3.3.65 \(\int \frac {(a+b x^3)^5}{x^4} \, dx\) [265]

Optimal. Leaf size=66 \[ -\frac {a^5}{3 x^3}+\frac {10}{3} a^3 b^2 x^3+\frac {5}{3} a^2 b^3 x^6+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{12}}{12}+5 a^4 b \log (x) \]

[Out]

-1/3*a^5/x^3+10/3*a^3*b^2*x^3+5/3*a^2*b^3*x^6+5/9*a*b^4*x^9+1/12*b^5*x^12+5*a^4*b*ln(x)

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Rubi [A]
time = 0.03, antiderivative size = 66, 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^5}{3 x^3}+5 a^4 b \log (x)+\frac {10}{3} a^3 b^2 x^3+\frac {5}{3} a^2 b^3 x^6+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{12}}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^4,x]

[Out]

-1/3*a^5/x^3 + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^9)/9 + (b^5*x^12)/12 + 5*a^4*b*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 )^5}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^5}{x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (10 a^3 b^2+\frac {a^5}{x^2}+\frac {5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx,x,x^3\right )\\ &=-\frac {a^5}{3 x^3}+\frac {10}{3} a^3 b^2 x^3+\frac {5}{3} a^2 b^3 x^6+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{12}}{12}+5 a^4 b \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 66, normalized size = 1.00 \begin {gather*} -\frac {a^5}{3 x^3}+\frac {10}{3} a^3 b^2 x^3+\frac {5}{3} a^2 b^3 x^6+\frac {5}{9} a b^4 x^9+\frac {b^5 x^{12}}{12}+5 a^4 b \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^4,x]

[Out]

-1/3*a^5/x^3 + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^9)/9 + (b^5*x^12)/12 + 5*a^4*b*Log[x]

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Maple [A]
time = 0.12, size = 57, normalized size = 0.86

method	result	size
default	\(-\frac {a^{5}}{3 x^{3}}+\frac {10 a^{3} b^{2} x^{3}}{3}+\frac {5 a^{2} b^{3} x^{6}}{3}+\frac {5 a \,b^{4} x^{9}}{9}+\frac {b^{5} x^{12}}{12}+5 a^{4} b \ln \left (x \right )\)	\(57\)
risch	\(-\frac {a^{5}}{3 x^{3}}+\frac {10 a^{3} b^{2} x^{3}}{3}+\frac {5 a^{2} b^{3} x^{6}}{3}+\frac {5 a \,b^{4} x^{9}}{9}+\frac {b^{5} x^{12}}{12}+5 a^{4} b \ln \left (x \right )\)	\(57\)
norman	\(\frac {-\frac {1}{3} a^{5}+\frac {1}{12} b^{5} x^{15}+\frac {5}{9} a \,b^{4} x^{12}+\frac {5}{3} a^{2} b^{3} x^{9}+\frac {10}{3} a^{3} b^{2} x^{6}}{x^{3}}+5 a^{4} b \ln \left (x \right )\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^5/x^3+10/3*a^3*b^2*x^3+5/3*a^2*b^3*x^6+5/9*a*b^4*x^9+1/12*b^5*x^12+5*a^4*b*ln(x)

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Maxima [A]
time = 0.30, size = 58, normalized size = 0.88 \begin {gather*} \frac {1}{12} \, b^{5} x^{12} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{3} \, a^{4} b \log \left (x^{3}\right ) - \frac {a^{5}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^4,x, algorithm="maxima")

[Out]

1/12*b^5*x^12 + 5/9*a*b^4*x^9 + 5/3*a^2*b^3*x^6 + 10/3*a^3*b^2*x^3 + 5/3*a^4*b*log(x^3) - 1/3*a^5/x^3

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Fricas [A]
time = 0.34, size = 61, normalized size = 0.92 \begin {gather*} \frac {3 \, b^{5} x^{15} + 20 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 120 \, a^{3} b^{2} x^{6} + 180 \, a^{4} b x^{3} \log \left (x\right ) - 12 \, a^{5}}{36 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^4,x, algorithm="fricas")

[Out]

1/36*(3*b^5*x^15 + 20*a*b^4*x^12 + 60*a^2*b^3*x^9 + 120*a^3*b^2*x^6 + 180*a^4*b*x^3*log(x) - 12*a^5)/x^3

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Sympy [A]
time = 0.06, size = 65, normalized size = 0.98 \begin {gather*} - \frac {a^{5}}{3 x^{3}} + 5 a^{4} b \log {\left (x \right )} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{6}}{3} + \frac {5 a b^{4} x^{9}}{9} + \frac {b^{5} x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**4,x)

[Out]

-a**5/(3*x**3) + 5*a**4*b*log(x) + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**6/3 + 5*a*b**4*x**9/9 + b**5*x**12/12

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Giac [A]
time = 1.89, size = 67, normalized size = 1.02 \begin {gather*} \frac {1}{12} \, b^{5} x^{12} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) - \frac {5 \, a^{4} b x^{3} + a^{5}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^4,x, algorithm="giac")

[Out]

1/12*b^5*x^12 + 5/9*a*b^4*x^9 + 5/3*a^2*b^3*x^6 + 10/3*a^3*b^2*x^3 + 5*a^4*b*log(abs(x)) - 1/3*(5*a^4*b*x^3 +
a^5)/x^3

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Mupad [B]
time = 0.03, size = 56, normalized size = 0.85 \begin {gather*} \frac {b^5\,x^{12}}{12}-\frac {a^5}{3\,x^3}+\frac {5\,a\,b^4\,x^9}{9}+5\,a^4\,b\,\ln \left (x\right )+\frac {10\,a^3\,b^2\,x^3}{3}+\frac {5\,a^2\,b^3\,x^6}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^5/x^4,x)

[Out]

(b^5*x^12)/12 - a^5/(3*x^3) + (5*a*b^4*x^9)/9 + 5*a^4*b*log(x) + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^6)/3

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