3.4.16 \(\int \frac {(a+b x^3)^8}{x^8} \, dx\) [316]

Optimal. Leaf size=98 \[ -\frac {a^8}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \]

[Out]

-1/7*a^8/x^7-2*a^7*b/x^4-28*a^6*b^2/x+28*a^5*b^3*x^2+14*a^4*b^4*x^5+7*a^3*b^5*x^8+28/11*a^2*b^6*x^11+4/7*a*b^7
*x^14+1/17*b^8*x^17

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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}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^8,x]

[Out]

-1/7*a^8/x^7 - (2*a^7*b)/x^4 - (28*a^6*b^2)/x + 28*a^5*b^3*x^2 + 14*a^4*b^4*x^5 + 7*a^3*b^5*x^8 + (28*a^2*b^6*
x^11)/11 + (4*a*b^7*x^14)/7 + (b^8*x^17)/17

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

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Mathematica [A]
time = 0.01, size = 98, normalized size = 1.00 \begin {gather*} -\frac {a^8}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^8,x]

[Out]

-1/7*a^8/x^7 - (2*a^7*b)/x^4 - (28*a^6*b^2)/x + 28*a^5*b^3*x^2 + 14*a^4*b^4*x^5 + 7*a^3*b^5*x^8 + (28*a^2*b^6*
x^11)/11 + (4*a*b^7*x^14)/7 + (b^8*x^17)/17

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Maple [A]
time = 0.12, size = 91, normalized size = 0.93

method	result	size
default	\(-\frac {a^{8}}{7 x^{7}}-\frac {2 a^{7} b}{x^{4}}-\frac {28 a^{6} b^{2}}{x}+28 a^{5} b^{3} x^{2}+14 a^{4} b^{4} x^{5}+7 a^{3} b^{5} x^{8}+\frac {28 a^{2} b^{6} x^{11}}{11}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {b^{8} x^{17}}{17}\)	\(91\)
norman	\(\frac {\frac {1}{17} b^{8} x^{24}+\frac {4}{7} a \,b^{7} x^{21}+14 a^{4} b^{4} x^{12}+7 a^{3} b^{5} x^{15}+\frac {28}{11} a^{2} b^{6} x^{18}+28 a^{5} b^{3} x^{9}-2 a^{7} b \,x^{3}-28 a^{6} b^{2} x^{6}-\frac {1}{7} a^{8}}{x^{7}}\)	\(92\)
gosper	\(-\frac {-77 b^{8} x^{24}-748 a \,b^{7} x^{21}-3332 a^{2} b^{6} x^{18}-9163 a^{3} b^{5} x^{15}-18326 a^{4} b^{4} x^{12}-36652 a^{5} b^{3} x^{9}+36652 a^{6} b^{2} x^{6}+2618 a^{7} b \,x^{3}+187 a^{8}}{1309 x^{7}}\)	\(93\)
risch	\(\frac {b^{8} x^{17}}{17}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {28 a^{2} b^{6} x^{11}}{11}+7 a^{3} b^{5} x^{8}+14 a^{4} b^{4} x^{5}+28 a^{5} b^{3} x^{2}+\frac {-28 a^{6} b^{2} x^{6}-2 a^{7} b \,x^{3}-\frac {1}{7} a^{8}}{x^{7}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^8/x^7-2*a^7*b/x^4-28*a^6*b^2/x+28*a^5*b^3*x^2+14*a^4*b^4*x^5+7*a^3*b^5*x^8+28/11*a^2*b^6*x^11+4/7*a*b^7
*x^14+1/17*b^8*x^17

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Maxima [A]
time = 0.30, size = 91, normalized size = 0.93 \begin {gather*} \frac {1}{17} \, b^{8} x^{17} + \frac {4}{7} \, a b^{7} x^{14} + \frac {28}{11} \, a^{2} b^{6} x^{11} + 7 \, a^{3} b^{5} x^{8} + 14 \, a^{4} b^{4} x^{5} + 28 \, a^{5} b^{3} x^{2} - \frac {196 \, a^{6} b^{2} x^{6} + 14 \, a^{7} b x^{3} + a^{8}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*b^8*x^17 + 4/7*a*b^7*x^14 + 28/11*a^2*b^6*x^11 + 7*a^3*b^5*x^8 + 14*a^4*b^4*x^5 + 28*a^5*b^3*x^2 - 1/7*(1
96*a^6*b^2*x^6 + 14*a^7*b*x^3 + a^8)/x^7

