3.5.67 \(\int x^4 (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=139 \[ \frac {a^4 (a+b x)^7 (A b-a B)}{7 b^6}-\frac {a^3 (a+b x)^8 (4 A b-5 a B)}{8 b^6}+\frac {2 a^2 (a+b x)^9 (3 A b-5 a B)}{9 b^6}+\frac {(a+b x)^{11} (A b-5 a B)}{11 b^6}-\frac {a (a+b x)^{10} (2 A b-5 a B)}{5 b^6}+\frac {B (a+b x)^{12}}{12 b^6} \]

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Rubi [A]  time = 0.10, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 76} \begin {gather*} \frac {2 a^2 (a+b x)^9 (3 A b-5 a B)}{9 b^6}-\frac {a^3 (a+b x)^8 (4 A b-5 a B)}{8 b^6}+\frac {a^4 (a+b x)^7 (A b-a B)}{7 b^6}+\frac {(a+b x)^{11} (A b-5 a B)}{11 b^6}-\frac {a (a+b x)^{10} (2 A b-5 a B)}{5 b^6}+\frac {B (a+b x)^{12}}{12 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a^4*(A*b - a*B)*(a + b*x)^7)/(7*b^6) - (a^3*(4*A*b - 5*a*B)*(a + b*x)^8)/(8*b^6) + (2*a^2*(3*A*b - 5*a*B)*(a
+ b*x)^9)/(9*b^6) - (a*(2*A*b - 5*a*B)*(a + b*x)^10)/(5*b^6) + ((A*b - 5*a*B)*(a + b*x)^11)/(11*b^6) + (B*(a +
b*x)^12)/(12*b^6)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^4 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (-\frac {a^4 (-A b+a B) (a+b x)^6}{b^5}+\frac {a^3 (-4 A b+5 a B) (a+b x)^7}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^8}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^9}{b^5}+\frac {(A b-5 a B) (a+b x)^{10}}{b^5}+\frac {B (a+b x)^{11}}{b^5}\right ) \, dx\\ &=\frac {a^4 (A b-a B) (a+b x)^7}{7 b^6}-\frac {a^3 (4 A b-5 a B) (a+b x)^8}{8 b^6}+\frac {2 a^2 (3 A b-5 a B) (a+b x)^9}{9 b^6}-\frac {a (2 A b-5 a B) (a+b x)^{10}}{5 b^6}+\frac {(A b-5 a B) (a+b x)^{11}}{11 b^6}+\frac {B (a+b x)^{12}}{12 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 143, normalized size = 1.03 \begin {gather*} \frac {1}{5} a^6 A x^5+\frac {1}{6} a^5 x^6 (a B+6 A b)+\frac {3}{7} a^4 b x^7 (2 a B+5 A b)+\frac {5}{8} a^3 b^2 x^8 (3 a B+4 A b)+\frac {5}{9} a^2 b^3 x^9 (4 a B+3 A b)+\frac {1}{11} b^5 x^{11} (6 a B+A b)+\frac {3}{10} a b^4 x^{10} (5 a B+2 A b)+\frac {1}{12} b^6 B x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a^6*A*x^5)/5 + (a^5*(6*A*b + a*B)*x^6)/6 + (3*a^4*b*(5*A*b + 2*a*B)*x^7)/7 + (5*a^3*b^2*(4*A*b + 3*a*B)*x^8)/
8 + (5*a^2*b^3*(3*A*b + 4*a*B)*x^9)/9 + (3*a*b^4*(2*A*b + 5*a*B)*x^10)/10 + (b^5*(A*b + 6*a*B)*x^11)/11 + (b^6
*B*x^12)/12

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [A]  time = 0.36, size = 148, normalized size = 1.06 \begin {gather*} \frac {1}{12} x^{12} b^{6} B + \frac {6}{11} x^{11} b^{5} a B + \frac {1}{11} x^{11} b^{6} A + \frac {3}{2} x^{10} b^{4} a^{2} B + \frac {3}{5} x^{10} b^{5} a A + \frac {20}{9} x^{9} b^{3} a^{3} B + \frac {5}{3} x^{9} b^{4} a^{2} A + \frac {15}{8} x^{8} b^{2} a^{4} B + \frac {5}{2} x^{8} b^{3} a^{3} A + \frac {6}{7} x^{7} b a^{5} B + \frac {15}{7} x^{7} b^{2} a^{4} A + \frac {1}{6} x^{6} a^{6} B + x^{6} b a^{5} A + \frac {1}{5} x^{5} a^{6} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/12*x^12*b^6*B + 6/11*x^11*b^5*a*B + 1/11*x^11*b^6*A + 3/2*x^10*b^4*a^2*B + 3/5*x^10*b^5*a*A + 20/9*x^9*b^3*a
^3*B + 5/3*x^9*b^4*a^2*A + 15/8*x^8*b^2*a^4*B + 5/2*x^8*b^3*a^3*A + 6/7*x^7*b*a^5*B + 15/7*x^7*b^2*a^4*A + 1/6
*x^6*a^6*B + x^6*b*a^5*A + 1/5*x^5*a^6*A

