3.121 \(\int x^3 (a+b x)^5 (A+B x) \, dx\)

Optimal. Leaf size=112 \[ \frac{a^2 (a+b x)^7 (3 A b-4 a B)}{7 b^5}-\frac{a^3 (a+b x)^6 (A b-a B)}{6 b^5}+\frac{(a+b x)^9 (A b-4 a B)}{9 b^5}-\frac{3 a (a+b x)^8 (A b-2 a B)}{8 b^5}+\frac{B (a+b x)^{10}}{10 b^5} \]

[Out]

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

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Rubi [A]  time = 0.0690141, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{a^2 (a+b x)^7 (3 A b-4 a B)}{7 b^5}-\frac{a^3 (a+b x)^6 (A b-a B)}{6 b^5}+\frac{(a+b x)^9 (A b-4 a B)}{9 b^5}-\frac{3 a (a+b x)^8 (A b-2 a B)}{8 b^5}+\frac{B (a+b x)^{10}}{10 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^5*(A + B*x),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0148077, size = 117, normalized size = 1.04 \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{5}{6} a^3 b x^6 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{4} a^5 A x^4+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^5*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0.002, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{10}}{10}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{9}}{9}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{8}}{8}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{5}}{5}}+{\frac{{a}^{5}A{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05512, size = 161, normalized size = 1.44 \begin{align*} \frac{1}{10} \, B b^{5} x^{10} + \frac{1}{4} \, A a^{5} x^{4} + \frac{1}{9} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac{5}{8} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac{10}{7} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + \frac{5}{6} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66054, size = 290, normalized size = 2.59 \begin{align*} \frac{1}{10} x^{10} b^{5} B + \frac{5}{9} x^{9} b^{4} a B + \frac{1}{9} x^{9} b^{5} A + \frac{5}{4} x^{8} b^{3} a^{2} B + \frac{5}{8} x^{8} b^{4} a A + \frac{10}{7} x^{7} b^{2} a^{3} B + \frac{10}{7} x^{7} b^{3} a^{2} A + \frac{5}{6} x^{6} b a^{4} B + \frac{5}{3} x^{6} b^{2} a^{3} A + \frac{1}{5} x^{5} a^{5} B + x^{5} b a^{4} A + \frac{1}{4} x^{4} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.094121, size = 134, normalized size = 1.2 \begin{align*} \frac{A a^{5} x^{4}}{4} + \frac{B b^{5} x^{10}}{10} + x^{9} \left (\frac{A b^{5}}{9} + \frac{5 B a b^{4}}{9}\right ) + x^{8} \left (\frac{5 A a b^{4}}{8} + \frac{5 B a^{2} b^{3}}{4}\right ) + x^{7} \left (\frac{10 A a^{2} b^{3}}{7} + \frac{10 B a^{3} b^{2}}{7}\right ) + x^{6} \left (\frac{5 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{6}\right ) + x^{5} \left (A a^{4} b + \frac{B a^{5}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17174, size = 167, normalized size = 1.49 \begin{align*} \frac{1}{10} \, B b^{5} x^{10} + \frac{5}{9} \, B a b^{4} x^{9} + \frac{1}{9} \, A b^{5} x^{9} + \frac{5}{4} \, B a^{2} b^{3} x^{8} + \frac{5}{8} \, A a b^{4} x^{8} + \frac{10}{7} \, B a^{3} b^{2} x^{7} + \frac{10}{7} \, A a^{2} b^{3} x^{7} + \frac{5}{6} \, B a^{4} b x^{6} + \frac{5}{3} \, A a^{3} b^{2} x^{6} + \frac{1}{5} \, B a^{5} x^{5} + A a^{4} b x^{5} + \frac{1}{4} \, A a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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