Optimal. Leaf size=121 \[ \frac {3 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{128 c^{3/2}}-\frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{3/2}}+\frac {5 d \sqrt {c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {446, 98, 151, 156, 63, 208, 206} \begin {gather*} \frac {3 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{128 c^{3/2}}-\frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{3/2}}+\frac {5 d \sqrt {c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 151
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{3/2}}{x^4 \left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^2 (8 c-d x)^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-14 c^2 d-\frac {19}{2} c d^2 x}{x (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{24 c}\\ &=\frac {5 d \sqrt {c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {126 c^3 d^2+45 c^2 d^3 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{1728 c^3 d}\\ &=\frac {5 d \sqrt {c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )}+\frac {(7 d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{768 c}+\frac {\left (9 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{256 c}\\ &=\frac {5 d \sqrt {c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{384 c}+\frac {(9 d) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{128 c}\\ &=\frac {5 d \sqrt {c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 x^3 \left (8 c-d x^3\right )}+\frac {3 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{128 c^{3/2}}-\frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 97, normalized size = 0.80 \begin {gather*} \frac {9 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )+\frac {4 \sqrt {c} \sqrt {c+d x^3} \left (4 c-5 d x^3\right )}{d x^6-8 c x^3}}{384 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 103, normalized size = 0.85 \begin {gather*} \frac {3 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{128 c^{3/2}}-\frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{3/2}}+\frac {\sqrt {c+d x^3} \left (5 d x^3-4 c\right )}{96 c x^3 \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 280, normalized size = 2.31 \begin {gather*} \left [\frac {9 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 7 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 8 \, {\left (5 \, c d x^{3} - 4 \, c^{2}\right )} \sqrt {d x^{3} + c}}{768 \, {\left (c^{2} d x^{6} - 8 \, c^{3} x^{3}\right )}}, \frac {7 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 9 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) - 4 \, {\left (5 \, c d x^{3} - 4 \, c^{2}\right )} \sqrt {d x^{3} + c}}{384 \, {\left (c^{2} d x^{6} - 8 \, c^{3} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 114, normalized size = 0.94 \begin {gather*} \frac {7 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{384 \, \sqrt {-c} c} - \frac {3 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{128 \, \sqrt {-c} c} - \frac {5 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d - 9 \, \sqrt {d x^{3} + c} c d}{96 \, {\left ({\left (d x^{3} + c\right )}^{2} - 10 \, {\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 1014, normalized size = 8.38
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 \begin {gather*} \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.24, size = 110, normalized size = 0.91 \begin {gather*} \frac {\frac {9\,d\,\sqrt {d\,x^3+c}}{32}-\frac {5\,d\,{\left (d\,x^3+c\right )}^{3/2}}{32\,c}}{3\,{\left (d\,x^3+c\right )}^2-30\,c\,\left (d\,x^3+c\right )+27\,c^2}+\frac {d\,\left (\mathrm {atanh}\left (\frac {c\,\sqrt {d\,x^3+c}}{\sqrt {c^3}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^3}}\right )\,9{}\mathrm {i}}{7}\right )\,7{}\mathrm {i}}{384\,\sqrt {c^3}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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