Optimal. Leaf size=20 \[ \frac {2 \left (a x^3-b\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {446, 74} \begin {gather*} \frac {2 \left (a x^3-b\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-b+a x} (2 b+a x)}{x^2} \, dx,x,x^3\right )\\ &=\frac {2 \left (-b+a x^3\right )^{3/2}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (a x^3-b\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (-b+a x^3\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 39, normalized size = 1.95 \begin {gather*} \frac {2 \, {\left (\sqrt {a x^{3} - b} a^{2} - \frac {\sqrt {a x^{3} - b} a b}{x^{3}}\right )}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 17, normalized size = 0.85
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
trager | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
risch | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
elliptic | \(-\frac {2 b \sqrt {a \,x^{3}-b}}{3 x^{3}}+\frac {2 a \sqrt {a \,x^{3}-b}}{3}\) | \(33\) |
default | \(a \left (\frac {2 \sqrt {a \,x^{3}-b}}{3}+\frac {2 b \arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\right )+2 b \left (-\frac {\sqrt {a \,x^{3}-b}}{3 x^{3}}-\frac {a \arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\right )\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.73, size = 79, normalized size = 3.95 \begin {gather*} -\frac {2}{3} \, {\left (\sqrt {b} \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right ) - \sqrt {a x^{3} - b}\right )} a + \frac {2}{3} \, {\left (\frac {a \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{\sqrt {b}} - \frac {\sqrt {a x^{3} - b}}{x^{3}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 16, normalized size = 0.80 \begin {gather*} \frac {2\,{\left (a\,x^3-b\right )}^{3/2}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.40, size = 309, normalized size = 15.45 \begin {gather*} a \left (\begin {cases} - \frac {2 i \sqrt {a} x^{\frac {3}{2}}}{3 \sqrt {-1 + \frac {b}{a x^{3}}}} - \frac {2 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3} + \frac {2 i b}{3 \sqrt {a} x^{\frac {3}{2}} \sqrt {-1 + \frac {b}{a x^{3}}}} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\\frac {2 \sqrt {a} x^{\frac {3}{2}}}{3 \sqrt {1 - \frac {b}{a x^{3}}}} + \frac {2 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3} - \frac {2 b}{3 \sqrt {a} x^{\frac {3}{2}} \sqrt {1 - \frac {b}{a x^{3}}}} & \text {otherwise} \end {cases}\right ) + 2 b \left (\begin {cases} \frac {i \sqrt {a}}{3 x^{\frac {3}{2}} \sqrt {-1 + \frac {b}{a x^{3}}}} + \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} - \frac {i b}{3 \sqrt {a} x^{\frac {9}{2}} \sqrt {-1 + \frac {b}{a x^{3}}}} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {1 - \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} - \frac {a \operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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