3.2.28 \(\int x^6 (a+b x)^{10} \, dx\)

Optimal. Leaf size=112 \[ \frac {a^6 (a+b x)^{11}}{11 b^7}-\frac {a^5 (a+b x)^{12}}{2 b^7}+\frac {15 a^4 (a+b x)^{13}}{13 b^7}-\frac {10 a^3 (a+b x)^{14}}{7 b^7}+\frac {a^2 (a+b x)^{15}}{b^7}+\frac {(a+b x)^{17}}{17 b^7}-\frac {3 a (a+b x)^{16}}{8 b^7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {a^2 (a+b x)^{15}}{b^7}-\frac {10 a^3 (a+b x)^{14}}{7 b^7}+\frac {15 a^4 (a+b x)^{13}}{13 b^7}-\frac {a^5 (a+b x)^{12}}{2 b^7}+\frac {a^6 (a+b x)^{11}}{11 b^7}+\frac {(a+b x)^{17}}{17 b^7}-\frac {3 a (a+b x)^{16}}{8 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^10,x]

[Out]

(a^6*(a + b*x)^11)/(11*b^7) - (a^5*(a + b*x)^12)/(2*b^7) + (15*a^4*(a + b*x)^13)/(13*b^7) - (10*a^3*(a + b*x)^
14)/(7*b^7) + (a^2*(a + b*x)^15)/b^7 - (3*a*(a + b*x)^16)/(8*b^7) + (a + b*x)^17/(17*b^7)

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 x^6 (a+b x)^{10} \, dx &=\int \left (\frac {a^6 (a+b x)^{10}}{b^6}-\frac {6 a^5 (a+b x)^{11}}{b^6}+\frac {15 a^4 (a+b x)^{12}}{b^6}-\frac {20 a^3 (a+b x)^{13}}{b^6}+\frac {15 a^2 (a+b x)^{14}}{b^6}-\frac {6 a (a+b x)^{15}}{b^6}+\frac {(a+b x)^{16}}{b^6}\right ) \, dx\\ &=\frac {a^6 (a+b x)^{11}}{11 b^7}-\frac {a^5 (a+b x)^{12}}{2 b^7}+\frac {15 a^4 (a+b x)^{13}}{13 b^7}-\frac {10 a^3 (a+b x)^{14}}{7 b^7}+\frac {a^2 (a+b x)^{15}}{b^7}-\frac {3 a (a+b x)^{16}}{8 b^7}+\frac {(a+b x)^{17}}{17 b^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 126, normalized size = 1.12 \begin {gather*} \frac {a^{10} x^7}{7}+\frac {5}{4} a^9 b x^8+5 a^8 b^2 x^9+12 a^7 b^3 x^{10}+\frac {210}{11} a^6 b^4 x^{11}+21 a^5 b^5 x^{12}+\frac {210}{13} a^4 b^6 x^{13}+\frac {60}{7} a^3 b^7 x^{14}+3 a^2 b^8 x^{15}+\frac {5}{8} a b^9 x^{16}+\frac {b^{10} x^{17}}{17} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^10,x]

[Out]

(a^10*x^7)/7 + (5*a^9*b*x^8)/4 + 5*a^8*b^2*x^9 + 12*a^7*b^3*x^10 + (210*a^6*b^4*x^11)/11 + 21*a^5*b^5*x^12 + (
210*a^4*b^6*x^13)/13 + (60*a^3*b^7*x^14)/7 + 3*a^2*b^8*x^15 + (5*a*b^9*x^16)/8 + (b^10*x^17)/17

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^6 (a+b x)^{10} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^6*(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[x^6*(a + b*x)^10, x]

________________________________________________________________________________________

fricas [A]  time = 0.97, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{17} x^{17} b^{10} + \frac {5}{8} x^{16} b^{9} a + 3 x^{15} b^{8} a^{2} + \frac {60}{7} x^{14} b^{7} a^{3} + \frac {210}{13} x^{13} b^{6} a^{4} + 21 x^{12} b^{5} a^{5} + \frac {210}{11} x^{11} b^{4} a^{6} + 12 x^{10} b^{3} a^{7} + 5 x^{9} b^{2} a^{8} + \frac {5}{4} x^{8} b a^{9} + \frac {1}{7} x^{7} a^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/17*x^17*b^10 + 5/8*x^16*b^9*a + 3*x^15*b^8*a^2 + 60/7*x^14*b^7*a^3 + 210/13*x^13*b^6*a^4 + 21*x^12*b^5*a^5 +
210/11*x^11*b^4*a^6 + 12*x^10*b^3*a^7 + 5*x^9*b^2*a^8 + 5/4*x^8*b*a^9 + 1/7*x^7*a^10

