Optimal. Leaf size=104 \[ \frac {(b c-2 a d) \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}+\frac {b x^4 \sqrt {c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \]
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Rubi [A] time = 0.09, 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 = {465, 382, 377, 205} \[ \frac {(b c-2 a d) \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 a^{3/2} (b c-a d)^{3/2}}+\frac {b x^4 \sqrt {c+d x^8}}{8 a \left (a+b x^8\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 382
Rule 465
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{4} \operatorname {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) \operatorname {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) \operatorname {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 [C] time = 0.94, size = 407, normalized size = 3.91 \[ \frac {x^4 \sqrt {c+d x^8} \left (-30 d x^8 \sqrt {\frac {a x^8 \left (c+d x^8\right ) (b c-a d)}{c^2 \left (a+b x^8\right )^2}}-45 c \sqrt {\frac {a x^8 \left (c+d x^8\right ) (b c-a d)}{c^2 \left (a+b x^8\right )^2}}+16 d x^8 \left (\frac {x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )^{5/2} \sqrt {\frac {a \left (c+d x^8\right )}{c \left (a+b x^8\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )+16 c \left (\frac {x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )^{5/2} \sqrt {\frac {a \left (c+d x^8\right )}{c \left (a+b x^8\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^8}{c \left (b x^8+a\right )}\right )+30 d x^8 \sin ^{-1}\left (\sqrt {\frac {x^8 (b c-a d)}{c \left (a+b x^8\right )}}\right )+45 c \sin ^{-1}\left (\sqrt {\frac {x^8 (b c-a d)}{c \left (a+b x^8\right )}}\right )\right )}{120 c^2 \left (a+b x^8\right )^2 \left (\frac {x^8 (b c-a d)}{c \left (a+b x^8\right )}\right )^{3/2} \sqrt {\frac {a \left (c+d x^8\right )}{c \left (a+b x^8\right )}}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.26, size = 467, normalized size = 4.49 \[ \left [\frac {4 \, \sqrt {d x^{8} + c} {\left (a b^{2} c - a^{2} b d\right )} x^{4} - {\left ({\left (b^{2} c - 2 \, a b d\right )} x^{8} + a b c - 2 \, a^{2} d\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{32 \, {\left ({\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{8} + a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{8} + c} {\left (a b^{2} c - a^{2} b d\right )} x^{4} + {\left ({\left (b^{2} c - 2 \, a b d\right )} x^{8} + a b c - 2 \, a^{2} d\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} + {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right )}{16 \, {\left ({\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{8} + a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 237, normalized size = 2.28 \[ -\frac {1}{8} \, d^{\frac {3}{2}} {\left (\frac {{\left (b c - 2 \, a d\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b c d - a^{2} d^{2}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, 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 \[ \int \frac {x^{3}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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