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Fricas [A]
time = 0.34, size = 92, normalized size = 0.94 \begin {gather*} \frac {77 \, b^{8} x^{24} + 748 \, a b^{7} x^{21} + 3332 \, a^{2} b^{6} x^{18} + 9163 \, a^{3} b^{5} x^{15} + 18326 \, a^{4} b^{4} x^{12} + 36652 \, a^{5} b^{3} x^{9} - 36652 \, a^{6} b^{2} x^{6} - 2618 \, a^{7} b x^{3} - 187 \, a^{8}}{1309 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1309*(77*b^8*x^24 + 748*a*b^7*x^21 + 3332*a^2*b^6*x^18 + 9163*a^3*b^5*x^15 + 18326*a^4*b^4*x^12 + 36652*a^5*
b^3*x^9 - 36652*a^6*b^2*x^6 - 2618*a^7*b*x^3 - 187*a^8)/x^7

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Sympy [A]
time = 0.12, size = 100, normalized size = 1.02 \begin {gather*} 28 a^{5} b^{3} x^{2} + 14 a^{4} b^{4} x^{5} + 7 a^{3} b^{5} x^{8} + \frac {28 a^{2} b^{6} x^{11}}{11} + \frac {4 a b^{7} x^{14}}{7} + \frac {b^{8} x^{17}}{17} + \frac {- a^{8} - 14 a^{7} b x^{3} - 196 a^{6} b^{2} x^{6}}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

28*a**5*b**3*x**2 + 14*a**4*b**4*x**5 + 7*a**3*b**5*x**8 + 28*a**2*b**6*x**11/11 + 4*a*b**7*x**14/7 + b**8*x**
17/17 + (-a**8 - 14*a**7*b*x**3 - 196*a**6*b**2*x**6)/(7*x**7)

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Giac [A]
time = 1.21, size = 91, normalized size = 0.93 \begin {gather*} \frac {1}{17} \, b^{8} x^{17} + \frac {4}{7} \, a b^{7} x^{14} + \frac {28}{11} \, a^{2} b^{6} x^{11} + 7 \, a^{3} b^{5} x^{8} + 14 \, a^{4} b^{4} x^{5} + 28 \, a^{5} b^{3} x^{2} - \frac {196 \, a^{6} b^{2} x^{6} + 14 \, a^{7} b x^{3} + a^{8}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*b^8*x^17 + 4/7*a*b^7*x^14 + 28/11*a^2*b^6*x^11 + 7*a^3*b^5*x^8 + 14*a^4*b^4*x^5 + 28*a^5*b^3*x^2 - 1/7*(1
96*a^6*b^2*x^6 + 14*a^7*b*x^3 + a^8)/x^7

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Mupad [B]
time = 0.95, size = 93, normalized size = 0.95 \begin {gather*} \frac {b^8\,x^{17}}{17}-\frac {\frac {a^8}{7}+2\,a^7\,b\,x^3+28\,a^6\,b^2\,x^6}{x^7}+\frac {4\,a\,b^7\,x^{14}}{7}+28\,a^5\,b^3\,x^2+14\,a^4\,b^4\,x^5+7\,a^3\,b^5\,x^8+\frac {28\,a^2\,b^6\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^8,x)

[Out]

(b^8*x^17)/17 - (a^8/7 + 2*a^7*b*x^3 + 28*a^6*b^2*x^6)/x^7 + (4*a*b^7*x^14)/7 + 28*a^5*b^3*x^2 + 14*a^4*b^4*x^
5 + 7*a^3*b^5*x^8 + (28*a^2*b^6*x^11)/11

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