Optimal. Leaf size=137 \[ \frac {256 b^4 \sqrt {a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac {128 b^3 \sqrt {a x+b x^{2/3}}}{105 a^4}+\frac {32 b^2 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{35 a^3}-\frac {16 b x^{2/3} \sqrt {a x+b x^{2/3}}}{21 a^2}+\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2016, 2002, 2014} \[ \frac {256 b^4 \sqrt {a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac {128 b^3 \sqrt {a x+b x^{2/3}}}{105 a^4}+\frac {32 b^2 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{35 a^3}-\frac {16 b x^{2/3} \sqrt {a x+b x^{2/3}}}{21 a^2}+\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx &=\frac {2 x \sqrt {b x^{2/3}+a x}}{3 a}-\frac {(8 b) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{9 a}\\ &=-\frac {16 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^2}+\frac {2 x \sqrt {b x^{2/3}+a x}}{3 a}+\frac {\left (16 b^2\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{21 a^2}\\ &=\frac {32 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{35 a^3}-\frac {16 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^2}+\frac {2 x \sqrt {b x^{2/3}+a x}}{3 a}-\frac {\left (64 b^3\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{105 a^3}\\ &=-\frac {128 b^3 \sqrt {b x^{2/3}+a x}}{105 a^4}+\frac {32 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{35 a^3}-\frac {16 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^2}+\frac {2 x \sqrt {b x^{2/3}+a x}}{3 a}+\frac {\left (128 b^4\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{315 a^4}\\ &=-\frac {128 b^3 \sqrt {b x^{2/3}+a x}}{105 a^4}+\frac {256 b^4 \sqrt {b x^{2/3}+a x}}{105 a^5 \sqrt [3]{x}}+\frac {32 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{35 a^3}-\frac {16 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^2}+\frac {2 x \sqrt {b x^{2/3}+a x}}{3 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 74, normalized size = 0.54 \[ \frac {2 \sqrt {a x+b x^{2/3}} \left (35 a^4 x^{4/3}-40 a^3 b x+48 a^2 b^2 x^{2/3}-64 a b^3 \sqrt [3]{x}+128 b^4\right )}{105 a^5 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 80, normalized size = 0.58 \[ -\frac {256 \, b^{\frac {9}{2}}}{105 \, a^{5}} + \frac {2 \, {\left (35 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {a x^{\frac {1}{3}} + b} b^{4}\right )}}{105 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 68, normalized size = 0.50 \[ \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (35 a^{4} x^{\frac {4}{3}}-40 a^{3} b x +48 a^{2} b^{2} x^{\frac {2}{3}}-64 a \,b^{3} x^{\frac {1}{3}}+128 b^{4}\right ) x^{\frac {1}{3}}}{105 \sqrt {a x +b \,x^{\frac {2}{3}}}\, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a x + b x^{\frac {2}{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________