Optimal. Leaf size=50 \[ -\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 270} \begin {gather*} 6 a^2 b^2 x-\frac {4 a^3 b}{x}-\frac {a^4}{3 x^3}+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 1593
Rubi steps
\begin {align*} \int \left (\frac {a}{x}+b x\right )^4 \, dx &=\int \frac {\left (a+b x^2\right )^4}{x^4} \, dx\\ &=\int \left (6 a^2 b^2+\frac {a^4}{x^4}+\frac {4 a^3 b}{x^2}+4 a b^3 x^2+b^4 x^4\right ) \, dx\\ &=-\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} -\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {a}{x}+b x\right )^4 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 48, normalized size = 0.96 \begin {gather*} \frac {3 \, b^{4} x^{8} + 20 \, a b^{3} x^{6} + 90 \, a^{2} b^{2} x^{4} - 60 \, a^{3} b x^{2} - 5 \, a^{4}}{15 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 45, normalized size = 0.90 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + \frac {4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac {12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 45, normalized size = 0.90 \begin {gather*} \frac {b^{4} x^{5}}{5}+\frac {4 a \,b^{3} x^{3}}{3}+6 a^{2} b^{2} x -\frac {4 a^{3} b}{x}-\frac {a^{4}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 44, normalized size = 0.88 \begin {gather*} \frac {1}{5} \, b^{4} x^{5} + \frac {4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac {4 \, a^{3} b}{x} - \frac {a^{4}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 47, normalized size = 0.94 \begin {gather*} \frac {b^4\,x^5}{5}-\frac {\frac {a^4}{3}+4\,b\,a^3\,x^2}{x^3}+6\,a^2\,b^2\,x+\frac {4\,a\,b^3\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.19, size = 49, normalized size = 0.98 \begin {gather*} 6 a^{2} b^{2} x + \frac {4 a b^{3} x^{3}}{3} + \frac {b^{4} x^{5}}{5} + \frac {- a^{4} - 12 a^{3} b x^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________