3.4.8 \(\int x^4 (a+b x^3)^8 \, dx\) [308]

Optimal. Leaf size=103 \[ \frac {a^8 x^5}{5}+a^7 b x^8+\frac {28}{11} a^6 b^2 x^{11}+4 a^5 b^3 x^{14}+\frac {70}{17} a^4 b^4 x^{17}+\frac {14}{5} a^3 b^5 x^{20}+\frac {28}{23} a^2 b^6 x^{23}+\frac {4}{13} a b^7 x^{26}+\frac {b^8 x^{29}}{29} \]

[Out]

1/5*a^8*x^5+a^7*b*x^8+28/11*a^6*b^2*x^11+4*a^5*b^3*x^14+70/17*a^4*b^4*x^17+14/5*a^3*b^5*x^20+28/23*a^2*b^6*x^2
3+4/13*a*b^7*x^26+1/29*b^8*x^29

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Rubi [A]
time = 0.03, antiderivative size = 103, 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^5}{5}+a^7 b x^8+\frac {28}{11} a^6 b^2 x^{11}+4 a^5 b^3 x^{14}+\frac {70}{17} a^4 b^4 x^{17}+\frac {14}{5} a^3 b^5 x^{20}+\frac {28}{23} a^2 b^6 x^{23}+\frac {4}{13} a b^7 x^{26}+\frac {b^8 x^{29}}{29} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^3)^8,x]

[Out]

(a^8*x^5)/5 + a^7*b*x^8 + (28*a^6*b^2*x^11)/11 + 4*a^5*b^3*x^14 + (70*a^4*b^4*x^17)/17 + (14*a^3*b^5*x^20)/5 +
(28*a^2*b^6*x^23)/23 + (4*a*b^7*x^26)/13 + (b^8*x^29)/29

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^4 \left (a+b x^3\right )^8 \, dx &=\int \left (a^8 x^4+8 a^7 b x^7+28 a^6 b^2 x^{10}+56 a^5 b^3 x^{13}+70 a^4 b^4 x^{16}+56 a^3 b^5 x^{19}+28 a^2 b^6 x^{22}+8 a b^7 x^{25}+b^8 x^{28}\right ) \, dx\\ &=\frac {a^8 x^5}{5}+a^7 b x^8+\frac {28}{11} a^6 b^2 x^{11}+4 a^5 b^3 x^{14}+\frac {70}{17} a^4 b^4 x^{17}+\frac {14}{5} a^3 b^5 x^{20}+\frac {28}{23} a^2 b^6 x^{23}+\frac {4}{13} a b^7 x^{26}+\frac {b^8 x^{29}}{29}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 103, normalized size = 1.00 \begin {gather*} \frac {a^8 x^5}{5}+a^7 b x^8+\frac {28}{11} a^6 b^2 x^{11}+4 a^5 b^3 x^{14}+\frac {70}{17} a^4 b^4 x^{17}+\frac {14}{5} a^3 b^5 x^{20}+\frac {28}{23} a^2 b^6 x^{23}+\frac {4}{13} a b^7 x^{26}+\frac {b^8 x^{29}}{29} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^3)^8,x]

[Out]

(a^8*x^5)/5 + a^7*b*x^8 + (28*a^6*b^2*x^11)/11 + 4*a^5*b^3*x^14 + (70*a^4*b^4*x^17)/17 + (14*a^3*b^5*x^20)/5 +
(28*a^2*b^6*x^23)/23 + (4*a*b^7*x^26)/13 + (b^8*x^29)/29

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Maple [A]
time = 0.14, size = 90, normalized size = 0.87

method	result	size
gosper	\(\frac {1}{5} a^{8} x^{5}+a^{7} b \,x^{8}+\frac {28}{11} a^{6} b^{2} x^{11}+4 a^{5} b^{3} x^{14}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {28}{23} a^{2} b^{6} x^{23}+\frac {4}{13} a \,b^{7} x^{26}+\frac {1}{29} b^{8} x^{29}\)	\(90\)
default	\(\frac {1}{5} a^{8} x^{5}+a^{7} b \,x^{8}+\frac {28}{11} a^{6} b^{2} x^{11}+4 a^{5} b^{3} x^{14}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {28}{23} a^{2} b^{6} x^{23}+\frac {4}{13} a \,b^{7} x^{26}+\frac {1}{29} b^{8} x^{29}\)	\(90\)
norman	\(\frac {1}{5} a^{8} x^{5}+a^{7} b \,x^{8}+\frac {28}{11} a^{6} b^{2} x^{11}+4 a^{5} b^{3} x^{14}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {28}{23} a^{2} b^{6} x^{23}+\frac {4}{13} a \,b^{7} x^{26}+\frac {1}{29} b^{8} x^{29}\)	\(90\)
risch	\(\frac {1}{5} a^{8} x^{5}+a^{7} b \,x^{8}+\frac {28}{11} a^{6} b^{2} x^{11}+4 a^{5} b^{3} x^{14}+\frac {70}{17} a^{4} b^{4} x^{17}+\frac {14}{5} a^{3} b^{5} x^{20}+\frac {28}{23} a^{2} b^{6} x^{23}+\frac {4}{13} a \,b^{7} x^{26}+\frac {1}{29} b^{8} x^{29}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^3+a)^8,x,method=_RETURNVERBOSE)

