3.4.14 \(\int \frac {(a+b x^3)^8}{x^5} \, dx\) [314]

Optimal. Leaf size=102 \[ -\frac {a^8}{4 x^4}-\frac {8 a^7 b}{x}+14 a^6 b^2 x^2+\frac {56}{5} a^5 b^3 x^5+\frac {35}{4} a^4 b^4 x^8+\frac {56}{11} a^3 b^5 x^{11}+2 a^2 b^6 x^{14}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{20}}{20} \]

[Out]

-1/4*a^8/x^4-8*a^7*b/x+14*a^6*b^2*x^2+56/5*a^5*b^3*x^5+35/4*a^4*b^4*x^8+56/11*a^3*b^5*x^11+2*a^2*b^6*x^14+8/17
*a*b^7*x^17+1/20*b^8*x^20

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 102, 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}{4 x^4}-\frac {8 a^7 b}{x}+14 a^6 b^2 x^2+\frac {56}{5} a^5 b^3 x^5+\frac {35}{4} a^4 b^4 x^8+\frac {56}{11} a^3 b^5 x^{11}+2 a^2 b^6 x^{14}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{20}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^5,x]

[Out]

-1/4*a^8/x^4 - (8*a^7*b)/x + 14*a^6*b^2*x^2 + (56*a^5*b^3*x^5)/5 + (35*a^4*b^4*x^8)/4 + (56*a^3*b^5*x^11)/11 +
2*a^2*b^6*x^14 + (8*a*b^7*x^17)/17 + (b^8*x^20)/20

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^5} \, dx &=\int \left (\frac {a^8}{x^5}+\frac {8 a^7 b}{x^2}+28 a^6 b^2 x+56 a^5 b^3 x^4+70 a^4 b^4 x^7+56 a^3 b^5 x^{10}+28 a^2 b^6 x^{13}+8 a b^7 x^{16}+b^8 x^{19}\right ) \, dx\\ &=-\frac {a^8}{4 x^4}-\frac {8 a^7 b}{x}+14 a^6 b^2 x^2+\frac {56}{5} a^5 b^3 x^5+\frac {35}{4} a^4 b^4 x^8+\frac {56}{11} a^3 b^5 x^{11}+2 a^2 b^6 x^{14}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{20}}{20}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 102, normalized size = 1.00 \begin {gather*} -\frac {a^8}{4 x^4}-\frac {8 a^7 b}{x}+14 a^6 b^2 x^2+\frac {56}{5} a^5 b^3 x^5+\frac {35}{4} a^4 b^4 x^8+\frac {56}{11} a^3 b^5 x^{11}+2 a^2 b^6 x^{14}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{20}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^5,x]

[Out]

-1/4*a^8/x^4 - (8*a^7*b)/x + 14*a^6*b^2*x^2 + (56*a^5*b^3*x^5)/5 + (35*a^4*b^4*x^8)/4 + (56*a^3*b^5*x^11)/11 +
2*a^2*b^6*x^14 + (8*a*b^7*x^17)/17 + (b^8*x^20)/20

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 91, normalized size = 0.89

method	result	size
default	\(-\frac {a^{8}}{4 x^{4}}-\frac {8 a^{7} b}{x}+14 a^{6} b^{2} x^{2}+\frac {56 a^{5} b^{3} x^{5}}{5}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {56 a^{3} b^{5} x^{11}}{11}+2 a^{2} b^{6} x^{14}+\frac {8 a \,b^{7} x^{17}}{17}+\frac {b^{8} x^{20}}{20}\)	\(91\)
norman	\(\frac {\frac {8}{17} a \,b^{7} x^{21}+\frac {1}{20} b^{8} x^{24}+\frac {56}{11} a^{3} b^{5} x^{15}+2 a^{2} b^{6} x^{18}+\frac {56}{5} a^{5} b^{3} x^{9}+\frac {35}{4} a^{4} b^{4} x^{12}-8 a^{7} b \,x^{3}+14 a^{6} b^{2} x^{6}-\frac {1}{4} a^{8}}{x^{4}}\)	\(92\)
gosper	\(-\frac {-187 b^{8} x^{24}-1760 a \,b^{7} x^{21}-7480 a^{2} b^{6} x^{18}-19040 a^{3} b^{5} x^{15}-32725 a^{4} b^{4} x^{12}-41888 a^{5} b^{3} x^{9}-52360 a^{6} b^{2} x^{6}+29920 a^{7} b \,x^{3}+935 a^{8}}{3740 x^{4}}\)	\(93\)
risch	\(\frac {b^{8} x^{20}}{20}+\frac {8 a \,b^{7} x^{17}}{17}+2 a^{2} b^{6} x^{14}+\frac {56 a^{3} b^{5} x^{11}}{11}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {56 a^{5} b^{3} x^{5}}{5}+14 a^{6} b^{2} x^{2}+\frac {-8 a^{7} b \,x^{3}-\frac {1}{4} a^{8}}{x^{4}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^8/x^4-8*a^7*b/x+14*a^6*b^2*x^2+56/5*a^5*b^3*x^5+35/4*a^4*b^4*x^8+56/11*a^3*b^5*x^11+2*a^2*b^6*x^14+8/17
*a*b^7*x^17+1/20*b^8*x^20

