Optimal. Leaf size=121 \[ \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 \sqrt {b} \sqrt {b c-a d}}-\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {\sqrt {c+d x^3}}{3 a \left (a+b x^3\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 121, 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 = {446, 99, 156, 63, 208} \[ \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 \sqrt {b} \sqrt {b c-a d}}-\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {\sqrt {c+d x^3}}{3 a \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 99
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {c+d x^3}}{3 a \left (a+b x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-c-\frac {d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a}\\ &=\frac {\sqrt {c+d x^3}}{3 a \left (a+b x^3\right )}+\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2}-\frac {(2 b c-a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^2}\\ &=\frac {\sqrt {c+d x^3}}{3 a \left (a+b x^3\right )}+\frac {(2 c) \operatorname {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 {(2 b c-a d) \operatorname {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 d}\\ &=\frac {\sqrt {c+d x^3}}{3 a \left (a+b x^3\right )}-\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2}+\frac {(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 \sqrt {b} \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 112, normalized size = 0.93 \[ \frac {\frac {a \sqrt {c+d x^3}}{a+b x^3}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \sqrt {b c-a d}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 856, normalized size = 7.07 \[ \left [-\frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) - 2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}\right )}}, -\frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) - {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}\right )}}, \frac {4 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{3}\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 {b^{2} c - a b d} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}\right )}}, -\frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) - 2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 114, normalized size = 0.94 \[ \frac {\sqrt {d x^{3} + c} d}{3 \, {\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} a} - \frac {{\left (2 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a^{2}} + \frac {2 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{2} \sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 934, normalized size = 7.72 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {d x^{3} + c}}{{\left (b x^{3} + a\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.28, size = 182, normalized size = 1.50 \[ \frac {\sqrt {c}\,\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a^2}-\frac {\left (\frac {b\,d}{3\,\left (b^2\,c-a\,b\,d\right )}-\frac {b^2\,c}{3\,a\,\left (b^2\,c-a\,b\,d\right )}\right )\,\sqrt {d\,x^3+c}}{b\,x^3+a}+\frac {\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {d\,x^3+c}\,\sqrt {a\,b\,d-b^2\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{6\,a^2\,\sqrt {a\,b\,d-b^2\,c}} \]
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 \[ \int \frac {\sqrt {c + d x^{3}}}{x \left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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