3.3.94 \(\int \frac {(a+b x^3)^8}{x^7} \, dx\) [294]

Optimal. Leaf size=105 \[ -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{3 x^3}+\frac {56}{3} a^5 b^3 x^3+\frac {35}{3} a^4 b^4 x^6+\frac {56}{9} a^3 b^5 x^9+\frac {7}{3} a^2 b^6 x^{12}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{18}}{18}+28 a^6 b^2 \log (x) \]

[Out]

-1/6*a^8/x^6-8/3*a^7*b/x^3+56/3*a^5*b^3*x^3+35/3*a^4*b^4*x^6+56/9*a^3*b^5*x^9+7/3*a^2*b^6*x^12+8/15*a*b^7*x^15
+1/18*b^8*x^18+28*a^6*b^2*ln(x)

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Rubi [A]
time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{3 x^3}+28 a^6 b^2 \log (x)+\frac {56}{3} a^5 b^3 x^3+\frac {35}{3} a^4 b^4 x^6+\frac {56}{9} a^3 b^5 x^9+\frac {7}{3} a^2 b^6 x^{12}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{18}}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^7,x]

[Out]

-1/6*a^8/x^6 - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56*a^3*b^5*x^9)/9 + (7*a^2*b^6*x
^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^6*b^2*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^7} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (56 a^5 b^3+\frac {a^8}{x^3}+\frac {8 a^7 b}{x^2}+\frac {28 a^6 b^2}{x}+70 a^4 b^4 x+56 a^3 b^5 x^2+28 a^2 b^6 x^3+8 a b^7 x^4+b^8 x^5\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{6 x^6}-\frac {8 a^7 b}{3 x^3}+\frac {56}{3} a^5 b^3 x^3+\frac {35}{3} a^4 b^4 x^6+\frac {56}{9} a^3 b^5 x^9+\frac {7}{3} a^2 b^6 x^{12}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{18}}{18}+28 a^6 b^2 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 105, normalized size = 1.00 \begin {gather*} -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{3 x^3}+\frac {56}{3} a^5 b^3 x^3+\frac {35}{3} a^4 b^4 x^6+\frac {56}{9} a^3 b^5 x^9+\frac {7}{3} a^2 b^6 x^{12}+\frac {8}{15} a b^7 x^{15}+\frac {b^8 x^{18}}{18}+28 a^6 b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^7,x]

[Out]

-1/6*a^8/x^6 - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56*a^3*b^5*x^9)/9 + (7*a^2*b^6*x
^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^6*b^2*Log[x]

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Maple [A]
time = 0.12, size = 90, normalized size = 0.86

method	result	size
default	\(-\frac {a^{8}}{6 x^{6}}-\frac {8 a^{7} b}{3 x^{3}}+\frac {56 a^{5} b^{3} x^{3}}{3}+\frac {35 a^{4} b^{4} x^{6}}{3}+\frac {56 a^{3} b^{5} x^{9}}{9}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {8 a \,b^{7} x^{15}}{15}+\frac {b^{8} x^{18}}{18}+28 a^{6} b^{2} \ln \left (x \right )\)	\(90\)
norman	\(\frac {-\frac {1}{6} a^{8}+\frac {1}{18} b^{8} x^{24}+\frac {8}{15} a \,b^{7} x^{21}+\frac {7}{3} a^{2} b^{6} x^{18}+\frac {56}{9} a^{3} b^{5} x^{15}+\frac {35}{3} a^{4} b^{4} x^{12}+\frac {56}{3} a^{5} b^{3} x^{9}-\frac {8}{3} a^{7} b \,x^{3}}{x^{6}}+28 a^{6} b^{2} \ln \left (x \right )\)	\(92\)
risch	\(\frac {b^{8} x^{18}}{18}+\frac {8 a \,b^{7} x^{15}}{15}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {56 a^{3} b^{5} x^{9}}{9}+\frac {35 a^{4} b^{4} x^{6}}{3}+\frac {56 a^{5} b^{3} x^{3}}{3}+\frac {-\frac {8}{3} a^{7} b \,x^{3}-\frac {1}{6} a^{8}}{x^{6}}+28 a^{6} b^{2} \ln \left (x \right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^8/x^6-8/3*a^7*b/x^3+56/3*a^5*b^3*x^3+35/3*a^4*b^4*x^6+56/9*a^3*b^5*x^9+7/3*a^2*b^6*x^12+8/15*a*b^7*x^15
+1/18*b^8*x^18+28*a^6*b^2*ln(x)

