3.4.15 \(\int \frac {(a+b x^3)^8}{x^6} \, dx\) [315]

Optimal. Leaf size=98 \[ -\frac {a^8}{5 x^5}-\frac {4 a^7 b}{x^2}+28 a^6 b^2 x+14 a^5 b^3 x^4+10 a^4 b^4 x^7+\frac {28}{5} a^3 b^5 x^{10}+\frac {28}{13} a^2 b^6 x^{13}+\frac {1}{2} a b^7 x^{16}+\frac {b^8 x^{19}}{19} \]

[Out]

-1/5*a^8/x^5-4*a^7*b/x^2+28*a^6*b^2*x+14*a^5*b^3*x^4+10*a^4*b^4*x^7+28/5*a^3*b^5*x^10+28/13*a^2*b^6*x^13+1/2*a
*b^7*x^16+1/19*b^8*x^19

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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}{5 x^5}-\frac {4 a^7 b}{x^2}+28 a^6 b^2 x+14 a^5 b^3 x^4+10 a^4 b^4 x^7+\frac {28}{5} a^3 b^5 x^{10}+\frac {28}{13} a^2 b^6 x^{13}+\frac {1}{2} a b^7 x^{16}+\frac {b^8 x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^6,x]

[Out]

-1/5*a^8/x^5 - (4*a^7*b)/x^2 + 28*a^6*b^2*x + 14*a^5*b^3*x^4 + 10*a^4*b^4*x^7 + (28*a^3*b^5*x^10)/5 + (28*a^2*
b^6*x^13)/13 + (a*b^7*x^16)/2 + (b^8*x^19)/19

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^6} \, dx &=\int \left (28 a^6 b^2+\frac {a^8}{x^6}+\frac {8 a^7 b}{x^3}+56 a^5 b^3 x^3+70 a^4 b^4 x^6+56 a^3 b^5 x^9+28 a^2 b^6 x^{12}+8 a b^7 x^{15}+b^8 x^{18}\right ) \, dx\\ &=-\frac {a^8}{5 x^5}-\frac {4 a^7 b}{x^2}+28 a^6 b^2 x+14 a^5 b^3 x^4+10 a^4 b^4 x^7+\frac {28}{5} a^3 b^5 x^{10}+\frac {28}{13} a^2 b^6 x^{13}+\frac {1}{2} a b^7 x^{16}+\frac {b^8 x^{19}}{19}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 98, normalized size = 1.00 \begin {gather*} -\frac {a^8}{5 x^5}-\frac {4 a^7 b}{x^2}+28 a^6 b^2 x+14 a^5 b^3 x^4+10 a^4 b^4 x^7+\frac {28}{5} a^3 b^5 x^{10}+\frac {28}{13} a^2 b^6 x^{13}+\frac {1}{2} a b^7 x^{16}+\frac {b^8 x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^6,x]

[Out]

-1/5*a^8/x^5 - (4*a^7*b)/x^2 + 28*a^6*b^2*x + 14*a^5*b^3*x^4 + 10*a^4*b^4*x^7 + (28*a^3*b^5*x^10)/5 + (28*a^2*
b^6*x^13)/13 + (a*b^7*x^16)/2 + (b^8*x^19)/19

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Maple [A]
time = 0.12, size = 89, normalized size = 0.91

method	result	size
default	\(-\frac {a^{8}}{5 x^{5}}-\frac {4 a^{7} b}{x^{2}}+28 a^{6} b^{2} x +14 a^{5} b^{3} x^{4}+10 a^{4} b^{4} x^{7}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {28 a^{2} b^{6} x^{13}}{13}+\frac {a \,b^{7} x^{16}}{2}+\frac {b^{8} x^{19}}{19}\)	\(89\)
risch	\(\frac {b^{8} x^{19}}{19}+\frac {a \,b^{7} x^{16}}{2}+\frac {28 a^{2} b^{6} x^{13}}{13}+\frac {28 a^{3} b^{5} x^{10}}{5}+10 a^{4} b^{4} x^{7}+14 a^{5} b^{3} x^{4}+28 a^{6} b^{2} x +\frac {-4 a^{7} b \,x^{3}-\frac {1}{5} a^{8}}{x^{5}}\)	\(91\)
norman	\(\frac {\frac {28}{13} a^{2} b^{6} x^{18}+\frac {1}{2} a \,b^{7} x^{21}+\frac {1}{19} b^{8} x^{24}+10 a^{4} b^{4} x^{12}+\frac {28}{5} a^{3} b^{5} x^{15}+14 a^{5} b^{3} x^{9}-4 a^{7} b \,x^{3}+28 a^{6} b^{2} x^{6}-\frac {1}{5} a^{8}}{x^{5}}\)	\(92\)
gosper	\(-\frac {-130 b^{8} x^{24}-1235 a \,b^{7} x^{21}-5320 a^{2} b^{6} x^{18}-13832 a^{3} b^{5} x^{15}-24700 a^{4} b^{4} x^{12}-34580 a^{5} b^{3} x^{9}-69160 a^{6} b^{2} x^{6}+9880 a^{7} b \,x^{3}+494 a^{8}}{2470 x^{5}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*a^8/x^5-4*a^7*b/x^2+28*a^6*b^2*x+14*a^5*b^3*x^4+10*a^4*b^4*x^7+28/5*a^3*b^5*x^10+28/13*a^2*b^6*x^13+1/2*a
*b^7*x^16+1/19*b^8*x^19

