Optimal. Leaf size=106 \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{7776 c^{7/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}}+\frac {5}{648 c^3 \sqrt {c+d x^3}}+\frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 106, 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, 103, 152, 156, 63, 208, 206} \begin {gather*} \frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {5}{648 c^3 \sqrt {c+d x^3}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{7776 c^{7/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 103
Rule 152
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {-9 c d-\frac {3 d^2 x}{2}}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{216 c^2 d}\\ &=\frac {5}{648 c^3 \sqrt {c+d x^3}}+\frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {81}{2} c^2 d^2+\frac {15}{4} c d^3 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{972 c^4 d^2}\\ &=\frac {5}{648 c^3 \sqrt {c+d x^3}}+\frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{192 c^3}+\frac {(7 d) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{5184 c^3}\\ &=\frac {5}{648 c^3 \sqrt {c+d x^3}}+\frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{2592 c^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{96 c^3 d}\\ &=\frac {5}{648 c^3 \sqrt {c+d x^3}}+\frac {1}{216 c^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{7776 c^{7/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{96 c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 97, normalized size = 0.92 \begin {gather*} \frac {\left (7 d x^3-56 c\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )+27 \left (8 c-d x^3\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )+12 c}{2592 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.10, size = 163, normalized size = 1.54 \begin {gather*} \frac {\sqrt {c+d x^3} \left (\frac {7}{972 c^{5/2}}-\frac {7 d x^3}{7776 c^{7/2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+\sqrt {c+d x^3} \left (\frac {d x^3}{96 c^{7/2}}-\frac {1}{12 c^{5/2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )-\frac {5 d x^3}{648 c^3}+\frac {43}{648 c^2}}{8 c \sqrt {c+d x^3}-d x^3 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 316, normalized size = 2.98 \begin {gather*} \left [\frac {7 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\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 ) + 81 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 24 \, {\left (5 \, c d x^{3} - 43 \, c^{2}\right )} \sqrt {d x^{3} + c}}{15552 \, {\left (c^{4} d^{2} x^{6} - 7 \, c^{5} d x^{3} - 8 \, c^{6}\right )}}, \frac {81 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 7 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (5 \, c d x^{3} - 43 \, c^{2}\right )} \sqrt {d x^{3} + c}}{7776 \, {\left (c^{4} d^{2} x^{6} - 7 \, c^{5} d x^{3} - 8 \, c^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 93, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{96 \, \sqrt {-c} c^{3}} - \frac {7 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{7776 \, \sqrt {-c} c^{3}} + \frac {5 \, d x^{3} - 43 \, c}{648 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c\right )} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.17, size = 953, normalized size = 8.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.33, size = 101, normalized size = 0.95 \begin {gather*} -\frac {\frac {5\,\left (d\,x^3+c\right )}{216\,c^3}-\frac {2}{9\,c^2}}{27\,c\,\sqrt {d\,x^3+c}-3\,{\left (d\,x^3+c\right )}^{3/2}}+\frac {\left (\mathrm {atanh}\left (\frac {c^3\,\sqrt {d\,x^3+c}}{\sqrt {c^7}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c^3\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^7}}\right )\,7{}\mathrm {i}}{81}\right )\,1{}\mathrm {i}}{96\,\sqrt {c^7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________