Optimal. Leaf size=154 \[ -\frac {b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac {a c-d \sqrt {a+b x^3}}{3 a x^3 \left (a c^2-d^2\right )}+\frac {2 b c^3 \log \left (c \sqrt {a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )^2}-\frac {b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]
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Rubi [A] time = 0.30, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2155, 741, 801, 635, 206, 260} \[ -\frac {b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac {a c-d \sqrt {a+b x^3}}{3 a x^3 \left (a c^2-d^2\right )}+\frac {2 b c^3 \log \left (c \sqrt {a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )^2}-\frac {b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 260
Rule 635
Rule 741
Rule 801
Rule 2155
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a c+b c x+d \sqrt {a+b x}\right )} \, dx,x,x^3\right )\\ &=\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {1}{(d+c x) \left (a-x^2\right )^2} \, dx,x,\sqrt {a+b x^3}\right )\\ &=-\frac {a c-d \sqrt {a+b x^3}}{3 a \left (a c^2-d^2\right ) x^3}-\frac {b \operatorname {Subst}\left (\int \frac {-2 a c^2+d^2+c d x}{(d+c x) \left (a-x^2\right )} \, dx,x,\sqrt {a+b x^3}\right )}{3 a \left (a c^2-d^2\right )}\\ &=-\frac {a c-d \sqrt {a+b x^3}}{3 a \left (a c^2-d^2\right ) x^3}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {2 a c^4}{\left (a c^2-d^2\right ) (d+c x)}+\frac {3 a c^2 d-d^3-2 a c^3 x}{\left (a c^2-d^2\right ) \left (a-x^2\right )}\right ) \, dx,x,\sqrt {a+b x^3}\right )}{3 a \left (a c^2-d^2\right )}\\ &=-\frac {a c-d \sqrt {a+b x^3}}{3 a \left (a c^2-d^2\right ) x^3}+\frac {2 b c^3 \log \left (d+c \sqrt {a+b x^3}\right )}{3 \left (a c^2-d^2\right )^2}-\frac {b \operatorname {Subst}\left (\int \frac {3 a c^2 d-d^3-2 a c^3 x}{a-x^2} \, dx,x,\sqrt {a+b x^3}\right )}{3 a \left (a c^2-d^2\right )^2}\\ &=-\frac {a c-d \sqrt {a+b x^3}}{3 a \left (a c^2-d^2\right ) x^3}+\frac {2 b c^3 \log \left (d+c \sqrt {a+b x^3}\right )}{3 \left (a c^2-d^2\right )^2}+\frac {\left (2 b c^3\right ) \operatorname {Subst}\left (\int \frac {x}{a-x^2} \, dx,x,\sqrt {a+b x^3}\right )}{3 \left (a c^2-d^2\right )^2}-\frac {\left (b d \left (3 a c^2-d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b x^3}\right )}{3 a \left (a c^2-d^2\right )^2}\\ &=-\frac {a c-d \sqrt {a+b x^3}}{3 a \left (a c^2-d^2\right ) x^3}-\frac {b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac {b c^3 \log (x)}{\left (a c^2-d^2\right )^2}+\frac {2 b c^3 \log \left (d+c \sqrt {a+b x^3}\right )}{3 \left (a c^2-d^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 307, normalized size = 1.99 \[ \frac {\sqrt {a} \left (-a^2 c^3 \sqrt {a+b x^3}+a^2 c^2 d+2 a b c^3 x^3 \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {c \sqrt {a+b x^3}}{d}\right )-3 a b c^3 x^3 \log (x) \sqrt {a+b x^3}+b d x^3 \sqrt {\frac {b x^3}{a}+1} \left (a c^2-d^2\right ) \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )+a b c^2 d x^3+a b c^3 x^3 \sqrt {a+b x^3} \log \left (a c^2+b c^2 x^3-d^2\right )+a c d^2 \sqrt {a+b x^3}-a d^3-b d^3 x^3\right )-2 b d x^3 \sqrt {a+b x^3} \left (2 a c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2} x^3 \sqrt {a+b x^3} \left (d^2-a c^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 445, normalized size = 2.89 \[ \left [\frac {2 \, a^{2} b c^{3} x^{3} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + 2 \, a^{2} b c^{3} x^{3} \log \left (\sqrt {b x^{3} + a} c + d\right ) - 2 \, a^{2} b c^{3} x^{3} \log \left (\sqrt {b x^{3} + a} c - d\right ) - 6 \, a^{2} b c^{3} x^{3} \log \relax (x) - 2 \, a^{3} c^{3} - {\left (3 \, a b c^{2} d - b d^{3}\right )} \sqrt {a} x^{3} \log \left (\frac {b x^{3} + 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, a^{2} c d^{2} + 2 \, {\left (a^{2} c^{2} d - a d^{3}\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4}\right )} x^{3}}, \frac {a^{2} b c^{3} x^{3} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + a^{2} b c^{3} x^{3} \log \left (\sqrt {b x^{3} + a} c + d\right ) - a^{2} b c^{3} x^{3} \log \left (\sqrt {b x^{3} + a} c - d\right ) - 3 \, a^{2} b c^{3} x^{3} \log \relax (x) - a^{3} c^{3} + {\left (3 \, a b c^{2} d - b d^{3}\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + a^{2} c d^{2} + {\left (a^{2} c^{2} d - a d^{3}\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4}\right )} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 211, normalized size = 1.37 \[ \frac {2 \, b c^{4} \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{3 \, {\left (a^{2} c^{5} - 2 \, a c^{3} d^{2} + c d^{4}\right )}} - \frac {b c^{3} \log \left (-b x^{3}\right )}{3 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )}} + \frac {{\left (3 \, a b c^{2} d - b d^{3}\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, {\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt {-a}} - \frac {a^{2} b c^{3} - a b c d^{2} - {\left (a b c^{2} d - b d^{3}\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a c^{2} - d^{2}\right )}^{2} a b x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 863, normalized size = 5.60 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (b c x^{3} + a c + \sqrt {b x^{3} + a} d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.46, size = 248, normalized size = 1.61 \[ \frac {b\,c^3\,\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )}{3\,a^2\,c^4-6\,a\,c^2\,d^2+3\,d^4}-\frac {b\,c^3\,\ln \relax (x)}{a^2\,c^4-2\,a\,c^2\,d^2+d^4}-\frac {c}{3\,x^3\,\left (a\,c^2-d^2\right )}+\frac {b\,c^3\,\ln \left (\frac {d+c\,\sqrt {b\,x^3+a}}{d-c\,\sqrt {b\,x^3+a}}\right )}{3\,{\left (a\,c^2-d^2\right )}^2}+\frac {d\,\sqrt {b\,x^3+a}}{3\,a\,x^3\,\left (a\,c^2-d^2\right )}+\frac {b\,d\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )\,\left (3\,a\,c^2-d^2\right )}{6\,a^{3/2}\,{\left (a\,c^2-d^2\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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