3.4.12 \(\int \frac {(a+b x^3)^8}{x^2} \, dx\) [312]

Optimal. Leaf size=100 \[ -\frac {a^8}{x}+4 a^7 b x^2+\frac {28}{5} a^6 b^2 x^5+7 a^5 b^3 x^8+\frac {70}{11} a^4 b^4 x^{11}+4 a^3 b^5 x^{14}+\frac {28}{17} a^2 b^6 x^{17}+\frac {2}{5} a b^7 x^{20}+\frac {b^8 x^{23}}{23} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 100, 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}+4 a^7 b x^2+\frac {28}{5} a^6 b^2 x^5+7 a^5 b^3 x^8+\frac {70}{11} a^4 b^4 x^{11}+4 a^3 b^5 x^{14}+\frac {28}{17} a^2 b^6 x^{17}+\frac {2}{5} a b^7 x^{20}+\frac {b^8 x^{23}}{23} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 100, normalized size = 1.00 \begin {gather*} -\frac {a^8}{x}+4 a^7 b x^2+\frac {28}{5} a^6 b^2 x^5+7 a^5 b^3 x^8+\frac {70}{11} a^4 b^4 x^{11}+4 a^3 b^5 x^{14}+\frac {28}{17} a^2 b^6 x^{17}+\frac {2}{5} a b^7 x^{20}+\frac {b^8 x^{23}}{23} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^2,x]

[Out]

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

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Maple [A]
time = 0.20, size = 91, normalized size = 0.91

method	result	size
default	\(-\frac {a^{8}}{x}+4 a^{7} b \,x^{2}+\frac {28 a^{6} b^{2} x^{5}}{5}+7 a^{5} b^{3} x^{8}+\frac {70 a^{4} b^{4} x^{11}}{11}+4 a^{3} b^{5} x^{14}+\frac {28 a^{2} b^{6} x^{17}}{17}+\frac {2 a \,b^{7} x^{20}}{5}+\frac {b^{8} x^{23}}{23}\)	\(91\)
risch	\(-\frac {a^{8}}{x}+4 a^{7} b \,x^{2}+\frac {28 a^{6} b^{2} x^{5}}{5}+7 a^{5} b^{3} x^{8}+\frac {70 a^{4} b^{4} x^{11}}{11}+4 a^{3} b^{5} x^{14}+\frac {28 a^{2} b^{6} x^{17}}{17}+\frac {2 a \,b^{7} x^{20}}{5}+\frac {b^{8} x^{23}}{23}\)	\(91\)
norman	\(\frac {\frac {70}{11} a^{4} b^{4} x^{12}+4 a^{3} b^{5} x^{15}+\frac {28}{17} a^{2} b^{6} x^{18}+\frac {2}{5} a \,b^{7} x^{21}+\frac {1}{23} b^{8} x^{24}+7 a^{5} b^{3} x^{9}+4 a^{7} b \,x^{3}+\frac {28}{5} a^{6} b^{2} x^{6}-a^{8}}{x}\)	\(92\)
gosper	\(-\frac {-935 b^{8} x^{24}-8602 a \,b^{7} x^{21}-35420 a^{2} b^{6} x^{18}-86020 a^{3} b^{5} x^{15}-136850 a^{4} b^{4} x^{12}-150535 a^{5} b^{3} x^{9}-120428 a^{6} b^{2} x^{6}-86020 a^{7} b \,x^{3}+21505 a^{8}}{21505 x}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 90, normalized size = 0.90 \begin {gather*} \frac {1}{23} \, b^{8} x^{23} + \frac {2}{5} \, a b^{7} x^{20} + \frac {28}{17} \, a^{2} b^{6} x^{17} + 4 \, a^{3} b^{5} x^{14} + \frac {70}{11} \, a^{4} b^{4} x^{11} + 7 \, a^{5} b^{3} x^{8} + \frac {28}{5} \, a^{6} b^{2} x^{5} + 4 \, a^{7} b x^{2} - \frac {a^{8}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 92, normalized size = 0.92 \begin {gather*} \frac {935 \, b^{8} x^{24} + 8602 \, a b^{7} x^{21} + 35420 \, a^{2} b^{6} x^{18} + 86020 \, a^{3} b^{5} x^{15} + 136850 \, a^{4} b^{4} x^{12} + 150535 \, a^{5} b^{3} x^{9} + 120428 \, a^{6} b^{2} x^{6} + 86020 \, a^{7} b x^{3} - 21505 \, a^{8}}{21505 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^2,x, algorithm="fricas")

[Out]

1/21505*(935*b^8*x^24 + 8602*a*b^7*x^21 + 35420*a^2*b^6*x^18 + 86020*a^3*b^5*x^15 + 136850*a^4*b^4*x^12 + 1505
35*a^5*b^3*x^9 + 120428*a^6*b^2*x^6 + 86020*a^7*b*x^3 - 21505*a^8)/x

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Sympy [A]
time = 0.05, size = 99, normalized size = 0.99 \begin {gather*} - \frac {a^{8}}{x} + 4 a^{7} b x^{2} + \frac {28 a^{6} b^{2} x^{5}}{5} + 7 a^{5} b^{3} x^{8} + \frac {70 a^{4} b^{4} x^{11}}{11} + 4 a^{3} b^{5} x^{14} + \frac {28 a^{2} b^{6} x^{17}}{17} + \frac {2 a b^{7} x^{20}}{5} + \frac {b^{8} x^{23}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**2,x)

[Out]

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

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Giac [A]
time = 1.42, size = 90, normalized size = 0.90 \begin {gather*} \frac {1}{23} \, b^{8} x^{23} + \frac {2}{5} \, a b^{7} x^{20} + \frac {28}{17} \, a^{2} b^{6} x^{17} + 4 \, a^{3} b^{5} x^{14} + \frac {70}{11} \, a^{4} b^{4} x^{11} + 7 \, a^{5} b^{3} x^{8} + \frac {28}{5} \, a^{6} b^{2} x^{5} + 4 \, a^{7} b x^{2} - \frac {a^{8}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.05, size = 90, normalized size = 0.90 \begin {gather*} \frac {b^8\,x^{23}}{23}-\frac {a^8}{x}+4\,a^7\,b\,x^2+\frac {2\,a\,b^7\,x^{20}}{5}+\frac {28\,a^6\,b^2\,x^5}{5}+7\,a^5\,b^3\,x^8+\frac {70\,a^4\,b^4\,x^{11}}{11}+4\,a^3\,b^5\,x^{14}+\frac {28\,a^2\,b^6\,x^{17}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^2,x)

[Out]

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

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