Optimal. Leaf size=104 \[ \frac {b x^4 \sqrt {c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac {(b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {476, 390, 385,
211} \begin {gather*} \frac {(b c-2 a d) \text {ArcTan}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}+\frac {b x^4 \sqrt {c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 385
Rule 390
Rule 476
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^4\right )\\ &=\frac {b x^4 \sqrt {c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac {(b c-2 a d) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{8 a (b c-a d)}\\ &=\frac {b x^4 \sqrt {c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac {(b c-2 a d) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^4}{\sqrt {c+d x^8}}\right )}{8 a (b c-a d)}\\ &=\frac {b x^4 \sqrt {c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac {(b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.73, size = 124, normalized size = 1.19 \begin {gather*} -\frac {b x^4 \sqrt {c+d x^8}}{8 a (-b c+a d) \left (a+b x^8\right )}+\frac {(b c-2 a d) \tan ^{-1}\left (\frac {a \sqrt {d}+b \sqrt {d} x^8+b x^4 \sqrt {c+d x^8}}{\sqrt {a} \sqrt {b c-a d}}\right )}{8 a^{3/2} (b c-a d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx\]
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. 213 vs.
\(2 (88) = 176\).
time = 3.20, size = 467, normalized size = 4.49 \begin {gather*} \left [\frac {4 \, \sqrt {d x^{8} + c} {\left (a b^{2} c - a^{2} b d\right )} x^{4} - {\left ({\left (b^{2} c - 2 \, a b d\right )} x^{8} + a b c - 2 \, a^{2} d\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^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \, {\left ({\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{8} + a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{8} + c} {\left (a b^{2} c - a^{2} b d\right )} x^{4} + {\left ({\left (b^{2} c - 2 \, a b d\right )} x^{8} + a b c - 2 \, a^{2} d\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} + {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right )}{16 \, {\left ({\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{8} + a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{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 {x^{3}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, 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. 237 vs.
\(2 (88) = 176\).
time = 1.45, size = 237, normalized size = 2.28 \begin {gather*} -\frac {1}{8} \, d^{\frac {3}{2}} {\left (\frac {{\left (b c - 2 \, a d\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b c d - a^{2} d^{2}\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 {x^3}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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