Optimal. Leaf size=149 \[ -\frac {(3 b c-2 a d) \sqrt {c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {476, 483, 597,
12, 385, 211} \begin {gather*} -\frac {b (3 b c-4 a d) \text {ArcTan}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^4} (3 b c-2 a d)}{4 a^2 c x^2 (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^2 \left (a+b x^4\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 385
Rule 476
Rule 483
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^2\right )\\ &=\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac {\text {Subst}\left (\int \frac {-3 b c+2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 a (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac {\text {Subst}\left (\int \frac {b c (3 b c-4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 a^2 c (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac {(b (3 b c-4 a d)) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 a^2 (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac {(b (3 b c-4 a d)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{4 a^2 (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 157, normalized size = 1.05 \begin {gather*} \frac {\sqrt {c+d x^4} \left (2 a b c-2 a^2 d+3 b^2 c x^4-2 a b d x^4\right )}{4 a^2 c (-b c+a d) x^2 \left (a+b x^4\right )}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {a \sqrt {d}+b \sqrt {d} x^4+b x^2 \sqrt {c+d x^4}}{\sqrt {a} \sqrt {b c-a d}}\right )}{4 a^{5/2} (b c-a d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1220\) vs.
\(2(129)=258\).
time = 0.40, size = 1221, normalized size = 8.19 Too large to display
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. 286 vs.
\(2 (129) = 258\).
time = 2.76, size = 612, normalized size = 4.11 \begin {gather*} \left [-\frac {{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{6} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, {\left (2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2} + {\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{16 \, {\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{6} + {\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{2}\right )}}, -\frac {{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{6} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2} + {\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{8 \, {\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{6} + {\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{2}\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^{3} \left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs.
\(2 (129) = 258\).
time = 3.17, size = 418, normalized size = 2.81 \begin {gather*} \frac {1}{4} \, d^{\frac {5}{2}} {\left (\frac {{\left (3 \, b^{2} c - 4 \, a b d\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b^{2} c - 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} a b d - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b^{2} c^{2} + 14 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a b c d - 8 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a^{2} d^{2} + 3 \, b^{2} c^{3} - 2 \, a b c^{2} d\right )}}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a c d - b c^{3}\right )} {\left (a^{2} b c d^{2} - a^{3} d^{3}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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