________________________________________________________________________________________

giac [A]  time = 1.10, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{17} \, b^{10} x^{17} + \frac {5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac {60}{7} \, a^{3} b^{7} x^{14} + \frac {210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac {210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac {5}{4} \, a^{9} b x^{8} + \frac {1}{7} \, a^{10} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10,x, algorithm="giac")

[Out]

1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210/13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 +
210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7

________________________________________________________________________________________

maple [A]  time = 0.00, size = 113, normalized size = 1.01 \begin {gather*} \frac {1}{17} b^{10} x^{17}+\frac {5}{8} a \,b^{9} x^{16}+3 a^{2} b^{8} x^{15}+\frac {60}{7} a^{3} b^{7} x^{14}+\frac {210}{13} a^{4} b^{6} x^{13}+21 a^{5} b^{5} x^{12}+\frac {210}{11} a^{6} b^{4} x^{11}+12 a^{7} b^{3} x^{10}+5 a^{8} b^{2} x^{9}+\frac {5}{4} a^{9} b \,x^{8}+\frac {1}{7} a^{10} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^10,x)

[Out]

1/17*b^10*x^17+5/8*a*b^9*x^16+3*a^2*b^8*x^15+60/7*a^3*b^7*x^14+210/13*a^4*b^6*x^13+21*a^5*b^5*x^12+210/11*a^6*
b^4*x^11+12*a^7*b^3*x^10+5*a^8*b^2*x^9+5/4*a^9*b*x^8+1/7*a^10*x^7

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{17} \, b^{10} x^{17} + \frac {5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac {60}{7} \, a^{3} b^{7} x^{14} + \frac {210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac {210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac {5}{4} \, a^{9} b x^{8} + \frac {1}{7} \, a^{10} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210/13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 +
210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 112, normalized size = 1.00 \begin {gather*} \frac {a^{10}\,x^7}{7}+\frac {5\,a^9\,b\,x^8}{4}+5\,a^8\,b^2\,x^9+12\,a^7\,b^3\,x^{10}+\frac {210\,a^6\,b^4\,x^{11}}{11}+21\,a^5\,b^5\,x^{12}+\frac {210\,a^4\,b^6\,x^{13}}{13}+\frac {60\,a^3\,b^7\,x^{14}}{7}+3\,a^2\,b^8\,x^{15}+\frac {5\,a\,b^9\,x^{16}}{8}+\frac {b^{10}\,x^{17}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*x)^10,x)

[Out]

(a^10*x^7)/7 + (b^10*x^17)/17 + (5*a^9*b*x^8)/4 + (5*a*b^9*x^16)/8 + 5*a^8*b^2*x^9 + 12*a^7*b^3*x^10 + (210*a^
6*b^4*x^11)/11 + 21*a^5*b^5*x^12 + (210*a^4*b^6*x^13)/13 + (60*a^3*b^7*x^14)/7 + 3*a^2*b^8*x^15

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 128, normalized size = 1.14 \begin {gather*} \frac {a^{10} x^{7}}{7} + \frac {5 a^{9} b x^{8}}{4} + 5 a^{8} b^{2} x^{9} + 12 a^{7} b^{3} x^{10} + \frac {210 a^{6} b^{4} x^{11}}{11} + 21 a^{5} b^{5} x^{12} + \frac {210 a^{4} b^{6} x^{13}}{13} + \frac {60 a^{3} b^{7} x^{14}}{7} + 3 a^{2} b^{8} x^{15} + \frac {5 a b^{9} x^{16}}{8} + \frac {b^{10} x^{17}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**10,x)

[Out]

a**10*x**7/7 + 5*a**9*b*x**8/4 + 5*a**8*b**2*x**9 + 12*a**7*b**3*x**10 + 210*a**6*b**4*x**11/11 + 21*a**5*b**5
*x**12 + 210*a**4*b**6*x**13/13 + 60*a**3*b**7*x**14/7 + 3*a**2*b**8*x**15 + 5*a*b**9*x**16/8 + b**10*x**17/17

________________________________________________________________________________________