^{11} + \frac{7}{2} \, a^{3} b^{4} x^{10} + \frac{35}{9} \, a^{4} b^{3} x^{9} + \frac{21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac{1}{6} \, a^{7} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^7,x, algorithm="giac")

[Out]

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