3.2.10 \(\int x^6 (a+b x^2)^8 \, dx\) [110]

Optimal. Leaf size=108 \[ \frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23} \]

[Out]

1/7*a^8*x^7+8/9*a^7*b*x^9+28/11*a^6*b^2*x^11+56/13*a^5*b^3*x^13+14/3*a^4*b^4*x^15+56/17*a^3*b^5*x^17+28/19*a^2
*b^6*x^19+8/21*a*b^7*x^21+1/23*b^8*x^23

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Rubi [A]
time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \begin {gather*} \frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x^2)^8,x]

[Out]

(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^
5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^6 \left (a+b x^2\right )^8 \, dx &=\int \left (a^8 x^6+8 a^7 b x^8+28 a^6 b^2 x^{10}+56 a^5 b^3 x^{12}+70 a^4 b^4 x^{14}+56 a^3 b^5 x^{16}+28 a^2 b^6 x^{18}+8 a b^7 x^{20}+b^8 x^{22}\right ) \, dx\\ &=\frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 108, normalized size = 1.00 \begin {gather*} \frac {a^8 x^7}{7}+\frac {8}{9} a^7 b x^9+\frac {28}{11} a^6 b^2 x^{11}+\frac {56}{13} a^5 b^3 x^{13}+\frac {14}{3} a^4 b^4 x^{15}+\frac {56}{17} a^3 b^5 x^{17}+\frac {28}{19} a^2 b^6 x^{19}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{23}}{23} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^2)^8,x]

[Out]

(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^
5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23

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Maple [A]
time = 0.06, size = 91, normalized size = 0.84

method	result	size
gosper	\(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\)	\(91\)
default	\(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\)	\(91\)
norman	\(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\)	\(91\)
risch	\(\frac {1}{7} a^{8} x^{7}+\frac {8}{9} a^{7} b \,x^{9}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {56}{13} a^{5} b^{3} x^{13}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {8}{21} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{23}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^2+a)^8,x,method=_RETURNVERBOSE)

[Out]

1/7*a^8*x^7+8/9*a^7*b*x^9+28/11*a^6*b^2*x^11+56/13*a^5*b^3*x^13+14/3*a^4*b^4*x^15+56/17*a^3*b^5*x^17+28/19*a^2
*b^6*x^19+8/21*a*b^7*x^21+1/23*b^8*x^23

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Maxima [A]
time = 0.29, size = 90, normalized size = 0.83 \begin {gather*} \frac {1}{23} \, b^{8} x^{23} + \frac {8}{21} \, a b^{7} x^{21} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {56}{17} \, a^{3} b^{5} x^{17} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{7} \, a^{8} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*
x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b*x^9 + 1/7*a^8*x^7

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Fricas [A]
time = 0.95, size = 90, normalized size = 0.83 \begin {gather*} \frac {1}{23} \, b^{8} x^{23} + \frac {8}{21} \, a b^{7} x^{21} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {56}{17} \, a^{3} b^{5} x^{17} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{7} \, a^{8} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*
x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b*x^9 + 1/7*a^8*x^7

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Sympy [A]
time = 0.01, size = 107, normalized size = 0.99 \begin {gather*} \frac {a^{8} x^{7}}{7} + \frac {8 a^{7} b x^{9}}{9} + \frac {28 a^{6} b^{2} x^{11}}{11} + \frac {56 a^{5} b^{3} x^{13}}{13} + \frac {14 a^{4} b^{4} x^{15}}{3} + \frac {56 a^{3} b^{5} x^{17}}{17} + \frac {28 a^{2} b^{6} x^{19}}{19} + \frac {8 a b^{7} x^{21}}{21} + \frac {b^{8} x^{23}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**2+a)**8,x)

[Out]

a**8*x**7/7 + 8*a**7*b*x**9/9 + 28*a**6*b**2*x**11/11 + 56*a**5*b**3*x**13/13 + 14*a**4*b**4*x**15/3 + 56*a**3
*b**5*x**17/17 + 28*a**2*b**6*x**19/19 + 8*a*b**7*x**21/21 + b**8*x**23/23

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Giac [A]
time = 0.71, size = 90, normalized size = 0.83 \begin {gather*} \frac {1}{23} \, b^{8} x^{23} + \frac {8}{21} \, a b^{7} x^{21} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {56}{17} \, a^{3} b^{5} x^{17} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{7} \, a^{8} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*
x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b*x^9 + 1/7*a^8*x^7

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Mupad [B]
time = 0.10, size = 90, normalized size = 0.83 \begin {gather*} \frac {a^8\,x^7}{7}+\frac {8\,a^7\,b\,x^9}{9}+\frac {28\,a^6\,b^2\,x^{11}}{11}+\frac {56\,a^5\,b^3\,x^{13}}{13}+\frac {14\,a^4\,b^4\,x^{15}}{3}+\frac {56\,a^3\,b^5\,x^{17}}{17}+\frac {28\,a^2\,b^6\,x^{19}}{19}+\frac {8\,a\,b^7\,x^{21}}{21}+\frac {b^8\,x^{23}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*x^2)^8,x)

[Out]

(a^8*x^7)/7 + (b^8*x^23)/23 + (8*a^7*b*x^9)/9 + (8*a*b^7*x^21)/21 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/1
3 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^5*x^17)/17 + (28*a^2*b^6*x^19)/19

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