3.2.13 \(\int (a+b x^2)^8 \, dx\) [113]

Optimal. Leaf size=101 \[ a^8 x+\frac {8}{3} a^7 b x^3+\frac {28}{5} a^6 b^2 x^5+8 a^5 b^3 x^7+\frac {70}{9} a^4 b^4 x^9+\frac {56}{11} a^3 b^5 x^{11}+\frac {28}{13} a^2 b^6 x^{13}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{17}}{17} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \begin {gather*} a^8 x+\frac {8}{3} a^7 b x^3+\frac {28}{5} a^6 b^2 x^5+8 a^5 b^3 x^7+\frac {70}{9} a^4 b^4 x^9+\frac {56}{11} a^3 b^5 x^{11}+\frac {28}{13} a^2 b^6 x^{13}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{17}}{17} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8,x]

[Out]

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

Rule 200

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 101, normalized size = 1.00 \begin {gather*} a^8 x+\frac {8}{3} a^7 b x^3+\frac {28}{5} a^6 b^2 x^5+8 a^5 b^3 x^7+\frac {70}{9} a^4 b^4 x^9+\frac {56}{11} a^3 b^5 x^{11}+\frac {28}{13} a^2 b^6 x^{13}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{17}}{17} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8,x]

[Out]

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

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Maple [A]
time = 0.02, size = 88, normalized size = 0.87

method	result	size
gosper	\(a^{8} x +\frac {8}{3} a^{7} b \,x^{3}+\frac {28}{5} a^{6} b^{2} x^{5}+8 a^{5} b^{3} x^{7}+\frac {70}{9} a^{4} b^{4} x^{9}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {28}{13} a^{2} b^{6} x^{13}+\frac {8}{15} a \,b^{7} x^{15}+\frac {1}{17} b^{8} x^{17}\)	\(88\)
default	\(a^{8} x +\frac {8}{3} a^{7} b \,x^{3}+\frac {28}{5} a^{6} b^{2} x^{5}+8 a^{5} b^{3} x^{7}+\frac {70}{9} a^{4} b^{4} x^{9}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {28}{13} a^{2} b^{6} x^{13}+\frac {8}{15} a \,b^{7} x^{15}+\frac {1}{17} b^{8} x^{17}\)	\(88\)
norman	\(a^{8} x +\frac {8}{3} a^{7} b \,x^{3}+\frac {28}{5} a^{6} b^{2} x^{5}+8 a^{5} b^{3} x^{7}+\frac {70}{9} a^{4} b^{4} x^{9}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {28}{13} a^{2} b^{6} x^{13}+\frac {8}{15} a \,b^{7} x^{15}+\frac {1}{17} b^{8} x^{17}\)	\(88\)
risch	\(a^{8} x +\frac {8}{3} a^{7} b \,x^{3}+\frac {28}{5} a^{6} b^{2} x^{5}+8 a^{5} b^{3} x^{7}+\frac {70}{9} a^{4} b^{4} x^{9}+\frac {56}{11} a^{3} b^{5} x^{11}+\frac {28}{13} a^{2} b^{6} x^{13}+\frac {8}{15} a \,b^{7} x^{15}+\frac {1}{17} b^{8} x^{17}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 87, normalized size = 0.86 \begin {gather*} \frac {1}{17} \, b^{8} x^{17} + \frac {8}{15} \, a b^{7} x^{15} + \frac {28}{13} \, a^{2} b^{6} x^{13} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 8 \, a^{5} b^{3} x^{7} + \frac {28}{5} \, a^{6} b^{2} x^{5} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.85, size = 87, normalized size = 0.86 \begin {gather*} \frac {1}{17} \, b^{8} x^{17} + \frac {8}{15} \, a b^{7} x^{15} + \frac {28}{13} \, a^{2} b^{6} x^{13} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 8 \, a^{5} b^{3} x^{7} + \frac {28}{5} \, a^{6} b^{2} x^{5} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 102, normalized size = 1.01 \begin {gather*} a^{8} x + \frac {8 a^{7} b x^{3}}{3} + \frac {28 a^{6} b^{2} x^{5}}{5} + 8 a^{5} b^{3} x^{7} + \frac {70 a^{4} b^{4} x^{9}}{9} + \frac {56 a^{3} b^{5} x^{11}}{11} + \frac {28 a^{2} b^{6} x^{13}}{13} + \frac {8 a b^{7} x^{15}}{15} + \frac {b^{8} x^{17}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.23, size = 87, normalized size = 0.86 \begin {gather*} \frac {1}{17} \, b^{8} x^{17} + \frac {8}{15} \, a b^{7} x^{15} + \frac {28}{13} \, a^{2} b^{6} x^{13} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 8 \, a^{5} b^{3} x^{7} + \frac {28}{5} \, a^{6} b^{2} x^{5} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 87, normalized size = 0.86 \begin {gather*} a^8\,x+\frac {8\,a^7\,b\,x^3}{3}+\frac {28\,a^6\,b^2\,x^5}{5}+8\,a^5\,b^3\,x^7+\frac {70\,a^4\,b^4\,x^9}{9}+\frac {56\,a^3\,b^5\,x^{11}}{11}+\frac {28\,a^2\,b^6\,x^{13}}{13}+\frac {8\,a\,b^7\,x^{15}}{15}+\frac {b^8\,x^{17}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8,x)

[Out]

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

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