3.103 \(\int x^3 (a+b x)^7 \, dx\)

Optimal. Leaf size=64 \[ \frac{a^2 (a+b x)^9}{3 b^4}-\frac{a^3 (a+b x)^8}{8 b^4}+\frac{(a+b x)^{11}}{11 b^4}-\frac{3 a (a+b x)^{10}}{10 b^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0299538, antiderivative size = 64, 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} \[ \frac{a^2 (a+b x)^9}{3 b^4}-\frac{a^3 (a+b x)^8}{8 b^4}+\frac{(a+b x)^{11}}{11 b^4}-\frac{3 a (a+b x)^{10}}{10 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0025907, size = 93, normalized size = 1.45 \[ \frac{7}{3} a^2 b^5 x^9+\frac{35}{8} a^3 b^4 x^8+5 a^4 b^3 x^7+\frac{7}{2} a^5 b^2 x^6+\frac{7}{5} a^6 b x^5+\frac{a^7 x^4}{4}+\frac{7}{10} a b^6 x^{10}+\frac{b^7 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^7,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 80, normalized size = 1.3 \begin{align*}{\frac{{b}^{7}{x}^{11}}{11}}+{\frac{7\,a{b}^{6}{x}^{10}}{10}}+{\frac{7\,{a}^{2}{b}^{5}{x}^{9}}{3}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{8}}{8}}+5\,{a}^{4}{b}^{3}{x}^{7}+{\frac{7\,{a}^{5}{b}^{2}{x}^{6}}{2}}+{\frac{7\,{a}^{6}b{x}^{5}}{5}}+{\frac{{a}^{7}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^7,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01883, size = 107, normalized size = 1.67 \begin{align*} \frac{1}{11} \, b^{7} x^{11} + \frac{7}{10} \, a b^{6} x^{10} + \frac{7}{3} \, a^{2} b^{5} x^{9} + \frac{35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac{7}{2} \, a^{5} b^{2} x^{6} + \frac{7}{5} \, a^{6} b x^{5} + \frac{1}{4} \, a^{7} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^7,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.49904, size = 181, normalized size = 2.83 \begin{align*} \frac{1}{11} x^{11} b^{7} + \frac{7}{10} x^{10} b^{6} a + \frac{7}{3} x^{9} b^{5} a^{2} + \frac{35}{8} x^{8} b^{4} a^{3} + 5 x^{7} b^{3} a^{4} + \frac{7}{2} x^{6} b^{2} a^{5} + \frac{7}{5} x^{5} b a^{6} + \frac{1}{4} x^{4} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^7,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.087945, size = 92, normalized size = 1.44 \begin{align*} \frac{a^{7} x^{4}}{4} + \frac{7 a^{6} b x^{5}}{5} + \frac{7 a^{5} b^{2} x^{6}}{2} + 5 a^{4} b^{3} x^{7} + \frac{35 a^{3} b^{4} x^{8}}{8} + \frac{7 a^{2} b^{5} x^{9}}{3} + \frac{7 a b^{6} x^{10}}{10} + \frac{b^{7} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**7,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16786, size = 107, normalized size = 1.67 \begin{align*} \frac{1}{11} \, b^{7} x^{11} + \frac{7}{10} \, a b^{6} x^{10} + \frac{7}{3} \, a^{2} b^{5} x^{9} + \frac{35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac{7}{2} \, a^{5} b^{2} x^{6} + \frac{7}{5} \, a^{6} b x^{5} + \frac{1}{4} \, a^{7} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^7,x, algorithm="giac")

[Out]

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