3.4.11 \(\int (a+b x^3)^8 \, dx\) [311]

Optimal. Leaf size=99 \[ a^8 x+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \]

[Out]

a^8*x+2*a^7*b*x^4+4*a^6*b^2*x^7+28/5*a^5*b^3*x^10+70/13*a^4*b^4*x^13+7/2*a^3*b^5*x^16+28/19*a^2*b^6*x^19+4/11*
a*b^7*x^22+1/25*b^8*x^25

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Rubi [A]
time = 0.03, antiderivative size = 99, 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+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8,x]

[Out]

a^8*x + 2*a^7*b*x^4 + 4*a^6*b^2*x^7 + (28*a^5*b^3*x^10)/5 + (70*a^4*b^4*x^13)/13 + (7*a^3*b^5*x^16)/2 + (28*a^
2*b^6*x^19)/19 + (4*a*b^7*x^22)/11 + (b^8*x^25)/25

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

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Mathematica [A]
time = 0.00, size = 99, normalized size = 1.00 \begin {gather*} a^8 x+2 a^7 b x^4+4 a^6 b^2 x^7+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{13} a^4 b^4 x^{13}+\frac {7}{2} a^3 b^5 x^{16}+\frac {28}{19} a^2 b^6 x^{19}+\frac {4}{11} a b^7 x^{22}+\frac {b^8 x^{25}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8,x]

[Out]

a^8*x + 2*a^7*b*x^4 + 4*a^6*b^2*x^7 + (28*a^5*b^3*x^10)/5 + (70*a^4*b^4*x^13)/13 + (7*a^3*b^5*x^16)/2 + (28*a^
2*b^6*x^19)/19 + (4*a*b^7*x^22)/11 + (b^8*x^25)/25

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Maple [A]
time = 0.13, size = 88, normalized size = 0.89

method	result	size
gosper	\(a^{8} x +2 a^{7} b \,x^{4}+4 a^{6} b^{2} x^{7}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\)	\(88\)
default	\(a^{8} x +2 a^{7} b \,x^{4}+4 a^{6} b^{2} x^{7}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\)	\(88\)
norman	\(a^{8} x +2 a^{7} b \,x^{4}+4 a^{6} b^{2} x^{7}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\)	\(88\)
risch	\(a^{8} x +2 a^{7} b \,x^{4}+4 a^{6} b^{2} x^{7}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {7}{2} a^{3} b^{5} x^{16}+\frac {28}{19} a^{2} b^{6} x^{19}+\frac {4}{11} a \,b^{7} x^{22}+\frac {1}{25} b^{8} x^{25}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8,x,method=_RETURNVERBOSE)

[Out]

a^8*x+2*a^7*b*x^4+4*a^6*b^2*x^7+28/5*a^5*b^3*x^10+70/13*a^4*b^4*x^13+7/2*a^3*b^5*x^16+28/19*a^2*b^6*x^19+4/11*
a*b^7*x^22+1/25*b^8*x^25

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Maxima [A]
time = 0.29, size = 87, normalized size = 0.88 \begin {gather*} \frac {1}{25} \, b^{8} x^{25} + \frac {4}{11} \, a b^{7} x^{22} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {7}{2} \, a^{3} b^{5} x^{16} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/25*b^8*x^25 + 4/11*a*b^7*x^22 + 28/19*a^2*b^6*x^19 + 7/2*a^3*b^5*x^16 + 70/13*a^4*b^4*x^13 + 28/5*a^5*b^3*x^
10 + 4*a^6*b^2*x^7 + 2*a^7*b*x^4 + a^8*x

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Fricas [A]
time = 0.33, size = 87, normalized size = 0.88 \begin {gather*} \frac {1}{25} \, b^{8} x^{25} + \frac {4}{11} \, a b^{7} x^{22} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {7}{2} \, a^{3} b^{5} x^{16} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/25*b^8*x^25 + 4/11*a*b^7*x^22 + 28/19*a^2*b^6*x^19 + 7/2*a^3*b^5*x^16 + 70/13*a^4*b^4*x^13 + 28/5*a^5*b^3*x^
10 + 4*a^6*b^2*x^7 + 2*a^7*b*x^4 + a^8*x

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Sympy [A]
time = 0.01, size = 100, normalized size = 1.01 \begin {gather*} a^{8} x + 2 a^{7} b x^{4} + 4 a^{6} b^{2} x^{7} + \frac {28 a^{5} b^{3} x^{10}}{5} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {7 a^{3} b^{5} x^{16}}{2} + \frac {28 a^{2} b^{6} x^{19}}{19} + \frac {4 a b^{7} x^{22}}{11} + \frac {b^{8} x^{25}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8,x)

[Out]

a**8*x + 2*a**7*b*x**4 + 4*a**6*b**2*x**7 + 28*a**5*b**3*x**10/5 + 70*a**4*b**4*x**13/13 + 7*a**3*b**5*x**16/2
+ 28*a**2*b**6*x**19/19 + 4*a*b**7*x**22/11 + b**8*x**25/25

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Giac [A]
time = 0.98, size = 87, normalized size = 0.88 \begin {gather*} \frac {1}{25} \, b^{8} x^{25} + \frac {4}{11} \, a b^{7} x^{22} + \frac {28}{19} \, a^{2} b^{6} x^{19} + \frac {7}{2} \, a^{3} b^{5} x^{16} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8,x, algorithm="giac")

[Out]

1/25*b^8*x^25 + 4/11*a*b^7*x^22 + 28/19*a^2*b^6*x^19 + 7/2*a^3*b^5*x^16 + 70/13*a^4*b^4*x^13 + 28/5*a^5*b^3*x^
10 + 4*a^6*b^2*x^7 + 2*a^7*b*x^4 + a^8*x

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Mupad [B]
time = 0.04, size = 87, normalized size = 0.88 \begin {gather*} a^8\,x+2\,a^7\,b\,x^4+4\,a^6\,b^2\,x^7+\frac {28\,a^5\,b^3\,x^{10}}{5}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {7\,a^3\,b^5\,x^{16}}{2}+\frac {28\,a^2\,b^6\,x^{19}}{19}+\frac {4\,a\,b^7\,x^{22}}{11}+\frac {b^8\,x^{25}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8,x)

[Out]

a^8*x + (b^8*x^25)/25 + 2*a^7*b*x^4 + (4*a*b^7*x^22)/11 + 4*a^6*b^2*x^7 + (28*a^5*b^3*x^10)/5 + (70*a^4*b^4*x^
13)/13 + (7*a^3*b^5*x^16)/2 + (28*a^2*b^6*x^19)/19

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