3.102 \(\int x^4 (a+b x)^7 \, dx\)

Optimal. Leaf size=81 \[ \frac{3 a^2 (a+b x)^{10}}{5 b^5}-\frac{4 a^3 (a+b x)^9}{9 b^5}+\frac{a^4 (a+b x)^8}{8 b^5}+\frac{(a+b x)^{12}}{12 b^5}-\frac{4 a (a+b x)^{11}}{11 b^5} \]

[Out]

(a^4*(a + b*x)^8)/(8*b^5) - (4*a^3*(a + b*x)^9)/(9*b^5) + (3*a^2*(a + b*x)^10)/(5*b^5) - (4*a*(a + b*x)^11)/(1
1*b^5) + (a + b*x)^12/(12*b^5)

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Rubi [A]  time = 0.0364159, antiderivative size = 81, 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} \[ \frac{3 a^2 (a+b x)^{10}}{5 b^5}-\frac{4 a^3 (a+b x)^9}{9 b^5}+\frac{a^4 (a+b x)^8}{8 b^5}+\frac{(a+b x)^{12}}{12 b^5}-\frac{4 a (a+b x)^{11}}{11 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^7,x]

[Out]

(a^4*(a + b*x)^8)/(8*b^5) - (4*a^3*(a + b*x)^9)/(9*b^5) + (3*a^2*(a + b*x)^10)/(5*b^5) - (4*a*(a + b*x)^11)/(1
1*b^5) + (a + b*x)^12/(12*b^5)

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

Mathematica [A]  time = 0.0048541, size = 93, normalized size = 1.15 \[ \frac{21}{10} a^2 b^5 x^{10}+\frac{35}{9} a^3 b^4 x^9+\frac{35}{8} a^4 b^3 x^8+3 a^5 b^2 x^7+\frac{7}{6} a^6 b x^6+\frac{a^7 x^5}{5}+\frac{7}{11} a b^6 x^{11}+\frac{b^7 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^7,x]

[Out]

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

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Maple [A]  time = 0.001, size = 80, normalized size = 1. \begin{align*}{\frac{{b}^{7}{x}^{12}}{12}}+{\frac{7\,a{b}^{6}{x}^{11}}{11}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{10}}{10}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{9}}{9}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{8}}{8}}+3\,{a}^{5}{b}^{2}{x}^{7}+{\frac{7\,{a}^{6}b{x}^{6}}{6}}+{\frac{{a}^{7}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^7,x)

[Out]

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

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Maxima [A]  time = 1.04485, size = 107, normalized size = 1.32 \begin{align*} \frac{1}{12} \, b^{7} x^{12} + \frac{7}{11} \, a b^{6} x^{11} + \frac{21}{10} \, a^{2} b^{5} x^{10} + \frac{35}{9} \, a^{3} b^{4} x^{9} + \frac{35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac{7}{6} \, a^{6} b x^{6} + \frac{1}{5} \, a^{7} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.36228, size = 186, normalized size = 2.3 \begin{align*} \frac{1}{12} x^{12} b^{7} + \frac{7}{11} x^{11} b^{6} a + \frac{21}{10} x^{10} b^{5} a^{2} + \frac{35}{9} x^{9} b^{4} a^{3} + \frac{35}{8} x^{8} b^{3} a^{4} + 3 x^{7} b^{2} a^{5} + \frac{7}{6} x^{6} b a^{6} + \frac{1}{5} x^{5} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.086854, size = 92, normalized size = 1.14 \begin{align*} \frac{a^{7} x^{5}}{5} + \frac{7 a^{6} b x^{6}}{6} + 3 a^{5} b^{2} x^{7} + \frac{35 a^{4} b^{3} x^{8}}{8} + \frac{35 a^{3} b^{4} x^{9}}{9} + \frac{21 a^{2} b^{5} x^{10}}{10} + \frac{7 a b^{6} x^{11}}{11} + \frac{b^{7} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**7,x)

[Out]

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

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Giac [A]  time = 1.2272, size = 107, normalized size = 1.32 \begin{align*} \frac{1}{12} \, b^{7} x^{12} + \frac{7}{11} \, a b^{6} x^{11} + \frac{21}{10} \, a^{2} b^{5} x^{10} + \frac{35}{9} \, a^{3} b^{4} x^{9} + \frac{35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac{7}{6} \, a^{6} b x^{6} + \frac{1}{5} \, a^{7} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^7,x, algorithm="giac")

[Out]

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