3.3.78 \(\int (a+b x^3)^5 \, dx\) [278]

Optimal. Leaf size=61 \[ a^5 x+\frac {5}{4} a^4 b x^4+\frac {10}{7} a^3 b^2 x^7+a^2 b^3 x^{10}+\frac {5}{13} a b^4 x^{13}+\frac {b^5 x^{16}}{16} \]

[Out]

a^5*x+5/4*a^4*b*x^4+10/7*a^3*b^2*x^7+a^2*b^3*x^10+5/13*a*b^4*x^13+1/16*b^5*x^16

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Rubi [A]
time = 0.01, antiderivative size = 61, 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^5 x+\frac {5}{4} a^4 b x^4+\frac {10}{7} a^3 b^2 x^7+a^2 b^3 x^{10}+\frac {5}{13} a b^4 x^{13}+\frac {b^5 x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5,x]

[Out]

a^5*x + (5*a^4*b*x^4)/4 + (10*a^3*b^2*x^7)/7 + a^2*b^3*x^10 + (5*a*b^4*x^13)/13 + (b^5*x^16)/16

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

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Mathematica [A]
time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} a^5 x+\frac {5}{4} a^4 b x^4+\frac {10}{7} a^3 b^2 x^7+a^2 b^3 x^{10}+\frac {5}{13} a b^4 x^{13}+\frac {b^5 x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5,x]

[Out]

a^5*x + (5*a^4*b*x^4)/4 + (10*a^3*b^2*x^7)/7 + a^2*b^3*x^10 + (5*a*b^4*x^13)/13 + (b^5*x^16)/16

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Maple [A]
time = 0.11, size = 54, normalized size = 0.89

method	result	size
gosper	\(a^{5} x +\frac {5}{4} a^{4} b \,x^{4}+\frac {10}{7} a^{3} b^{2} x^{7}+a^{2} b^{3} x^{10}+\frac {5}{13} a \,b^{4} x^{13}+\frac {1}{16} b^{5} x^{16}\)	\(54\)
default	\(a^{5} x +\frac {5}{4} a^{4} b \,x^{4}+\frac {10}{7} a^{3} b^{2} x^{7}+a^{2} b^{3} x^{10}+\frac {5}{13} a \,b^{4} x^{13}+\frac {1}{16} b^{5} x^{16}\)	\(54\)
norman	\(a^{5} x +\frac {5}{4} a^{4} b \,x^{4}+\frac {10}{7} a^{3} b^{2} x^{7}+a^{2} b^{3} x^{10}+\frac {5}{13} a \,b^{4} x^{13}+\frac {1}{16} b^{5} x^{16}\)	\(54\)
risch	\(a^{5} x +\frac {5}{4} a^{4} b \,x^{4}+\frac {10}{7} a^{3} b^{2} x^{7}+a^{2} b^{3} x^{10}+\frac {5}{13} a \,b^{4} x^{13}+\frac {1}{16} b^{5} x^{16}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5,x,method=_RETURNVERBOSE)

[Out]

a^5*x+5/4*a^4*b*x^4+10/7*a^3*b^2*x^7+a^2*b^3*x^10+5/13*a*b^4*x^13+1/16*b^5*x^16

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Maxima [A]
time = 0.29, size = 53, normalized size = 0.87 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{4} \, a^{4} b x^{4} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/16*b^5*x^16 + 5/13*a*b^4*x^13 + a^2*b^3*x^10 + 10/7*a^3*b^2*x^7 + 5/4*a^4*b*x^4 + a^5*x

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Fricas [A]
time = 0.33, size = 53, normalized size = 0.87 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{4} \, a^{4} b x^{4} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/16*b^5*x^16 + 5/13*a*b^4*x^13 + a^2*b^3*x^10 + 10/7*a^3*b^2*x^7 + 5/4*a^4*b*x^4 + a^5*x

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Sympy [A]
time = 0.01, size = 60, normalized size = 0.98 \begin {gather*} a^{5} x + \frac {5 a^{4} b x^{4}}{4} + \frac {10 a^{3} b^{2} x^{7}}{7} + a^{2} b^{3} x^{10} + \frac {5 a b^{4} x^{13}}{13} + \frac {b^{5} x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5,x)

[Out]

a**5*x + 5*a**4*b*x**4/4 + 10*a**3*b**2*x**7/7 + a**2*b**3*x**10 + 5*a*b**4*x**13/13 + b**5*x**16/16

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Giac [A]
time = 1.14, size = 53, normalized size = 0.87 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{4} \, a^{4} b x^{4} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5,x, algorithm="giac")

[Out]

1/16*b^5*x^16 + 5/13*a*b^4*x^13 + a^2*b^3*x^10 + 10/7*a^3*b^2*x^7 + 5/4*a^4*b*x^4 + a^5*x

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Mupad [B]
time = 0.02, size = 53, normalized size = 0.87 \begin {gather*} a^5\,x+\frac {5\,a^4\,b\,x^4}{4}+\frac {10\,a^3\,b^2\,x^7}{7}+a^2\,b^3\,x^{10}+\frac {5\,a\,b^4\,x^{13}}{13}+\frac {b^5\,x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^5,x)

[Out]

a^5*x + (b^5*x^16)/16 + (5*a^4*b*x^4)/4 + (5*a*b^4*x^13)/13 + (10*a^3*b^2*x^7)/7 + a^2*b^3*x^10

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