3.2.9 \(\int x^8 (a+b x^2)^8 \, dx\) [109]

Optimal. Leaf size=108 \[ \frac {a^8 x^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25} \]

[Out]

1/9*a^8*x^9+8/11*a^7*b*x^11+28/13*a^6*b^2*x^13+56/15*a^5*b^3*x^15+70/17*a^4*b^4*x^17+56/19*a^3*b^5*x^19+4/3*a^
2*b^6*x^21+8/23*a*b^7*x^23+1/25*b^8*x^25

________________________________________________________________________________________

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^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^2)^8,x]

[Out]

(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3
*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25

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^8 \left (a+b x^2\right )^8 \, dx &=\int \left (a^8 x^8+8 a^7 b x^{10}+28 a^6 b^2 x^{12}+56 a^5 b^3 x^{14}+70 a^4 b^4 x^{16}+56 a^3 b^5 x^{18}+28 a^2 b^6 x^{20}+8 a b^7 x^{22}+b^8 x^{24}\right ) \, dx\\ &=\frac {a^8 x^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 108, normalized size = 1.00 \begin {gather*} \frac {a^8 x^9}{9}+\frac {8}{11} a^7 b x^{11}+\frac {28}{13} a^6 b^2 x^{13}+\frac {56}{15} a^5 b^3 x^{15}+\frac {70}{17} a^4 b^4 x^{17}+\frac {56}{19} a^3 b^5 x^{19}+\frac {4}{3} a^2 b^6 x^{21}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{25}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^2)^8,x]

[Out]

(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3
*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 91, normalized size = 0.84

method	result	size
gosper	\(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\)	\(91\)
default	\(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\)	\(91\)
norman	\(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\)	\(91\)
risch	\(\frac {1}{9} a^{8} x^{9}+\frac {8}{11} a^{7} b \,x^{11}+\frac {28}{13} a^{6} b^{2} x^{13}+\frac {56}{15} a^{5} b^{3} x^{15}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{25} b^{8} x^{25}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^8*x^9+8/11*a^7*b*x^11+28/13*a^6*b^2*x^13+56/15*a^5*b^3*x^15+70/17*a^4*b^4*x^17+56/19*a^3*b^5*x^19+4/3*a^
2*b^6*x^21+8/23*a*b^7*x^23+1/25*b^8*x^25

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 90, normalized size = 0.83 \begin {gather*} \frac {1}{25} \, b^{8} x^{25} + \frac {8}{23} \, a b^{7} x^{23} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {70}{17} \, a^{4} b^{4} x^{17} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {28}{13} \, a^{6} b^{2} x^{13} + \frac {8}{11} \, a^{7} b x^{11} + \frac {1}{9} \, a^{8} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x
^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b*x^11 + 1/9*a^8*x^9

________________________________________________________________________________________

Fricas [A]
time = 0.62, size = 90, normalized size = 0.83 \begin {gather*} \frac {1}{25} \, b^{8} x^{25} + \frac {8}{23} \, a b^{7} x^{23} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {70}{17} \, a^{4} b^{4} x^{17} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {28}{13} \, a^{6} b^{2} x^{13} + \frac {8}{11} \, a^{7} b x^{11} + \frac {1}{9} \, a^{8} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x
^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b*x^11 + 1/9*a^8*x^9

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 107, normalized size = 0.99 \begin {gather*} \frac {a^{8} x^{9}}{9} + \frac {8 a^{7} b x^{11}}{11} + \frac {28 a^{6} b^{2} x^{13}}{13} + \frac {56 a^{5} b^{3} x^{15}}{15} + \frac {70 a^{4} b^{4} x^{17}}{17} + \frac {56 a^{3} b^{5} x^{19}}{19} + \frac {4 a^{2} b^{6} x^{21}}{3} + \frac {8 a b^{7} x^{23}}{23} + \frac {b^{8} x^{25}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**9/9 + 8*a**7*b*x**11/11 + 28*a**6*b**2*x**13/13 + 56*a**5*b**3*x**15/15 + 70*a**4*b**4*x**17/17 + 56*a
**3*b**5*x**19/19 + 4*a**2*b**6*x**21/3 + 8*a*b**7*x**23/23 + b**8*x**25/25

________________________________________________________________________________________

Giac [A]
time = 0.81, size = 90, normalized size = 0.83 \begin {gather*} \frac {1}{25} \, b^{8} x^{25} + \frac {8}{23} \, a b^{7} x^{23} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {70}{17} \, a^{4} b^{4} x^{17} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {28}{13} \, a^{6} b^{2} x^{13} + \frac {8}{11} \, a^{7} b x^{11} + \frac {1}{9} \, a^{8} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x
^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b*x^11 + 1/9*a^8*x^9

________________________________________________________________________________________

Mupad [B]
time = 4.94, size = 90, normalized size = 0.83 \begin {gather*} \frac {a^8\,x^9}{9}+\frac {8\,a^7\,b\,x^{11}}{11}+\frac {28\,a^6\,b^2\,x^{13}}{13}+\frac {56\,a^5\,b^3\,x^{15}}{15}+\frac {70\,a^4\,b^4\,x^{17}}{17}+\frac {56\,a^3\,b^5\,x^{19}}{19}+\frac {4\,a^2\,b^6\,x^{21}}{3}+\frac {8\,a\,b^7\,x^{23}}{23}+\frac {b^8\,x^{25}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^2)^8,x)

[Out]

(a^8*x^9)/9 + (b^8*x^25)/25 + (8*a^7*b*x^11)/11 + (8*a*b^7*x^23)/23 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)
/15 + (70*a^4*b^4*x^17)/17 + (56*a^3*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3

________________________________________________________________________________________