3.80 \(\int x^3 (a+b x)^5 \, dx\)

Optimal. Leaf size=64 \[ \frac{3 a^2 (a+b x)^7}{7 b^4}-\frac{a^3 (a+b x)^6}{6 b^4}+\frac{(a+b x)^9}{9 b^4}-\frac{3 a (a+b x)^8}{8 b^4} \]

[Out]

-(a^3*(a + b*x)^6)/(6*b^4) + (3*a^2*(a + b*x)^7)/(7*b^4) - (3*a*(a + b*x)^8)/(8*b^4) + (a + b*x)^9/(9*b^4)

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Rubi [A]  time = 0.0264511, 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} \[ \frac{3 a^2 (a+b x)^7}{7 b^4}-\frac{a^3 (a+b x)^6}{6 b^4}+\frac{(a+b x)^9}{9 b^4}-\frac{3 a (a+b x)^8}{8 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.0025793, size = 66, normalized size = 1.03 \[ \frac{10}{7} a^2 b^3 x^7+\frac{5}{3} a^3 b^2 x^6+a^4 b x^5+\frac{a^5 x^4}{4}+\frac{5}{8} a b^4 x^8+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^5,x]

[Out]

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

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Maple [A]  time = 0., size = 57, normalized size = 0.9 \begin{align*}{\frac{{b}^{5}{x}^{9}}{9}}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{7}}{7}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{6}}{3}}+{a}^{4}b{x}^{5}+{\frac{{a}^{5}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 1.07099, size = 76, normalized size = 1.19 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{8} \, a b^{4} x^{8} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.35036, size = 124, normalized size = 1.94 \begin{align*} \frac{1}{9} x^{9} b^{5} + \frac{5}{8} x^{8} b^{4} a + \frac{10}{7} x^{7} b^{3} a^{2} + \frac{5}{3} x^{6} b^{2} a^{3} + x^{5} b a^{4} + \frac{1}{4} x^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.076684, size = 63, normalized size = 0.98 \begin{align*} \frac{a^{5} x^{4}}{4} + a^{4} b x^{5} + \frac{5 a^{3} b^{2} x^{6}}{3} + \frac{10 a^{2} b^{3} x^{7}}{7} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**5,x)

[Out]

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

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Giac [A]  time = 1.15571, size = 76, normalized size = 1.19 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{8} \, a b^{4} x^{8} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^5,x, algorithm="giac")

[Out]

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