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{20} \, b^{8} x^{20} + \frac {8}{17} \, a b^{7} x^{17} + 2 \, a^{2} b^{6} x^{14} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {56}{5} \, a^{5} b^{3} x^{5} + 14 \, a^{6} b^{2} x^{2} - \frac {32 \, a^{7} b x^{3} + a^{8}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^5,x, algorithm="maxima")

[Out]

1/20*b^8*x^20 + 8/17*a*b^7*x^17 + 2*a^2*b^6*x^14 + 56/11*a^3*b^5*x^11 + 35/4*a^4*b^4*x^8 + 56/5*a^5*b^3*x^5 +
14*a^6*b^2*x^2 - 1/4*(32*a^7*b*x^3 + a^8)/x^4

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 92, normalized size = 0.90 \begin {gather*} \frac {187 \, b^{8} x^{24} + 1760 \, a b^{7} x^{21} + 7480 \, a^{2} b^{6} x^{18} + 19040 \, a^{3} b^{5} x^{15} + 32725 \, a^{4} b^{4} x^{12} + 41888 \, a^{5} b^{3} x^{9} + 52360 \, a^{6} b^{2} x^{6} - 29920 \, a^{7} b x^{3} - 935 \, a^{8}}{3740 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^5,x, algorithm="fricas")

[Out]

1/3740*(187*b^8*x^24 + 1760*a*b^7*x^21 + 7480*a^2*b^6*x^18 + 19040*a^3*b^5*x^15 + 32725*a^4*b^4*x^12 + 41888*a
^5*b^3*x^9 + 52360*a^6*b^2*x^6 - 29920*a^7*b*x^3 - 935*a^8)/x^4

________________________________________________________________________________________

Sympy [A]
time = 0.08, size = 104, normalized size = 1.02 \begin {gather*} 14 a^{6} b^{2} x^{2} + \frac {56 a^{5} b^{3} x^{5}}{5} + \frac {35 a^{4} b^{4} x^{8}}{4} + \frac {56 a^{3} b^{5} x^{11}}{11} + 2 a^{2} b^{6} x^{14} + \frac {8 a b^{7} x^{17}}{17} + \frac {b^{8} x^{20}}{20} + \frac {- a^{8} - 32 a^{7} b x^{3}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**5,x)

[Out]

14*a**6*b**2*x**2 + 56*a**5*b**3*x**5/5 + 35*a**4*b**4*x**8/4 + 56*a**3*b**5*x**11/11 + 2*a**2*b**6*x**14 + 8*
a*b**7*x**17/17 + b**8*x**20/20 + (-a**8 - 32*a**7*b*x**3)/(4*x**4)

________________________________________________________________________________________

Giac [A]
time = 1.28, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{20} \, b^{8} x^{20} + \frac {8}{17} \, a b^{7} x^{17} + 2 \, a^{2} b^{6} x^{14} + \frac {56}{11} \, a^{3} b^{5} x^{11} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {56}{5} \, a^{5} b^{3} x^{5} + 14 \, a^{6} b^{2} x^{2} - \frac {32 \, a^{7} b x^{3} + a^{8}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^5,x, algorithm="giac")

[Out]

1/20*b^8*x^20 + 8/17*a*b^7*x^17 + 2*a^2*b^6*x^14 + 56/11*a^3*b^5*x^11 + 35/4*a^4*b^4*x^8 + 56/5*a^5*b^3*x^5 +
14*a^6*b^2*x^2 - 1/4*(32*a^7*b*x^3 + a^8)/x^4

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 93, normalized size = 0.91 \begin {gather*} \frac {b^8\,x^{20}}{20}-\frac {\frac {a^8}{4}+8\,b\,a^7\,x^3}{x^4}+\frac {8\,a\,b^7\,x^{17}}{17}+14\,a^6\,b^2\,x^2+\frac {56\,a^5\,b^3\,x^5}{5}+\frac {35\,a^4\,b^4\,x^8}{4}+\frac {56\,a^3\,b^5\,x^{11}}{11}+2\,a^2\,b^6\,x^{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^5,x)

[Out]

(b^8*x^20)/20 - (a^8/4 + 8*a^7*b*x^3)/x^4 + (8*a*b^7*x^17)/17 + 14*a^6*b^2*x^2 + (56*a^5*b^3*x^5)/5 + (35*a^4*
b^4*x^8)/4 + (56*a^3*b^5*x^11)/11 + 2*a^2*b^6*x^14

________________________________________________________________________________________