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Maxima [A]
time = 0.30, size = 92, normalized size = 0.88 \begin {gather*} \frac {1}{18} \, b^{8} x^{18} + \frac {8}{15} \, a b^{7} x^{15} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {35}{3} \, a^{4} b^{4} x^{6} + \frac {56}{3} \, a^{5} b^{3} x^{3} + \frac {28}{3} \, a^{6} b^{2} \log \left (x^{3}\right ) - \frac {16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^7,x, algorithm="maxima")

[Out]

1/18*b^8*x^18 + 8/15*a*b^7*x^15 + 7/3*a^2*b^6*x^12 + 56/9*a^3*b^5*x^9 + 35/3*a^4*b^4*x^6 + 56/3*a^5*b^3*x^3 +
28/3*a^6*b^2*log(x^3) - 1/6*(16*a^7*b*x^3 + a^8)/x^6

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Fricas [A]
time = 0.35, size = 94, normalized size = 0.90 \begin {gather*} \frac {5 \, b^{8} x^{24} + 48 \, a b^{7} x^{21} + 210 \, a^{2} b^{6} x^{18} + 560 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 1680 \, a^{5} b^{3} x^{9} + 2520 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 240 \, a^{7} b x^{3} - 15 \, a^{8}}{90 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^7,x, algorithm="fricas")

[Out]

1/90*(5*b^8*x^24 + 48*a*b^7*x^21 + 210*a^2*b^6*x^18 + 560*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 1680*a^5*b^3*x^9
+ 2520*a^6*b^2*x^6*log(x) - 240*a^7*b*x^3 - 15*a^8)/x^6

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Sympy [A]
time = 0.11, size = 105, normalized size = 1.00 \begin {gather*} 28 a^{6} b^{2} \log {\left (x \right )} + \frac {56 a^{5} b^{3} x^{3}}{3} + \frac {35 a^{4} b^{4} x^{6}}{3} + \frac {56 a^{3} b^{5} x^{9}}{9} + \frac {7 a^{2} b^{6} x^{12}}{3} + \frac {8 a b^{7} x^{15}}{15} + \frac {b^{8} x^{18}}{18} + \frac {- a^{8} - 16 a^{7} b x^{3}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**7,x)

[Out]

28*a**6*b**2*log(x) + 56*a**5*b**3*x**3/3 + 35*a**4*b**4*x**6/3 + 56*a**3*b**5*x**9/9 + 7*a**2*b**6*x**12/3 +
8*a*b**7*x**15/15 + b**8*x**18/18 + (-a**8 - 16*a**7*b*x**3)/(6*x**6)

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Giac [A]
time = 1.91, size = 102, normalized size = 0.97 \begin {gather*} \frac {1}{18} \, b^{8} x^{18} + \frac {8}{15} \, a b^{7} x^{15} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {56}{9} \, a^{3} b^{5} x^{9} + \frac {35}{3} \, a^{4} b^{4} x^{6} + \frac {56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} \log \left ({\left | x \right |}\right ) - \frac {84 \, a^{6} b^{2} x^{6} + 16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^7,x, algorithm="giac")

[Out]

1/18*b^8*x^18 + 8/15*a*b^7*x^15 + 7/3*a^2*b^6*x^12 + 56/9*a^3*b^5*x^9 + 35/3*a^4*b^4*x^6 + 56/3*a^5*b^3*x^3 +
28*a^6*b^2*log(abs(x)) - 1/6*(84*a^6*b^2*x^6 + 16*a^7*b*x^3 + a^8)/x^6

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Mupad [B]
time = 0.05, size = 92, normalized size = 0.88 \begin {gather*} \frac {b^8\,x^{18}}{18}-\frac {\frac {a^8}{6}+\frac {8\,b\,a^7\,x^3}{3}}{x^6}+\frac {8\,a\,b^7\,x^{15}}{15}+\frac {56\,a^5\,b^3\,x^3}{3}+\frac {35\,a^4\,b^4\,x^6}{3}+\frac {56\,a^3\,b^5\,x^9}{9}+\frac {7\,a^2\,b^6\,x^{12}}{3}+28\,a^6\,b^2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^7,x)

[Out]

(b^8*x^18)/18 - (a^8/6 + (8*a^7*b*x^3)/3)/x^6 + (8*a*b^7*x^15)/15 + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 +
(56*a^3*b^5*x^9)/9 + (7*a^2*b^6*x^12)/3 + 28*a^6*b^2*log(x)

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