3.6.39 \(\int x^5 (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\) [539]

Optimal. Leaf size=143 \[ \frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 (6 A b+a B) x^7+\frac {3}{8} a^4 b (5 A b+2 a B) x^8+\frac {5}{9} a^3 b^2 (4 A b+3 a B) x^9+\frac {1}{2} a^2 b^3 (3 A b+4 a B) x^{10}+\frac {3}{11} a b^4 (2 A b+5 a B) x^{11}+\frac {1}{12} b^5 (A b+6 a B) x^{12}+\frac {1}{13} b^6 B x^{13} \]

[Out]

1/6*a^6*A*x^6+1/7*a^5*(6*A*b+B*a)*x^7+3/8*a^4*b*(5*A*b+2*B*a)*x^8+5/9*a^3*b^2*(4*A*b+3*B*a)*x^9+1/2*a^2*b^3*(3
*A*b+4*B*a)*x^10+3/11*a*b^4*(2*A*b+5*B*a)*x^11+1/12*b^5*(A*b+6*B*a)*x^12+1/13*b^6*B*x^13

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Rubi [A]
time = 0.10, antiderivative size = 143, 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, 77} \begin {gather*} \frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 x^7 (a B+6 A b)+\frac {3}{8} a^4 b x^8 (2 a B+5 A b)+\frac {5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac {1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac {1}{12} b^5 x^{12} (6 a B+A b)+\frac {3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac {1}{13} b^6 B x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a^6*A*x^6)/6 + (a^5*(6*A*b + a*B)*x^7)/7 + (3*a^4*b*(5*A*b + 2*a*B)*x^8)/8 + (5*a^3*b^2*(4*A*b + 3*a*B)*x^9)/
9 + (a^2*b^3*(3*A*b + 4*a*B)*x^10)/2 + (3*a*b^4*(2*A*b + 5*a*B)*x^11)/11 + (b^5*(A*b + 6*a*B)*x^12)/12 + (b^6*
B*x^13)/13

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 77

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^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^5 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A x^5+a^5 (6 A b+a B) x^6+3 a^4 b (5 A b+2 a B) x^7+5 a^3 b^2 (4 A b+3 a B) x^8+5 a^2 b^3 (3 A b+4 a B) x^9+3 a b^4 (2 A b+5 a B) x^{10}+b^5 (A b+6 a B) x^{11}+b^6 B x^{12}\right ) \, dx\\ &=\frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 (6 A b+a B) x^7+\frac {3}{8} a^4 b (5 A b+2 a B) x^8+\frac {5}{9} a^3 b^2 (4 A b+3 a B) x^9+\frac {1}{2} a^2 b^3 (3 A b+4 a B) x^{10}+\frac {3}{11} a b^4 (2 A b+5 a B) x^{11}+\frac {1}{12} b^5 (A b+6 a B) x^{12}+\frac {1}{13} b^6 B x^{13}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 143, normalized size = 1.00 \begin {gather*} \frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 (6 A b+a B) x^7+\frac {3}{8} a^4 b (5 A b+2 a B) x^8+\frac {5}{9} a^3 b^2 (4 A b+3 a B) x^9+\frac {1}{2} a^2 b^3 (3 A b+4 a B) x^{10}+\frac {3}{11} a b^4 (2 A b+5 a B) x^{11}+\frac {1}{12} b^5 (A b+6 a B) x^{12}+\frac {1}{13} b^6 B x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a^6*A*x^6)/6 + (a^5*(6*A*b + a*B)*x^7)/7 + (3*a^4*b*(5*A*b + 2*a*B)*x^8)/8 + (5*a^3*b^2*(4*A*b + 3*a*B)*x^9)/
9 + (a^2*b^3*(3*A*b + 4*a*B)*x^10)/2 + (3*a*b^4*(2*A*b + 5*a*B)*x^11)/11 + (b^5*(A*b + 6*a*B)*x^12)/12 + (b^6*
B*x^13)/13

