3.1.90 \(\int x^4 (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=75 \[ \frac {1}{5} a^3 A x^5+\frac {1}{6} a^2 x^6 (a B+3 A b)+\frac {1}{8} b^2 x^8 (3 a B+A b)+\frac {3}{7} a b x^7 (a B+A b)+\frac {1}{9} b^3 B x^9 \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{6} a^2 x^6 (a B+3 A b)+\frac {1}{5} a^3 A x^5+\frac {1}{8} b^2 x^8 (3 a B+A b)+\frac {3}{7} a b x^7 (a B+A b)+\frac {1}{9} b^3 B x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^3*(A + B*x),x]

[Out]

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

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^4 (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A x^4+a^2 (3 A b+a B) x^5+3 a b (A b+a B) x^6+b^2 (A b+3 a B) x^7+b^3 B x^8\right ) \, dx\\ &=\frac {1}{5} a^3 A x^5+\frac {1}{6} a^2 (3 A b+a B) x^6+\frac {3}{7} a b (A b+a B) x^7+\frac {1}{8} b^2 (A b+3 a B) x^8+\frac {1}{9} b^3 B x^9\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{5} a^3 A x^5+\frac {1}{6} a^2 x^6 (a B+3 A b)+\frac {1}{8} b^2 x^8 (3 a B+A b)+\frac {3}{7} a b x^7 (a B+A b)+\frac {1}{9} b^3 B x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^3*(A + B*x),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 (a+b x)^3 (A+B x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4*(a + b*x)^3*(A + B*x),x]

[Out]

IntegrateAlgebraic[x^4*(a + b*x)^3*(A + B*x), x]

________________________________________________________________________________________

fricas [A]  time = 1.12, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{9} x^{9} b^{3} B + \frac {3}{8} x^{8} b^{2} a B + \frac {1}{8} x^{8} b^{3} A + \frac {3}{7} x^{7} b a^{2} B + \frac {3}{7} x^{7} b^{2} a A + \frac {1}{6} x^{6} a^{3} B + \frac {1}{2} x^{6} b a^{2} A + \frac {1}{5} x^{5} a^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^3*(B*x+A),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.25, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} + \frac {3}{8} \, B a b^{2} x^{8} + \frac {1}{8} \, A b^{3} x^{8} + \frac {3}{7} \, B a^{2} b x^{7} + \frac {3}{7} \, A a b^{2} x^{7} + \frac {1}{6} \, B a^{3} x^{6} + \frac {1}{2} \, A a^{2} b x^{6} + \frac {1}{5} \, A a^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^3*(B*x+A),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 76, normalized size = 1.01 \begin {gather*} \frac {B \,b^{3} x^{9}}{9}+\frac {A \,a^{3} x^{5}}{5}+\frac {\left (b^{3} A +3 a \,b^{2} B \right ) x^{8}}{8}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{7}}{7}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^3*(B*x+A),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.08, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} + \frac {1}{5} \, A a^{3} x^{5} + \frac {1}{8} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + \frac {3}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^3*(B*x+A),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 69, normalized size = 0.92 \begin {gather*} x^6\,\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )+x^8\,\left (\frac {A\,b^3}{8}+\frac {3\,B\,a\,b^2}{8}\right )+\frac {A\,a^3\,x^5}{5}+\frac {B\,b^3\,x^9}{9}+\frac {3\,a\,b\,x^7\,\left (A\,b+B\,a\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(A + B*x)*(a + b*x)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 82, normalized size = 1.09 \begin {gather*} \frac {A a^{3} x^{5}}{5} + \frac {B b^{3} x^{9}}{9} + x^{8} \left (\frac {A b^{3}}{8} + \frac {3 B a b^{2}}{8}\right ) + x^{7} \left (\frac {3 A a b^{2}}{7} + \frac {3 B a^{2} b}{7}\right ) + x^{6} \left (\frac {A a^{2} b}{2} + \frac {B a^{3}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**3*(B*x+A),x)

[Out]

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

________________________________________________________________________________________