3.10.20 \(\int (a+b x)^6 (A+B x) \, dx\)

Optimal. Leaf size=38 \[ \frac {(a+b x)^7 (A b-a B)}{7 b^2}+\frac {B (a+b x)^8}{8 b^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} \frac {(a+b x)^7 (A b-a B)}{7 b^2}+\frac {B (a+b x)^8}{8 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(A + B*x),x]

[Out]

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

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

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Mathematica [B]  time = 0.04, size = 122, normalized size = 3.21 \begin {gather*} \frac {1}{56} x \left (28 a^6 (2 A+B x)+56 a^5 b x (3 A+2 B x)+70 a^4 b^2 x^2 (4 A+3 B x)+56 a^3 b^3 x^3 (5 A+4 B x)+28 a^2 b^4 x^4 (6 A+5 B x)+8 a b^5 x^5 (7 A+6 B x)+b^6 x^6 (8 A+7 B x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(A + B*x),x]

[Out]

(x*(28*a^6*(2*A + B*x) + 56*a^5*b*x*(3*A + 2*B*x) + 70*a^4*b^2*x^2*(4*A + 3*B*x) + 56*a^3*b^3*x^3*(5*A + 4*B*x
) + 28*a^2*b^4*x^4*(6*A + 5*B*x) + 8*a*b^5*x^5*(7*A + 6*B*x) + b^6*x^6*(8*A + 7*B*x)))/56

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^6 (A+B x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^6*(A + B*x),x]

[Out]

IntegrateAlgebraic[(a + b*x)^6*(A + B*x), x]

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fricas [B]  time = 1.11, size = 145, normalized size = 3.82 \begin {gather*} \frac {1}{8} x^{8} b^{6} B + \frac {6}{7} x^{7} b^{5} a B + \frac {1}{7} x^{7} b^{6} A + \frac {5}{2} x^{6} b^{4} a^{2} B + x^{6} b^{5} a A + 4 x^{5} b^{3} a^{3} B + 3 x^{5} b^{4} a^{2} A + \frac {15}{4} x^{4} b^{2} a^{4} B + 5 x^{4} b^{3} a^{3} A + 2 x^{3} b a^{5} B + 5 x^{3} b^{2} a^{4} A + \frac {1}{2} x^{2} a^{6} B + 3 x^{2} b a^{5} A + x a^{6} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A),x, algorithm="fricas")

[Out]

1/8*x^8*b^6*B + 6/7*x^7*b^5*a*B + 1/7*x^7*b^6*A + 5/2*x^6*b^4*a^2*B + x^6*b^5*a*A + 4*x^5*b^3*a^3*B + 3*x^5*b^
4*a^2*A + 15/4*x^4*b^2*a^4*B + 5*x^4*b^3*a^3*A + 2*x^3*b*a^5*B + 5*x^3*b^2*a^4*A + 1/2*x^2*a^6*B + 3*x^2*b*a^5
*A + x*a^6*A

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giac [B]  time = 1.18, size = 145, normalized size = 3.82 \begin {gather*} \frac {1}{8} \, B b^{6} x^{8} + \frac {6}{7} \, B a b^{5} x^{7} + \frac {1}{7} \, A b^{6} x^{7} + \frac {5}{2} \, B a^{2} b^{4} x^{6} + A a b^{5} x^{6} + 4 \, B a^{3} b^{3} x^{5} + 3 \, A a^{2} b^{4} x^{5} + \frac {15}{4} \, B a^{4} b^{2} x^{4} + 5 \, A a^{3} b^{3} x^{4} + 2 \, B a^{5} b x^{3} + 5 \, A a^{4} b^{2} x^{3} + \frac {1}{2} \, B a^{6} x^{2} + 3 \, A a^{5} b x^{2} + A a^{6} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A),x, algorithm="giac")

[Out]

1/8*B*b^6*x^8 + 6/7*B*a*b^5*x^7 + 1/7*A*b^6*x^7 + 5/2*B*a^2*b^4*x^6 + A*a*b^5*x^6 + 4*B*a^3*b^3*x^5 + 3*A*a^2*
b^4*x^5 + 15/4*B*a^4*b^2*x^4 + 5*A*a^3*b^3*x^4 + 2*B*a^5*b*x^3 + 5*A*a^4*b^2*x^3 + 1/2*B*a^6*x^2 + 3*A*a^5*b*x
^2 + A*a^6*x

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maple [B]  time = 0.00, size = 145, normalized size = 3.82 \begin {gather*} \frac {B \,b^{6} x^{8}}{8}+A \,a^{6} x +\frac {\left (b^{6} A +6 a \,b^{5} B \right ) x^{7}}{7}+\frac {\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) x^{6}}{6}+\frac {\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) x^{5}}{5}+\frac {\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) x^{4}}{4}+\frac {\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) x^{3}}{3}+\frac {\left (6 a^{5} b A +a^{6} B \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A),x)

[Out]

1/8*b^6*B*x^8+1/7*(A*b^6+6*B*a*b^5)*x^7+1/6*(6*A*a*b^5+15*B*a^2*b^4)*x^6+1/5*(15*A*a^2*b^4+20*B*a^3*b^3)*x^5+1
/4*(20*A*a^3*b^3+15*B*a^4*b^2)*x^4+1/3*(15*A*a^4*b^2+6*B*a^5*b)*x^3+1/2*(6*A*a^5*b+B*a^6)*x^2+a^6*A*x

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maxima [B]  time = 0.54, size = 142, normalized size = 3.74 \begin {gather*} \frac {1}{8} \, B b^{6} x^{8} + A a^{6} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 126, normalized size = 3.32 \begin {gather*} x^2\,\left (\frac {B\,a^6}{2}+3\,A\,b\,a^5\right )+x^7\,\left (\frac {A\,b^6}{7}+\frac {6\,B\,a\,b^5}{7}\right )+\frac {B\,b^6\,x^8}{8}+A\,a^6\,x+\frac {5\,a^3\,b^2\,x^4\,\left (4\,A\,b+3\,B\,a\right )}{4}+a^2\,b^3\,x^5\,\left (3\,A\,b+4\,B\,a\right )+a^4\,b\,x^3\,\left (5\,A\,b+2\,B\,a\right )+\frac {a\,b^4\,x^6\,\left (2\,A\,b+5\,B\,a\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^6,x)

[Out]

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

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sympy [B]  time = 0.10, size = 148, normalized size = 3.89 \begin {gather*} A a^{6} x + \frac {B b^{6} x^{8}}{8} + x^{7} \left (\frac {A b^{6}}{7} + \frac {6 B a b^{5}}{7}\right ) + x^{6} \left (A a b^{5} + \frac {5 B a^{2} b^{4}}{2}\right ) + x^{5} \left (3 A a^{2} b^{4} + 4 B a^{3} b^{3}\right ) + x^{4} \left (5 A a^{3} b^{3} + \frac {15 B a^{4} b^{2}}{4}\right ) + x^{3} \left (5 A a^{4} b^{2} + 2 B a^{5} b\right ) + x^{2} \left (3 A a^{5} b + \frac {B a^{6}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A),x)

[Out]

A*a**6*x + B*b**6*x**8/8 + x**7*(A*b**6/7 + 6*B*a*b**5/7) + x**6*(A*a*b**5 + 5*B*a**2*b**4/2) + x**5*(3*A*a**2
*b**4 + 4*B*a**3*b**3) + x**4*(5*A*a**3*b**3 + 15*B*a**4*b**2/4) + x**3*(5*A*a**4*b**2 + 2*B*a**5*b) + x**2*(3
*A*a**5*b + B*a**6/2)

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