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Maxima [A]
time = 0.29, size = 89, normalized size = 0.91 \begin {gather*} \frac {1}{19} \, b^{8} x^{19} + \frac {1}{2} \, a b^{7} x^{16} + \frac {28}{13} \, a^{2} b^{6} x^{13} + \frac {28}{5} \, a^{3} b^{5} x^{10} + 10 \, a^{4} b^{4} x^{7} + 14 \, a^{5} b^{3} x^{4} + 28 \, a^{6} b^{2} x - \frac {20 \, a^{7} b x^{3} + a^{8}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^6,x, algorithm="maxima")

[Out]

1/19*b^8*x^19 + 1/2*a*b^7*x^16 + 28/13*a^2*b^6*x^13 + 28/5*a^3*b^5*x^10 + 10*a^4*b^4*x^7 + 14*a^5*b^3*x^4 + 28
*a^6*b^2*x - 1/5*(20*a^7*b*x^3 + a^8)/x^5

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Fricas [A]
time = 0.34, size = 92, normalized size = 0.94 \begin {gather*} \frac {130 \, b^{8} x^{24} + 1235 \, a b^{7} x^{21} + 5320 \, a^{2} b^{6} x^{18} + 13832 \, a^{3} b^{5} x^{15} + 24700 \, a^{4} b^{4} x^{12} + 34580 \, a^{5} b^{3} x^{9} + 69160 \, a^{6} b^{2} x^{6} - 9880 \, a^{7} b x^{3} - 494 \, a^{8}}{2470 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^6,x, algorithm="fricas")

[Out]

1/2470*(130*b^8*x^24 + 1235*a*b^7*x^21 + 5320*a^2*b^6*x^18 + 13832*a^3*b^5*x^15 + 24700*a^4*b^4*x^12 + 34580*a
^5*b^3*x^9 + 69160*a^6*b^2*x^6 - 9880*a^7*b*x^3 - 494*a^8)/x^5

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Sympy [A]
time = 0.08, size = 99, normalized size = 1.01 \begin {gather*} 28 a^{6} b^{2} x + 14 a^{5} b^{3} x^{4} + 10 a^{4} b^{4} x^{7} + \frac {28 a^{3} b^{5} x^{10}}{5} + \frac {28 a^{2} b^{6} x^{13}}{13} + \frac {a b^{7} x^{16}}{2} + \frac {b^{8} x^{19}}{19} + \frac {- a^{8} - 20 a^{7} b x^{3}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**6,x)

[Out]

28*a**6*b**2*x + 14*a**5*b**3*x**4 + 10*a**4*b**4*x**7 + 28*a**3*b**5*x**10/5 + 28*a**2*b**6*x**13/13 + a*b**7
*x**16/2 + b**8*x**19/19 + (-a**8 - 20*a**7*b*x**3)/(5*x**5)

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Giac [A]
time = 1.80, size = 89, normalized size = 0.91 \begin {gather*} \frac {1}{19} \, b^{8} x^{19} + \frac {1}{2} \, a b^{7} x^{16} + \frac {28}{13} \, a^{2} b^{6} x^{13} + \frac {28}{5} \, a^{3} b^{5} x^{10} + 10 \, a^{4} b^{4} x^{7} + 14 \, a^{5} b^{3} x^{4} + 28 \, a^{6} b^{2} x - \frac {20 \, a^{7} b x^{3} + a^{8}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^6,x, algorithm="giac")

[Out]

1/19*b^8*x^19 + 1/2*a*b^7*x^16 + 28/13*a^2*b^6*x^13 + 28/5*a^3*b^5*x^10 + 10*a^4*b^4*x^7 + 14*a^5*b^3*x^4 + 28
*a^6*b^2*x - 1/5*(20*a^7*b*x^3 + a^8)/x^5

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Mupad [B]
time = 0.05, size = 91, normalized size = 0.93 \begin {gather*} \frac {b^8\,x^{19}}{19}-\frac {\frac {a^8}{5}+4\,b\,a^7\,x^3}{x^5}+28\,a^6\,b^2\,x+\frac {a\,b^7\,x^{16}}{2}+14\,a^5\,b^3\,x^4+10\,a^4\,b^4\,x^7+\frac {28\,a^3\,b^5\,x^{10}}{5}+\frac {28\,a^2\,b^6\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^6,x)

[Out]

(b^8*x^19)/19 - (a^8/5 + 4*a^7*b*x^3)/x^5 + 28*a^6*b^2*x + (a*b^7*x^16)/2 + 14*a^5*b^3*x^4 + 10*a^4*b^4*x^7 +
(28*a^3*b^5*x^10)/5 + (28*a^2*b^6*x^13)/13

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