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Maple [A]
time = 0.74, size = 148, normalized size = 1.03

method	result	size
norman	\(\frac {b^{6} B \,x^{13}}{13}+\left (\frac {1}{12} A \,b^{6}+\frac {1}{2} B a \,b^{5}\right ) x^{12}+\left (\frac {6}{11} A a \,b^{5}+\frac {15}{11} B \,a^{2} b^{4}\right ) x^{11}+\left (\frac {3}{2} A \,a^{2} b^{4}+2 B \,a^{3} b^{3}\right ) x^{10}+\left (\frac {20}{9} A \,a^{3} b^{3}+\frac {5}{3} B \,a^{4} b^{2}\right ) x^{9}+\left (\frac {15}{8} A \,a^{4} b^{2}+\frac {3}{4} B \,a^{5} b \right ) x^{8}+\left (\frac {6}{7} A \,a^{5} b +\frac {1}{7} B \,a^{6}\right ) x^{7}+\frac {a^{6} A \,x^{6}}{6}\)	\(144\)
gosper	\(\frac {x^{6} \left (5544 b^{6} B \,x^{7}+6006 A \,b^{6} x^{6}+36036 x^{6} B a \,b^{5}+39312 a A \,b^{5} x^{5}+98280 x^{5} B \,a^{2} b^{4}+108108 a^{2} A \,b^{4} x^{4}+144144 x^{4} B \,a^{3} b^{3}+160160 a^{3} A \,b^{3} x^{3}+120120 x^{3} B \,a^{4} b^{2}+135135 a^{4} A \,b^{2} x^{2}+54054 x^{2} B \,a^{5} b +61776 a^{5} A b x +10296 x B \,a^{6}+12012 A \,a^{6}\right )}{72072}\)	\(148\)
default	\(\frac {b^{6} B \,x^{13}}{13}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{12}}{12}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{11}}{11}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{10}}{10}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{9}}{9}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{8}}{8}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{7}}{7}+\frac {a^{6} A \,x^{6}}{6}\)	\(148\)
risch	\(\frac {1}{13} b^{6} B \,x^{13}+\frac {1}{12} x^{12} A \,b^{6}+\frac {1}{2} x^{12} B a \,b^{5}+\frac {6}{11} x^{11} A a \,b^{5}+\frac {15}{11} x^{11} B \,a^{2} b^{4}+\frac {3}{2} x^{10} A \,a^{2} b^{4}+2 x^{10} B \,a^{3} b^{3}+\frac {20}{9} x^{9} A \,a^{3} b^{3}+\frac {5}{3} x^{9} B \,a^{4} b^{2}+\frac {15}{8} x^{8} A \,a^{4} b^{2}+\frac {3}{4} x^{8} B \,a^{5} b +\frac {6}{7} x^{7} A \,a^{5} b +\frac {1}{7} x^{7} B \,a^{6}+\frac {1}{6} a^{6} A \,x^{6}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/13*b^6*B*x^13+1/12*(A*b^6+6*B*a*b^5)*x^12+1/11*(6*A*a*b^5+15*B*a^2*b^4)*x^11+1/10*(15*A*a^2*b^4+20*B*a^3*b^3
)*x^10+1/9*(20*A*a^3*b^3+15*B*a^4*b^2)*x^9+1/8*(15*A*a^4*b^2+6*B*a^5*b)*x^8+1/7*(6*A*a^5*b+B*a^6)*x^7+1/6*a^6*
A*x^6

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Maxima [A]
time = 0.26, size = 147, normalized size = 1.03 \begin {gather*} \frac {1}{13} \, B b^{6} x^{13} + \frac {1}{6} \, A a^{6} x^{6} + \frac {1}{12} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{10} + \frac {5}{9} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{9} + \frac {3}{8} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/13*B*b^6*x^13 + 1/6*A*a^6*x^6 + 1/12*(6*B*a*b^5 + A*b^6)*x^12 + 3/11*(5*B*a^2*b^4 + 2*A*a*b^5)*x^11 + 1/2*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*x^10 + 5/9*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^9 + 3/8*(2*B*a^5*b + 5*A*a^4*b^2)*x^8 + 1/7
*(B*a^6 + 6*A*a^5*b)*x^7

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Fricas [A]
time = 2.18, size = 147, normalized size = 1.03 \begin {gather*} \frac {1}{13} \, B b^{6} x^{13} + \frac {1}{6} \, A a^{6} x^{6} + \frac {1}{12} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{10} + \frac {5}{9} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{9} + \frac {3}{8} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/13*B*b^6*x^13 + 1/6*A*a^6*x^6 + 1/12*(6*B*a*b^5 + A*b^6)*x^12 + 3/11*(5*B*a^2*b^4 + 2*A*a*b^5)*x^11 + 1/2*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*x^10 + 5/9*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^9 + 3/8*(2*B*a^5*b + 5*A*a^4*b^2)*x^8 + 1/7
*(B*a^6 + 6*A*a^5*b)*x^7

