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

Optimal. Leaf size=61 \[ \frac{(a+b x)^7 (A b-2 a B)}{7 b^3}-\frac{a (a+b x)^6 (A b-a B)}{6 b^3}+\frac{B (a+b x)^8}{8 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0419063, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {76} \[ \frac{(a+b x)^7 (A b-2 a B)}{7 b^3}-\frac{a (a+b x)^6 (A b-a B)}{6 b^3}+\frac{B (a+b x)^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^5*(A + B*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0153193, size = 115, normalized size = 1.89 \[ 2 a^2 b^2 x^5 (a B+A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{1}{3} a^4 x^3 (a B+5 A b)+\frac{1}{2} a^5 A x^2+\frac{1}{7} b^4 x^7 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^5*(A + B*x),x]

[Out]

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

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Maple [B]  time = 0.001, size = 124, normalized size = 2. \begin{align*}{\frac{{b}^{5}B{x}^{8}}{8}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{4}}{4}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{3}}{3}}+{\frac{{a}^{5}A{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03205, size = 161, normalized size = 2.64 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} + \frac{1}{2} \, A a^{5} x^{2} + \frac{1}{7} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + \frac{5}{6} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac{5}{4} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.48071, size = 285, normalized size = 4.67 \begin{align*} \frac{1}{8} x^{8} b^{5} B + \frac{5}{7} x^{7} b^{4} a B + \frac{1}{7} x^{7} b^{5} A + \frac{5}{3} x^{6} b^{3} a^{2} B + \frac{5}{6} x^{6} b^{4} a A + 2 x^{5} b^{2} a^{3} B + 2 x^{5} b^{3} a^{2} A + \frac{5}{4} x^{4} b a^{4} B + \frac{5}{2} x^{4} b^{2} a^{3} A + \frac{1}{3} x^{3} a^{5} B + \frac{5}{3} x^{3} b a^{4} A + \frac{1}{2} x^{2} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.090437, size = 134, normalized size = 2.2 \begin{align*} \frac{A a^{5} x^{2}}{2} + \frac{B b^{5} x^{8}}{8} + x^{7} \left (\frac{A b^{5}}{7} + \frac{5 B a b^{4}}{7}\right ) + x^{6} \left (\frac{5 A a b^{4}}{6} + \frac{5 B a^{2} b^{3}}{3}\right ) + x^{5} \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{4} \left (\frac{5 A a^{3} b^{2}}{2} + \frac{5 B a^{4} b}{4}\right ) + x^{3} \left (\frac{5 A a^{4} b}{3} + \frac{B a^{5}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17076, size = 169, normalized size = 2.77 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} + \frac{5}{7} \, B a b^{4} x^{7} + \frac{1}{7} \, A b^{5} x^{7} + \frac{5}{3} \, B a^{2} b^{3} x^{6} + \frac{5}{6} \, A a b^{4} x^{6} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac{5}{4} \, B a^{4} b x^{4} + \frac{5}{2} \, A a^{3} b^{2} x^{4} + \frac{1}{3} \, B a^{5} x^{3} + \frac{5}{3} \, A a^{4} b x^{3} + \frac{1}{2} \, A a^{5} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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