3.4.13 \(\int \frac {(a+b x^3)^8}{x^3} \, dx\) [313]

Optimal. Leaf size=98 \[ -\frac {a^8}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \]

[Out]

-1/2*a^8/x^2+8*a^7*b*x+7*a^6*b^2*x^4+8*a^5*b^3*x^7+7*a^4*b^4*x^10+56/13*a^3*b^5*x^13+7/4*a^2*b^6*x^16+8/19*a*b
^7*x^19+1/22*b^8*x^22

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Rubi [A]
time = 0.03, antiderivative size = 98, 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}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^3,x]

[Out]

-1/2*a^8/x^2 + 8*a^7*b*x + 7*a^6*b^2*x^4 + 8*a^5*b^3*x^7 + 7*a^4*b^4*x^10 + (56*a^3*b^5*x^13)/13 + (7*a^2*b^6*
x^16)/4 + (8*a*b^7*x^19)/19 + (b^8*x^22)/22

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

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Mathematica [A]
time = 0.00, size = 98, normalized size = 1.00 \begin {gather*} -\frac {a^8}{2 x^2}+8 a^7 b x+7 a^6 b^2 x^4+8 a^5 b^3 x^7+7 a^4 b^4 x^{10}+\frac {56}{13} a^3 b^5 x^{13}+\frac {7}{4} a^2 b^6 x^{16}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{22}}{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^3,x]

[Out]

-1/2*a^8/x^2 + 8*a^7*b*x + 7*a^6*b^2*x^4 + 8*a^5*b^3*x^7 + 7*a^4*b^4*x^10 + (56*a^3*b^5*x^13)/13 + (7*a^2*b^6*
x^16)/4 + (8*a*b^7*x^19)/19 + (b^8*x^22)/22

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Maple [A]
time = 0.19, size = 89, normalized size = 0.91

method	result	size
default	\(-\frac {a^{8}}{2 x^{2}}+8 a^{7} b x +7 a^{6} b^{2} x^{4}+8 a^{5} b^{3} x^{7}+7 a^{4} b^{4} x^{10}+\frac {56 a^{3} b^{5} x^{13}}{13}+\frac {7 a^{2} b^{6} x^{16}}{4}+\frac {8 a \,b^{7} x^{19}}{19}+\frac {b^{8} x^{22}}{22}\)	\(89\)
risch	\(-\frac {a^{8}}{2 x^{2}}+8 a^{7} b x +7 a^{6} b^{2} x^{4}+8 a^{5} b^{3} x^{7}+7 a^{4} b^{4} x^{10}+\frac {56 a^{3} b^{5} x^{13}}{13}+\frac {7 a^{2} b^{6} x^{16}}{4}+\frac {8 a \,b^{7} x^{19}}{19}+\frac {b^{8} x^{22}}{22}\)	\(89\)
norman	\(\frac {\frac {1}{22} b^{8} x^{24}+\frac {8}{19} a \,b^{7} x^{21}+\frac {56}{13} a^{3} b^{5} x^{15}+\frac {7}{4} a^{2} b^{6} x^{18}+8 a^{5} b^{3} x^{9}+7 a^{4} b^{4} x^{12}+8 a^{7} b \,x^{3}+7 a^{6} b^{2} x^{6}-\frac {1}{2} a^{8}}{x^{2}}\)	\(92\)
gosper	\(-\frac {-494 b^{8} x^{24}-4576 a \,b^{7} x^{21}-19019 a^{2} b^{6} x^{18}-46816 a^{3} b^{5} x^{15}-76076 a^{4} b^{4} x^{12}-86944 a^{5} b^{3} x^{9}-76076 a^{6} b^{2} x^{6}-86944 a^{7} b \,x^{3}+5434 a^{8}}{10868 x^{2}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^8/x^2+8*a^7*b*x+7*a^6*b^2*x^4+8*a^5*b^3*x^7+7*a^4*b^4*x^10+56/13*a^3*b^5*x^13+7/4*a^2*b^6*x^16+8/19*a*b
^7*x^19+1/22*b^8*x^22