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Sympy [A]
time = 0.02, size = 162, normalized size = 1.13 \begin {gather*} \frac {A a^{6} x^{6}}{6} + \frac {B b^{6} x^{13}}{13} + x^{12} \left (\frac {A b^{6}}{12} + \frac {B a b^{5}}{2}\right ) + x^{11} \cdot \left (\frac {6 A a b^{5}}{11} + \frac {15 B a^{2} b^{4}}{11}\right ) + x^{10} \cdot \left (\frac {3 A a^{2} b^{4}}{2} + 2 B a^{3} b^{3}\right ) + x^{9} \cdot \left (\frac {20 A a^{3} b^{3}}{9} + \frac {5 B a^{4} b^{2}}{3}\right ) + x^{8} \cdot \left (\frac {15 A a^{4} b^{2}}{8} + \frac {3 B a^{5} b}{4}\right ) + x^{7} \cdot \left (\frac {6 A a^{5} b}{7} + \frac {B a^{6}}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

A*a**6*x**6/6 + B*b**6*x**13/13 + x**12*(A*b**6/12 + B*a*b**5/2) + x**11*(6*A*a*b**5/11 + 15*B*a**2*b**4/11) +
x**10*(3*A*a**2*b**4/2 + 2*B*a**3*b**3) + x**9*(20*A*a**3*b**3/9 + 5*B*a**4*b**2/3) + x**8*(15*A*a**4*b**2/8
+ 3*B*a**5*b/4) + x**7*(6*A*a**5*b/7 + B*a**6/7)

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Giac [A]
time = 0.83, size = 149, normalized size = 1.04 \begin {gather*} \frac {1}{13} \, B b^{6} x^{13} + \frac {1}{2} \, B a b^{5} x^{12} + \frac {1}{12} \, A b^{6} x^{12} + \frac {15}{11} \, B a^{2} b^{4} x^{11} + \frac {6}{11} \, A a b^{5} x^{11} + 2 \, B a^{3} b^{3} x^{10} + \frac {3}{2} \, A a^{2} b^{4} x^{10} + \frac {5}{3} \, B a^{4} b^{2} x^{9} + \frac {20}{9} \, A a^{3} b^{3} x^{9} + \frac {3}{4} \, B a^{5} b x^{8} + \frac {15}{8} \, A a^{4} b^{2} x^{8} + \frac {1}{7} \, B a^{6} x^{7} + \frac {6}{7} \, A a^{5} b x^{7} + \frac {1}{6} \, A a^{6} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/13*B*b^6*x^13 + 1/2*B*a*b^5*x^12 + 1/12*A*b^6*x^12 + 15/11*B*a^2*b^4*x^11 + 6/11*A*a*b^5*x^11 + 2*B*a^3*b^3*
x^10 + 3/2*A*a^2*b^4*x^10 + 5/3*B*a^4*b^2*x^9 + 20/9*A*a^3*b^3*x^9 + 3/4*B*a^5*b*x^8 + 15/8*A*a^4*b^2*x^8 + 1/
7*B*a^6*x^7 + 6/7*A*a^5*b*x^7 + 1/6*A*a^6*x^6

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Mupad [B]
time = 0.06, size = 131, normalized size = 0.92 \begin {gather*} x^7\,\left (\frac {B\,a^6}{7}+\frac {6\,A\,b\,a^5}{7}\right )+x^{12}\,\left (\frac {A\,b^6}{12}+\frac {B\,a\,b^5}{2}\right )+\frac {A\,a^6\,x^6}{6}+\frac {B\,b^6\,x^{13}}{13}+\frac {5\,a^3\,b^2\,x^9\,\left (4\,A\,b+3\,B\,a\right )}{9}+\frac {a^2\,b^3\,x^{10}\,\left (3\,A\,b+4\,B\,a\right )}{2}+\frac {3\,a^4\,b\,x^8\,\left (5\,A\,b+2\,B\,a\right )}{8}+\frac {3\,a\,b^4\,x^{11}\,\left (2\,A\,b+5\,B\,a\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^7*((B*a^6)/7 + (6*A*a^5*b)/7) + x^12*((A*b^6)/12 + (B*a*b^5)/2) + (A*a^6*x^6)/6 + (B*b^6*x^13)/13 + (5*a^3*b
^2*x^9*(4*A*b + 3*B*a))/9 + (a^2*b^3*x^10*(3*A*b + 4*B*a))/2 + (3*a^4*b*x^8*(5*A*b + 2*B*a))/8 + (3*a*b^4*x^11
*(2*A*b + 5*B*a))/11

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