Optimal. Leaf size=143 \[ \frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{31104 c^{9/2}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{9/2}}-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {446, 103, 151, 152, 156, 63, 208, 206} \[ \frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{31104 c^{9/2}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 103
Rule 151
Rule 152
Rule 156
Rule 206
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x^4 \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^2 (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {10 c d-\frac {5 d^2 x}{2}}{x (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{24 c^2}\\ &=\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {-90 c^2 d^2+15 c d^3 x}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{1728 c^4 d}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {-405 c^3 d^3+\frac {105}{2} c^2 d^4 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{7776 c^6 d^2}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{768 c^4}+\frac {\left (5 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{20736 c^4}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{384 c^4}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{10368 c^4}\\ &=-\frac {35 d}{2592 c^4 \sqrt {c+d x^3}}+\frac {5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{31104 c^{9/2}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{384 c^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 117, normalized size = 0.82 \[ \frac {5 d x^3 \left (d x^3-8 c\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )+135 d x^3 \left (d x^3-8 c\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )+12 c \left (5 d x^3-36 c\right )}{10368 c^4 x^3 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 368, normalized size = 2.57 \[ \left [\frac {5 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 405 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} d x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 24 \, {\left (35 \, c d^{2} x^{6} - 265 \, c^{2} d x^{3} - 108 \, c^{3}\right )} \sqrt {d x^{3} + c}}{62208 \, {\left (c^{5} d^{2} x^{9} - 7 \, c^{6} d x^{6} - 8 \, c^{7} x^{3}\right )}}, -\frac {405 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + 5 \, {\left (d^{3} x^{9} - 7 \, c d^{2} x^{6} - 8 \, c^{2} d x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (35 \, c d^{2} x^{6} - 265 \, c^{2} d x^{3} - 108 \, c^{3}\right )} \sqrt {d x^{3} + c}}{31104 \, {\left (c^{5} d^{2} x^{9} - 7 \, c^{6} d x^{6} - 8 \, c^{7} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 129, normalized size = 0.90 \[ -\frac {5 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{384 \, \sqrt {-c} c^{4}} - \frac {5 \, d \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{31104 \, \sqrt {-c} c^{4}} - \frac {35 \, {\left (d x^{3} + c\right )}^{2} d - 335 \, {\left (d x^{3} + c\right )} c d + 192 \, c^{2} d}{2592 \, {\left ({\left (d x^{3} + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 9 \, \sqrt {d x^{3} + c} c^{2}\right )} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.20, size = 1019, normalized size = 7.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.56, size = 133, normalized size = 0.93 \[ -\frac {\frac {2\,d}{9\,c^2}+\frac {35\,d\,{\left (d\,x^3+c\right )}^2}{864\,c^4}-\frac {335\,d\,\left (d\,x^3+c\right )}{864\,c^3}}{3\,{\left (d\,x^3+c\right )}^{5/2}-30\,c\,{\left (d\,x^3+c\right )}^{3/2}+27\,c^2\,\sqrt {d\,x^3+c}}-\frac {d\,\left (\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{\sqrt {c^9}}\right )\,1{}\mathrm {i}+\frac {\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^9}}\right )\,1{}\mathrm {i}}{81}\right )\,5{}\mathrm {i}}{384\,\sqrt {c^9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________