Optimal. Leaf size=161 \[ \frac {15 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2048 c^{5/2}}-\frac {17 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{5/2}}+\frac {7 d^2 \sqrt {c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {7 d^2 \sqrt {c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}+\frac {15 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2048 c^{5/2}}-\frac {17 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{5/2}}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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^7 \left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^3 (8 c-d x)^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-23 c^2 d-\frac {37}{2} c d^2 x}{x^2 (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{48 c}\\ &=-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {102 c^3 d^2+\frac {69}{2} c^2 d^3 x}{x (8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{384 c^3}\\ &=\frac {7 d^2 \sqrt {c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {-918 c^4 d^3-189 c^3 d^4 x}{x (8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27648 c^5 d}\\ &=\frac {7 d^2 \sqrt {c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}+\frac {\left (17 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{4096 c^2}+\frac {\left (45 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{4096 c^2}\\ &=\frac {7 d^2 \sqrt {c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}+\frac {(17 d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{2048 c^2}+\frac {\left (45 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{2048 c^2}\\ &=\frac {7 d^2 \sqrt {c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 x^6 \left (8 c-d x^3\right )}-\frac {23 d \sqrt {c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}+\frac {15 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2048 c^{5/2}}-\frac {17 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.24, size = 112, normalized size = 0.70 \begin {gather*} \frac {\frac {4 \sqrt {c} \sqrt {c+d x^3} \left (32 c^2+92 c d x^3-21 d^2 x^6\right )}{d x^9-8 c x^6}+45 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-51 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 118, normalized size = 0.73 \begin {gather*} \frac {15 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2048 c^{5/2}}-\frac {17 d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2048 c^{5/2}}+\frac {\sqrt {c+d x^3} \left (-32 c^2-92 c d x^3+21 d^2 x^6\right )}{1536 c^2 x^6 \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 310, normalized size = 1.93 \begin {gather*} \left [\frac {45 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\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 ) + 51 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) - 8 \, {\left (21 \, c d^{2} x^{6} - 92 \, c^{2} d x^{3} - 32 \, c^{3}\right )} \sqrt {d x^{3} + c}}{12288 \, {\left (c^{3} d x^{9} - 8 \, c^{4} x^{6}\right )}}, \frac {51 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 45 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) - 4 \, {\left (21 \, c d^{2} x^{6} - 92 \, c^{2} d x^{3} - 32 \, c^{3}\right )} \sqrt {d x^{3} + c}}{6144 \, {\left (c^{3} d x^{9} - 8 \, c^{4} x^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 129, normalized size = 0.80 \begin {gather*} \frac {17 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{2048 \, \sqrt {-c} c^{2}} - \frac {15 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{2048 \, \sqrt {-c} c^{2}} - \frac {3 \, \sqrt {d x^{3} + c} d^{2}}{512 \, {\left (d x^{3} - 8 \, c\right )} c^{2}} - \frac {3 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 2 \, \sqrt {d x^{3} + c} c d^{2}}{384 \, c^{2} d^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.19, size = 1075, normalized size = 6.68
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 {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.62, size = 151, normalized size = 0.94 \begin {gather*} \frac {\frac {81\,d^2\,\sqrt {d\,x^3+c}}{512}-\frac {67\,d^2\,{\left (d\,x^3+c\right )}^{3/2}}{256\,c}+\frac {21\,d^2\,{\left (d\,x^3+c\right )}^{5/2}}{512\,c^2}}{33\,c\,{\left (d\,x^3+c\right )}^2-57\,c^2\,\left (d\,x^3+c\right )-3\,{\left (d\,x^3+c\right )}^3+27\,c^3}+\frac {d^2\,\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 )\,15{}\mathrm {i}}{17}\right )\,17{}\mathrm {i}}{2048\,\sqrt {c^5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________