[Out]

1/5*a^8*x^5+a^7*b*x^8+28/11*a^6*b^2*x^11+4*a^5*b^3*x^14+70/17*a^4*b^4*x^17+14/5*a^3*b^5*x^20+28/23*a^2*b^6*x^2
3+4/13*a*b^7*x^26+1/29*b^8*x^29

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Maxima [A]
time = 0.29, size = 89, normalized size = 0.86 \begin {gather*} \frac {1}{29} \, b^{8} x^{29} + \frac {4}{13} \, a b^{7} x^{26} + \frac {28}{23} \, a^{2} b^{6} x^{23} + \frac {14}{5} \, a^{3} b^{5} x^{20} + \frac {70}{17} \, a^{4} b^{4} x^{17} + 4 \, a^{5} b^{3} x^{14} + \frac {28}{11} \, a^{6} b^{2} x^{11} + a^{7} b x^{8} + \frac {1}{5} \, a^{8} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/29*b^8*x^29 + 4/13*a*b^7*x^26 + 28/23*a^2*b^6*x^23 + 14/5*a^3*b^5*x^20 + 70/17*a^4*b^4*x^17 + 4*a^5*b^3*x^14
+ 28/11*a^6*b^2*x^11 + a^7*b*x^8 + 1/5*a^8*x^5

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Fricas [A]
time = 0.33, size = 89, normalized size = 0.86 \begin {gather*} \frac {1}{29} \, b^{8} x^{29} + \frac {4}{13} \, a b^{7} x^{26} + \frac {28}{23} \, a^{2} b^{6} x^{23} + \frac {14}{5} \, a^{3} b^{5} x^{20} + \frac {70}{17} \, a^{4} b^{4} x^{17} + 4 \, a^{5} b^{3} x^{14} + \frac {28}{11} \, a^{6} b^{2} x^{11} + a^{7} b x^{8} + \frac {1}{5} \, a^{8} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/29*b^8*x^29 + 4/13*a*b^7*x^26 + 28/23*a^2*b^6*x^23 + 14/5*a^3*b^5*x^20 + 70/17*a^4*b^4*x^17 + 4*a^5*b^3*x^14
+ 28/11*a^6*b^2*x^11 + a^7*b*x^8 + 1/5*a^8*x^5

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Sympy [A]
time = 0.02, size = 102, normalized size = 0.99 \begin {gather*} \frac {a^{8} x^{5}}{5} + a^{7} b x^{8} + \frac {28 a^{6} b^{2} x^{11}}{11} + 4 a^{5} b^{3} x^{14} + \frac {70 a^{4} b^{4} x^{17}}{17} + \frac {14 a^{3} b^{5} x^{20}}{5} + \frac {28 a^{2} b^{6} x^{23}}{23} + \frac {4 a b^{7} x^{26}}{13} + \frac {b^{8} x^{29}}{29} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**3+a)**8,x)

[Out]

a**8*x**5/5 + a**7*b*x**8 + 28*a**6*b**2*x**11/11 + 4*a**5*b**3*x**14 + 70*a**4*b**4*x**17/17 + 14*a**3*b**5*x
**20/5 + 28*a**2*b**6*x**23/23 + 4*a*b**7*x**26/13 + b**8*x**29/29

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Giac [A]
time = 0.91, size = 89, normalized size = 0.86 \begin {gather*} \frac {1}{29} \, b^{8} x^{29} + \frac {4}{13} \, a b^{7} x^{26} + \frac {28}{23} \, a^{2} b^{6} x^{23} + \frac {14}{5} \, a^{3} b^{5} x^{20} + \frac {70}{17} \, a^{4} b^{4} x^{17} + 4 \, a^{5} b^{3} x^{14} + \frac {28}{11} \, a^{6} b^{2} x^{11} + a^{7} b x^{8} + \frac {1}{5} \, a^{8} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/29*b^8*x^29 + 4/13*a*b^7*x^26 + 28/23*a^2*b^6*x^23 + 14/5*a^3*b^5*x^20 + 70/17*a^4*b^4*x^17 + 4*a^5*b^3*x^14
+ 28/11*a^6*b^2*x^11 + a^7*b*x^8 + 1/5*a^8*x^5

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Mupad [B]
time = 1.01, size = 89, normalized size = 0.86 \begin {gather*} \frac {a^8\,x^5}{5}+a^7\,b\,x^8+\frac {28\,a^6\,b^2\,x^{11}}{11}+4\,a^5\,b^3\,x^{14}+\frac {70\,a^4\,b^4\,x^{17}}{17}+\frac {14\,a^3\,b^5\,x^{20}}{5}+\frac {28\,a^2\,b^6\,x^{23}}{23}+\frac {4\,a\,b^7\,x^{26}}{13}+\frac {b^8\,x^{29}}{29} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^3)^8,x)

[Out]

(a^8*x^5)/5 + (b^8*x^29)/29 + a^7*b*x^8 + (4*a*b^7*x^26)/13 + (28*a^6*b^2*x^11)/11 + 4*a^5*b^3*x^14 + (70*a^4*
b^4*x^17)/17 + (14*a^3*b^5*x^20)/5 + (28*a^2*b^6*x^23)/23

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