Optimal. Leaf size=185 \[ -\frac {b (2 b c-a d) \sqrt {c+d x^3}}{3 a^2 c (b c-a d) \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 185, 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, 156,
162, 65, 214} \begin {gather*} -\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 (b c-a d)^{3/2}}+\frac {(a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{3/2}}-\frac {b \sqrt {c+d x^3} (2 b c-a d)}{3 a^2 c \left (a+b x^3\right ) (b c-a d)}-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (4 b c+a d)+\frac {3 b d x}{2}}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^3}}{3 a^2 c (b c-a d) \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (b c-a d) (4 b c+a d)+\frac {1}{2} b d (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2 c (b c-a d)}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^3}}{3 a^2 c (b c-a d) \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )}+\frac {\left (b^2 (4 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3 (b c-a d)}-\frac {(4 b c+a d) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3 c}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^3}}{3 a^2 c (b c-a d) \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )}+\frac {\left (b^2 (4 b c-5 a d)\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^3 d (b c-a d)}-\frac {(4 b c+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^3 c d}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^3}}{3 a^2 c (b c-a d) \left (a+b x^3\right )}-\frac {\sqrt {c+d x^3}}{3 a c x^3 \left (a+b x^3\right )}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 163, normalized size = 0.88 \begin {gather*} \frac {\frac {a \sqrt {c+d x^3} \left (-a^2 d+2 b^2 c x^3+a b \left (c-d x^3\right )\right )}{c (-b c+a d) x^3 \left (a+b x^3\right )}-\frac {b^{3/2} (4 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{c^{3/2}}}{3 a^3} \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 = 0.49, size = 961, normalized size = 5.19
method | result | size |
risch | \(\text {Expression too large to display}\) | \(955\) |
default | \(\text {Expression too large to display}\) | \(961\) |
elliptic | \(\text {Expression too large to display}\) | \(1733\) |
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 [A]
time = 2.70, size = 1236, normalized size = 6.68 \begin {gather*} \left [\frac {{\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{6} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} 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 (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{6} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} 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^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{6} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} 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 (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{6} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} 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^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{6} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - {\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{6} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} 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 ) + 2 \, {\left (a^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )}}, -\frac {{\left ({\left (4 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d\right )} x^{6} + {\left (4 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} 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 (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{6} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) + {\left (a^{2} b c^{2} - a^{3} c d + {\left (2 \, a b^{2} c^{2} - a^{2} b c d\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} + {\left (a^{4} b c^{3} - a^{5} c^{2} 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 )^{2} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 257, normalized size = 1.39 \begin {gather*} \frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x^{3} + c} b^{2} c^{2} d - {\left (d x^{3} + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x^{3} + c} a b c d^{2} - \sqrt {d x^{3} + c} a^{2} d^{3}}{3 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x^{3} + c\right )}^{2} b - 2 \, {\left (d x^{3} + c\right )} b c + b c^{2} + {\left (d x^{3} + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{3} \sqrt {-c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.55, size = 355, normalized size = 1.92 \begin {gather*} \frac {\sqrt {d\,x^3+c}\,\left (\frac {d\,a^2+4\,b\,c\,a}{2\,a^3\,c^2}-\frac {a\,\left (\frac {2\,c\,b^2+2\,a\,d\,b}{2\,a^3\,c^2}-\frac {a\,\left (\frac {b^2\,d}{2\,a^3\,c^2}+\frac {b\,\left (2\,c\,b^2+2\,a\,d\,b\right )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}-\frac {b^2\,d\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^2\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}+\frac {b\,\left (d\,a^2+4\,b\,c\,a\right )\,\left (3\,a\,d-4\,b\,c\right )}{6\,a^3\,c^2\,\left (a^2\,d-a\,b\,c\right )}\right )}{b}\right )}{b\,x^3+a}-\frac {\sqrt {d\,x^3+c}}{3\,a^2\,c\,x^3}+\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 (a\,d+4\,b\,c\right )}{6\,a^3\,c^{3/2}}+\frac {b^{3/2}\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (5\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,a^3\,{\left (a\,d-b\,c\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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