Optimal. Leaf size=124 \[ \frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{1152 c^{5/2}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{128 c^{5/2}}+\frac {d \sqrt {c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 124, 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, 99, 151, 156, 63, 208, 206} \[ \frac {d \sqrt {c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}+\frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{1152 c^{5/2}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{128 c^{5/2}}-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 99
Rule 151
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{x^4 \left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x^2 (8 c-d x)^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {6 c d+\frac {3 d^2 x}{2}}{x (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{24 c}\\ &=\frac {d \sqrt {c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-54 c^2 d^2-9 c d^3 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{1728 c^3 d}\\ &=\frac {d \sqrt {c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )}+\frac {d \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{256 c^2}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{768 c^2}\\ &=\frac {d \sqrt {c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{128 c^2}+\frac {(7 d) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{384 c^2}\\ &=\frac {d \sqrt {c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )}+\frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{1152 c^{5/2}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{128 c^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 97, normalized size = 0.78 \[ \frac {7 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-9 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )+\frac {12 \sqrt {c} \sqrt {c+d x^3} \left (4 c-d x^3\right )}{d x^6-8 c x^3}}{1152 c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 278, normalized size = 2.24 \[ \left [\frac {7 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 9 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 24 \, {\left (c d x^{3} - 4 \, c^{2}\right )} \sqrt {d x^{3} + c}}{2304 \, {\left (c^{3} d x^{6} - 8 \, c^{4} x^{3}\right )}}, \frac {9 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 7 \, {\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) - 12 \, {\left (c d x^{3} - 4 \, c^{2}\right )} \sqrt {d x^{3} + c}}{1152 \, {\left (c^{3} d x^{6} - 8 \, c^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 113, normalized size = 0.91 \[ \frac {d \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{128 \, \sqrt {-c} c^{2}} - \frac {7 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{1152 \, \sqrt {-c} c^{2}} - \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} d - 5 \, \sqrt {d x^{3} + c} c d}{96 \, {\left ({\left (d x^{3} + c\right )}^{2} - 10 \, {\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.21, size = 957, normalized size = 7.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.21, size = 117, normalized size = 0.94 \[ \frac {\frac {5\,d\,\sqrt {d\,x^3+c}}{32\,c}-\frac {d\,{\left (d\,x^3+c\right )}^{3/2}}{32\,c^2}}{3\,{\left (d\,x^3+c\right )}^2-30\,c\,\left (d\,x^3+c\right )+27\,c^2}+\frac {d\,\left (\mathrm {atanh}\left (\frac {c^2\,\sqrt {d\,x^3+c}}{\sqrt {c^5}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c^2\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^5}}\right )\,7{}\mathrm {i}}{9}\right )\,1{}\mathrm {i}}{128\,\sqrt {c^5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________