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Maxima [A]
time = 0.30, size = 88, normalized size = 0.90 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {8}{19} \, a b^{7} x^{19} + \frac {7}{4} \, a^{2} b^{6} x^{16} + \frac {56}{13} \, a^{3} b^{5} x^{13} + 7 \, a^{4} b^{4} x^{10} + 8 \, a^{5} b^{3} x^{7} + 7 \, a^{6} b^{2} x^{4} + 8 \, a^{7} b x - \frac {a^{8}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*b^8*x^22 + 8/19*a*b^7*x^19 + 7/4*a^2*b^6*x^16 + 56/13*a^3*b^5*x^13 + 7*a^4*b^4*x^10 + 8*a^5*b^3*x^7 + 7*a
^6*b^2*x^4 + 8*a^7*b*x - 1/2*a^8/x^2

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Fricas [A]
time = 0.34, size = 92, normalized size = 0.94 \begin {gather*} \frac {494 \, b^{8} x^{24} + 4576 \, a b^{7} x^{21} + 19019 \, a^{2} b^{6} x^{18} + 46816 \, a^{3} b^{5} x^{15} + 76076 \, a^{4} b^{4} x^{12} + 86944 \, a^{5} b^{3} x^{9} + 76076 \, a^{6} b^{2} x^{6} + 86944 \, a^{7} b x^{3} - 5434 \, a^{8}}{10868 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10868*(494*b^8*x^24 + 4576*a*b^7*x^21 + 19019*a^2*b^6*x^18 + 46816*a^3*b^5*x^15 + 76076*a^4*b^4*x^12 + 86944
*a^5*b^3*x^9 + 76076*a^6*b^2*x^6 + 86944*a^7*b*x^3 - 5434*a^8)/x^2

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Sympy [A]
time = 0.05, size = 99, normalized size = 1.01 \begin {gather*} - \frac {a^{8}}{2 x^{2}} + 8 a^{7} b x + 7 a^{6} b^{2} x^{4} + 8 a^{5} b^{3} x^{7} + 7 a^{4} b^{4} x^{10} + \frac {56 a^{3} b^{5} x^{13}}{13} + \frac {7 a^{2} b^{6} x^{16}}{4} + \frac {8 a b^{7} x^{19}}{19} + \frac {b^{8} x^{22}}{22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**8/(2*x**2) + 8*a**7*b*x + 7*a**6*b**2*x**4 + 8*a**5*b**3*x**7 + 7*a**4*b**4*x**10 + 56*a**3*b**5*x**13/13
+ 7*a**2*b**6*x**16/4 + 8*a*b**7*x**19/19 + b**8*x**22/22

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Giac [A]
time = 1.33, size = 88, normalized size = 0.90 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {8}{19} \, a b^{7} x^{19} + \frac {7}{4} \, a^{2} b^{6} x^{16} + \frac {56}{13} \, a^{3} b^{5} x^{13} + 7 \, a^{4} b^{4} x^{10} + 8 \, a^{5} b^{3} x^{7} + 7 \, a^{6} b^{2} x^{4} + 8 \, a^{7} b x - \frac {a^{8}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*b^8*x^22 + 8/19*a*b^7*x^19 + 7/4*a^2*b^6*x^16 + 56/13*a^3*b^5*x^13 + 7*a^4*b^4*x^10 + 8*a^5*b^3*x^7 + 7*a
^6*b^2*x^4 + 8*a^7*b*x - 1/2*a^8/x^2

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Mupad [B]
time = 0.05, size = 88, normalized size = 0.90 \begin {gather*} \frac {b^8\,x^{22}}{22}-\frac {a^8}{2\,x^2}+\frac {8\,a\,b^7\,x^{19}}{19}+7\,a^6\,b^2\,x^4+8\,a^5\,b^3\,x^7+7\,a^4\,b^4\,x^{10}+\frac {56\,a^3\,b^5\,x^{13}}{13}+\frac {7\,a^2\,b^6\,x^{16}}{4}+8\,a^7\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^3,x)

[Out]

(b^8*x^22)/22 - a^8/(2*x^2) + (8*a*b^7*x^19)/19 + 7*a^6*b^2*x^4 + 8*a^5*b^3*x^7 + 7*a^4*b^4*x^10 + (56*a^3*b^5
*x^13)/13 + (7*a^2*b^6*x^16)/4 + 8*a^7*b*x

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