Optimal. Leaf size=131 \[ \frac {(b c-a d) \sqrt {c+d x^3}}{3 a b \left (a+b x^3\right )}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {\sqrt {b c-a d} (2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 b^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 100, 162,
65, 214} \begin {gather*} \frac {\sqrt {b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 b^{3/2}}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {\sqrt {c+d x^3} (b c-a d)}{3 a b \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {(b c-a d) \sqrt {c+d x^3}}{3 a b \left (a+b x^3\right )}+\frac {\text {Subst}\left (\int \frac {b c^2+\frac {1}{2} d (b c+a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a b}\\ &=\frac {(b c-a d) \sqrt {c+d x^3}}{3 a b \left (a+b x^3\right )}+\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2}-\frac {((b c-a d) (2 b c+a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^2 b}\\ &=\frac {(b c-a d) \sqrt {c+d x^3}}{3 a b \left (a+b x^3\right )}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^2 d}-\frac {((b c-a d) (2 b c+a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^2 b d}\\ &=\frac {(b c-a d) \sqrt {c+d x^3}}{3 a b \left (a+b x^3\right )}-\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {\sqrt {b c-a d} (2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 122, normalized size = 0.93 \begin {gather*} \frac {\frac {a (b c-a d) \sqrt {c+d x^3}}{b \left (a+b x^3\right )}+\frac {\sqrt {-b c+a d} (2 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}-2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.46, size = 1036, normalized size = 7.91
method | result | size |
default | \(\text {Expression too large to display}\) | \(1036\) |
elliptic | \(\text {Expression too large to display}\) | \(1632\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.08, size = 686, normalized size = 5.24 \begin {gather*} \left [\frac {{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{6 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (b^{2} c x^{3} + a b c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {4 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \, \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{6 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + 2 \, {\left (b^{2} c x^{3} + a b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + \sqrt {d x^{3} + c} {\left (a b c - a^{2} d\right )}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{x \left (a + b x^{3}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.75, size = 155, normalized size = 1.18 \begin {gather*} \frac {2 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{2} \sqrt {-c}} - \frac {{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a^{2} b} + \frac {\sqrt {d x^{3} + c} b c d - \sqrt {d x^{3} + c} a d^{2}}{3 \, {\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.14, size = 214, normalized size = 1.63 \begin {gather*} \frac {c^{3/2}\,\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a^2}+\frac {\sqrt {d\,x^3+c}\,\left (\frac {a\,\left (\frac {b\,d^2}{3\,\left (b^2\,c-a\,b\,d\right )}-\frac {2\,b^2\,c\,d}{3\,a\,\left (b^2\,c-a\,b\,d\right )}\right )}{b}+\frac {b^2\,c^2}{3\,a\,\left (b^2\,c-a\,b\,d\right )}\right )}{b\,x^3+a}+\frac {\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\sqrt {a\,d-b\,c}\,\left (a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{6\,a^2\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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