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

Optimal. Leaf size=112 \[ -\frac {a^3 (a+b x)^7 (A b-a B)}{7 b^5}+\frac {a^2 (a+b x)^8 (3 A b-4 a B)}{8 b^5}+\frac {(a+b x)^{10} (A b-4 a B)}{10 b^5}-\frac {a (a+b x)^9 (A b-2 a B)}{3 b^5}+\frac {B (a+b x)^{11}}{11 b^5} \]

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Rubi [A]  time = 0.08, antiderivative size = 112, 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 {a^2 (a+b x)^8 (3 A b-4 a B)}{8 b^5}-\frac {a^3 (a+b x)^7 (A b-a B)}{7 b^5}+\frac {(a+b x)^{10} (A b-4 a B)}{10 b^5}-\frac {a (a+b x)^9 (A b-2 a B)}{3 b^5}+\frac {B (a+b x)^{11}}{11 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 143, normalized size = 1.28 \begin {gather*} \frac {1}{4} a^6 A x^4+\frac {1}{5} a^5 x^5 (a B+6 A b)+\frac {1}{2} a^4 b x^6 (2 a B+5 A b)+\frac {5}{7} a^3 b^2 x^7 (3 a B+4 A b)+\frac {5}{8} a^2 b^3 x^8 (4 a B+3 A b)+\frac {1}{10} b^5 x^{10} (6 a B+A b)+\frac {1}{3} a b^4 x^9 (5 a B+2 A b)+\frac {1}{11} b^6 B x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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fricas [A]  time = 0.41, size = 148, normalized size = 1.32 \begin {gather*} \frac {1}{11} x^{11} b^{6} B + \frac {3}{5} x^{10} b^{5} a B + \frac {1}{10} x^{10} b^{6} A + \frac {5}{3} x^{9} b^{4} a^{2} B + \frac {2}{3} x^{9} b^{5} a A + \frac {5}{2} x^{8} b^{3} a^{3} B + \frac {15}{8} x^{8} b^{4} a^{2} A + \frac {15}{7} x^{7} b^{2} a^{4} B + \frac {20}{7} x^{7} b^{3} a^{3} A + x^{6} b a^{5} B + \frac {5}{2} x^{6} b^{2} a^{4} A + \frac {1}{5} x^{5} a^{6} B + \frac {6}{5} x^{5} b a^{5} A + \frac {1}{4} x^{4} a^{6} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 148, normalized size = 1.32 \begin {gather*} \frac {1}{11} \, B b^{6} x^{11} + \frac {3}{5} \, B a b^{5} x^{10} + \frac {1}{10} \, A b^{6} x^{10} + \frac {5}{3} \, B a^{2} b^{4} x^{9} + \frac {2}{3} \, A a b^{5} x^{9} + \frac {5}{2} \, B a^{3} b^{3} x^{8} + \frac {15}{8} \, A a^{2} b^{4} x^{8} + \frac {15}{7} \, B a^{4} b^{2} x^{7} + \frac {20}{7} \, A a^{3} b^{3} x^{7} + B a^{5} b x^{6} + \frac {5}{2} \, A a^{4} b^{2} x^{6} + \frac {1}{5} \, B a^{6} x^{5} + \frac {6}{5} \, A a^{5} b x^{5} + \frac {1}{4} \, A a^{6} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 148, normalized size = 1.32 \begin {gather*} \frac {B \,b^{6} x^{11}}{11}+\frac {A \,a^{6} x^{4}}{4}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{10}}{10}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{9}}{9}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{8}}{8}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{7}}{7}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{6}}{6}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 147, normalized size = 1.31 \begin {gather*} \frac {1}{11} \, B b^{6} x^{11} + \frac {1}{4} \, A a^{6} x^{4} + \frac {1}{10} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{9} + \frac {5}{8} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{8} + \frac {5}{7} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 131, normalized size = 1.17 \begin {gather*} x^5\,\left (\frac {B\,a^6}{5}+\frac {6\,A\,b\,a^5}{5}\right )+x^{10}\,\left (\frac {A\,b^6}{10}+\frac {3\,B\,a\,b^5}{5}\right )+\frac {A\,a^6\,x^4}{4}+\frac {B\,b^6\,x^{11}}{11}+\frac {5\,a^3\,b^2\,x^7\,\left (4\,A\,b+3\,B\,a\right )}{7}+\frac {5\,a^2\,b^3\,x^8\,\left (3\,A\,b+4\,B\,a\right )}{8}+\frac {a^4\,b\,x^6\,\left (5\,A\,b+2\,B\,a\right )}{2}+\frac {a\,b^4\,x^9\,\left (2\,A\,b+5\,B\,a\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.10, size = 162, normalized size = 1.45 \begin {gather*} \frac {A a^{6} x^{4}}{4} + \frac {B b^{6} x^{11}}{11} + x^{10} \left (\frac {A b^{6}}{10} + \frac {3 B a b^{5}}{5}\right ) + x^{9} \left (\frac {2 A a b^{5}}{3} + \frac {5 B a^{2} b^{4}}{3}\right ) + x^{8} \left (\frac {15 A a^{2} b^{4}}{8} + \frac {5 B a^{3} b^{3}}{2}\right ) + x^{7} \left (\frac {20 A a^{3} b^{3}}{7} + \frac {15 B a^{4} b^{2}}{7}\right ) + x^{6} \left (\frac {5 A a^{4} b^{2}}{2} + B a^{5} b\right ) + x^{5} \left (\frac {6 A a^{5} b}{5} + \frac {B a^{6}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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