Optimal. Leaf size=93 \[ \frac {c \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 \sqrt {a} (b c-a d)^{3/2}}-\frac {x^4 \sqrt {c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]
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Rubi [A] time = 0.09, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {465, 471, 12, 377, 205} \begin {gather*} \frac {c \tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 \sqrt {a} (b c-a d)^{3/2}}-\frac {x^4 \sqrt {c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 465
Rule 471
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^4\right )\\ &=-\frac {x^4 \sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {\operatorname {Subst}\left (\int \frac {c}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{8 (b c-a d)}\\ &=-\frac {x^4 \sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {c \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^4\right )}{8 (b c-a d)}\\ &=-\frac {x^4 \sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {c \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 (b c-a d)}\\ &=-\frac {x^4 \sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {c \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^4}{\sqrt {a} \sqrt {c+d x^8}}\right )}{8 \sqrt {a} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 124, normalized size = 1.33 \begin {gather*} \frac {\sqrt {c+d x^8} \left (-\frac {x^8 (b c-a d)}{a+b x^8}-\frac {c \sqrt {x^8 \left (\frac {d}{c}-\frac {b}{a}\right )} \tanh ^{-1}\left (\frac {\sqrt {x^8 \left (\frac {d}{c}-\frac {b}{a}\right )}}{\sqrt {\frac {d x^8}{c}+1}}\right )}{\sqrt {\frac {d x^8}{c}+1}}\right )}{8 x^4 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.82, size = 145, normalized size = 1.56 \begin {gather*} \frac {c \tan ^{-1}\left (\frac {b \sqrt {d} x^8}{\sqrt {a} \sqrt {b c-a d}}+\frac {b x^4 \sqrt {c+d x^8}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{8 \sqrt {a} (b c-a d)^{3/2}}-\frac {x^4 \sqrt {c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 426, normalized size = 4.58 \begin {gather*} \left [-\frac {4 \, \sqrt {d x^{8} + c} {\left (a b c - a^{2} d\right )} x^{4} - {\left (b c x^{8} + a c\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 b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{8} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}, -\frac {2 \, \sqrt {d x^{8} + c} {\left (a b c - a^{2} d\right )} x^{4} - {\left (b c x^{8} + a c\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 b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{8} + a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.59, size = 244, normalized size = 2.62 \begin {gather*} \frac {c \sqrt {d} \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 )}{8 \, \sqrt {a b c d - a^{2} d^{2}} {\left (b c - a d\right )}} + \frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{4 \, {\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 (b^{2} c - a b d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{11}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx \end {gather*}
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*} \int \frac {x^{11}}{{\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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{11}}{{\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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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