Optimal. Leaf size=158 \[ -\frac {d (b c-3 a d)}{3 a c^2 (b c-a d) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \sqrt {c+d x^3}}+\frac {(2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2 c^{5/2}}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 157,
162, 65, 214} \begin {gather*} -\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}}+\frac {(3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2 c^{5/2}}-\frac {d (b c-3 a d)}{3 a c^2 \sqrt {c+d x^3} (b c-a d)}-\frac {1}{3 a c x^3 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 157
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{3 a c x^3 \sqrt {c+d x^3}}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (2 b c+3 a d)+\frac {3 b d x}{2}}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac {d (b c-3 a d)}{3 a c^2 (b c-a d) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \sqrt {c+d x^3}}+\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{4} (b c-a d) (2 b c+3 a d)-\frac {1}{4} b d (b c-3 a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a c^2 (b c-a d)}\\ &=-\frac {d (b c-3 a d)}{3 a c^2 (b c-a d) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \sqrt {c+d x^3}}+\frac {b^3 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2 (b c-a d)}-\frac {(2 b c+3 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^2 c^2}\\ &=-\frac {d (b c-3 a d)}{3 a c^2 (b c-a d) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \sqrt {c+d x^3}}+\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^2 d (b c-a d)}-\frac {(2 b c+3 a d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^2 c^2 d}\\ &=-\frac {d (b c-3 a d)}{3 a c^2 (b c-a d) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \sqrt {c+d x^3}}+\frac {(2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^2 c^{5/2}}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 142, normalized size = 0.90 \begin {gather*} \frac {\frac {a \left (-b c \left (c+d x^3\right )+a d \left (c+3 d x^3\right )\right )}{c^2 (b c-a d) x^3 \sqrt {c+d x^3}}-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{c^{5/2}}}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.45, size = 575, normalized size = 3.64
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{3}+c}}{3 c^{2} a \,x^{3}}-\frac {-\frac {2 \left (3 a d +2 b c \right ) \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 a \sqrt {c}}-\frac {2 i b^{3} c^{2} \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {b \left (2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d \right )}{2 d \left (a d -b c \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \left (a d -b c \right ) \sqrt {d \,x^{3}+c}}\right )}{3 a \left (a d -b c \right ) d^{2}}+\frac {4 a \,d^{2}}{3 \left (a d -b c \right ) \sqrt {d \,x^{3}+c}}}{2 a \,c^{2}}\) | \(531\) |
default | \(\frac {b^{2} \left (-\frac {2}{3 \left (a d -b c \right ) \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {i b \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {b \left (2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d \right )}{2 d \left (a d -b c \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \left (-a d +b c \right ) \left (a d -b c \right ) \sqrt {d \,x^{3}+c}}\right )}{3 d^{2}}\right )}{a^{2}}+\frac {-\frac {2 d}{3 c^{2} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {\sqrt {d \,x^{3}+c}}{3 c^{2} x^{3}}+\frac {d \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}}}}{a}-\frac {b \left (\frac {2}{3 c \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {2 \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{a^{2}}\) | \(575\) |
elliptic | \(\text {Expression too large to display}\) | \(1682\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs.
\(2 (130) = 260\).
time = 3.88, size = 1120, normalized size = 7.09 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c^{3} d x^{6} + b^{2} c^{4} x^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) - {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{6} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{6} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{3}\right )}}, -\frac {4 \, {\left (b^{2} c^{3} d x^{6} + b^{2} c^{4} x^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) - {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{6} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 2 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{6} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{3}\right )}}, -\frac {{\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{6} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (b^{2} c^{3} d x^{6} + b^{2} c^{4} x^{3}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) + {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{6} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{3}\right )}}, -\frac {2 \, {\left (b^{2} c^{3} d x^{6} + b^{2} c^{4} x^{3}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) + {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{6} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{6} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a + b x^{3}\right ) \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 173, normalized size = 1.09 \begin {gather*} \frac {2 \, b^{3} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {{\left (d x^{3} + c\right )} b c d - 3 \, {\left (d x^{3} + c\right )} a d^{2} + 2 \, a c d^{2}}{3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - \sqrt {d x^{3} + c} c\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{2} \sqrt {-c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.47, size = 597, normalized size = 3.78 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )\,{\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}^3}{x^6}\right )\,\left (3\,a\,d+2\,b\,c\right )}{6\,a^2\,c^{5/2}}-\frac {\sqrt {d\,x^3+c}}{3\,a\,c^2\,x^3}-\frac {\frac {c\,\left (\frac {c\,\left (\frac {c\,\left (\frac {3\,a^2\,d^4+24\,a\,b\,c\,d^3+15\,b^2\,c^2\,d^2}{8\,a^3\,c^5}+\frac {c\,\left (\frac {c\,\left (\frac {3\,b^2\,d^4}{8\,a^3\,c^5}+\frac {b^2\,d^4\,\left (5\,a\,d-3\,b\,c\right )}{8\,a^3\,c^4\,\left (b\,c^2-a\,c\,d\right )}-\frac {b\,d^4\,\left (a\,d+2\,b\,c\right )\,\left (5\,a\,d-3\,b\,c\right )}{4\,a^3\,c^5\,\left (b\,c^2-a\,c\,d\right )}\right )}{d}-\frac {3\,b\,d^3\,\left (a\,d+2\,b\,c\right )}{4\,a^3\,c^5}+\frac {d\,\left (5\,a\,d-3\,b\,c\right )\,\left (3\,a^2\,d^4+24\,a\,b\,c\,d^3+15\,b^2\,c^2\,d^2\right )}{24\,a^3\,c^5\,\left (b\,c^2-a\,c\,d\right )}\right )}{d}-\frac {d^2\,\left (5\,a\,d-3\,b\,c\right )\,\left (6\,a^2\,d^2+14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{12\,a^3\,c^4\,\left (b\,c^2-a\,c\,d\right )}\right )}{d}-\frac {d\,\left (6\,a^2\,d^2+14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{4\,a^3\,c^4}+\frac {d^2\,\left (5\,a\,d-3\,b\,c\right )\,\left (13\,a\,d+18\,b\,c\right )}{24\,a^2\,c^3\,\left (b\,c^2-a\,c\,d\right )}\right )}{d}+\frac {d\,\left (13\,a\,d+18\,b\,c\right )}{8\,a^2\,c^3}-\frac {d\,\left (3\,a\,d+2\,b\,c\right )\,\left (5\,a\,d-3\,b\,c\right )}{6\,a^2\,c^2\,\left (b\,c^2-a\,c\,d\right )}\right )}{d}-\frac {3\,a\,d+2\,b\,c}{2\,a^2\,c^2}}{\sqrt {d\,x^3+c}}+\frac {b^{5/2}\,\ln \left (\frac {a\,d-2\,b\,c-b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,1{}\mathrm {i}}{3\,a^2\,{\left (a\,d-b\,c\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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