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giac [A]  time = 0.16, size = 148, normalized size = 1.06 \begin {gather*} \frac {1}{12} \, B b^{6} x^{12} + \frac {6}{11} \, B a b^{5} x^{11} + \frac {1}{11} \, A b^{6} x^{11} + \frac {3}{2} \, B a^{2} b^{4} x^{10} + \frac {3}{5} \, A a b^{5} x^{10} + \frac {20}{9} \, B a^{3} b^{3} x^{9} + \frac {5}{3} \, A a^{2} b^{4} x^{9} + \frac {15}{8} \, B a^{4} b^{2} x^{8} + \frac {5}{2} \, A a^{3} b^{3} x^{8} + \frac {6}{7} \, B a^{5} b x^{7} + \frac {15}{7} \, A a^{4} b^{2} x^{7} + \frac {1}{6} \, B a^{6} x^{6} + A a^{5} b x^{6} + \frac {1}{5} \, A a^{6} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/12*B*b^6*x^12 + 6/11*B*a*b^5*x^11 + 1/11*A*b^6*x^11 + 3/2*B*a^2*b^4*x^10 + 3/5*A*a*b^5*x^10 + 20/9*B*a^3*b^3
*x^9 + 5/3*A*a^2*b^4*x^9 + 15/8*B*a^4*b^2*x^8 + 5/2*A*a^3*b^3*x^8 + 6/7*B*a^5*b*x^7 + 15/7*A*a^4*b^2*x^7 + 1/6
*B*a^6*x^6 + A*a^5*b*x^6 + 1/5*A*a^6*x^5

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maple [A]  time = 0.04, size = 148, normalized size = 1.06 \begin {gather*} \frac {B \,b^{6} x^{12}}{12}+\frac {A \,a^{6} x^{5}}{5}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{11}}{11}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{10}}{10}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{9}}{9}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{8}}{8}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{7}}{7}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/12*B*b^6*x^12+1/11*(A*b^6+6*B*a*b^5)*x^11+1/10*(6*A*a*b^5+15*B*a^2*b^4)*x^10+1/9*(15*A*a^2*b^4+20*B*a^3*b^3)
*x^9+1/8*(20*A*a^3*b^3+15*B*a^4*b^2)*x^8+1/7*(15*A*a^4*b^2+6*B*a^5*b)*x^7+1/6*(6*A*a^5*b+B*a^6)*x^6+1/5*A*a^6*
x^5

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maxima [A]  time = 0.50, size = 147, normalized size = 1.06 \begin {gather*} \frac {1}{12} \, B b^{6} x^{12} + \frac {1}{5} \, A a^{6} x^{5} + \frac {1}{11} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{11} + \frac {3}{10} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{10} + \frac {5}{9} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{9} + \frac {5}{8} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{8} + \frac {3}{7} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/12*B*b^6*x^12 + 1/5*A*a^6*x^5 + 1/11*(6*B*a*b^5 + A*b^6)*x^11 + 3/10*(5*B*a^2*b^4 + 2*A*a*b^5)*x^10 + 5/9*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*x^9 + 5/8*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^8 + 3/7*(2*B*a^5*b + 5*A*a^4*b^2)*x^7 + 1/6*
(B*a^6 + 6*A*a^5*b)*x^6

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mupad [B]  time = 1.07, size = 130, normalized size = 0.94 \begin {gather*} x^6\,\left (\frac {B\,a^6}{6}+A\,b\,a^5\right )+x^{11}\,\left (\frac {A\,b^6}{11}+\frac {6\,B\,a\,b^5}{11}\right )+\frac {A\,a^6\,x^5}{5}+\frac {B\,b^6\,x^{12}}{12}+\frac {5\,a^3\,b^2\,x^8\,\left (4\,A\,b+3\,B\,a\right )}{8}+\frac {5\,a^2\,b^3\,x^9\,\left (3\,A\,b+4\,B\,a\right )}{9}+\frac {3\,a^4\,b\,x^7\,\left (5\,A\,b+2\,B\,a\right )}{7}+\frac {3\,a\,b^4\,x^{10}\,\left (2\,A\,b+5\,B\,a\right )}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^6*((B*a^6)/6 + A*a^5*b) + x^11*((A*b^6)/11 + (6*B*a*b^5)/11) + (A*a^6*x^5)/5 + (B*b^6*x^12)/12 + (5*a^3*b^2*
x^8*(4*A*b + 3*B*a))/8 + (5*a^2*b^3*x^9*(3*A*b + 4*B*a))/9 + (3*a^4*b*x^7*(5*A*b + 2*B*a))/7 + (3*a*b^4*x^10*(
2*A*b + 5*B*a))/10

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sympy [A]  time = 0.10, size = 162, normalized size = 1.17 \begin {gather*} \frac {A a^{6} x^{5}}{5} + \frac {B b^{6} x^{12}}{12} + x^{11} \left (\frac {A b^{6}}{11} + \frac {6 B a b^{5}}{11}\right ) + x^{10} \left (\frac {3 A a b^{5}}{5} + \frac {3 B a^{2} b^{4}}{2}\right ) + x^{9} \left (\frac {5 A a^{2} b^{4}}{3} + \frac {20 B a^{3} b^{3}}{9}\right ) + x^{8} \left (\frac {5 A a^{3} b^{3}}{2} + \frac {15 B a^{4} b^{2}}{8}\right ) + x^{7} \left (\frac {15 A a^{4} b^{2}}{7} + \frac {6 B a^{5} b}{7}\right ) + x^{6} \left (A a^{5} b + \frac {B a^{6}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

A*a**6*x**5/5 + B*b**6*x**12/12 + x**11*(A*b**6/11 + 6*B*a*b**5/11) + x**10*(3*A*a*b**5/5 + 3*B*a**2*b**4/2) +
x**9*(5*A*a**2*b**4/3 + 20*B*a**3*b**3/9) + x**8*(5*A*a**3*b**3/2 + 15*B*a**4*b**2/8) + x**7*(15*A*a**4*b**2/
7 + 6*B*a**5*b/7) + x**6*(A*a**5*b + B*a**6/6)

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