Optimal. Leaf size=123 \[ -\frac {a^2 \sqrt {c+d x^8}}{8 b^2 \left (a+b x^8\right ) (b c-a d)}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^8}}{4 b^2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 123, 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 = {446, 89, 80, 63, 208} \[ -\frac {a^2 \sqrt {c+d x^8}}{8 b^2 \left (a+b x^8\right ) (b c-a d)}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^8}}{4 b^2 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{23}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=-\frac {a^2 \sqrt {c+d x^8}}{8 b^2 (b c-a d) \left (a+b x^8\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} a (2 b c-a d)+b (b c-a d) x}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{8 b^2 (b c-a d)}\\ &=\frac {\sqrt {c+d x^8}}{4 b^2 d}-\frac {a^2 \sqrt {c+d x^8}}{8 b^2 (b c-a d) \left (a+b x^8\right )}-\frac {(a (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{16 b^2 (b c-a d)}\\ &=\frac {\sqrt {c+d x^8}}{4 b^2 d}-\frac {a^2 \sqrt {c+d x^8}}{8 b^2 (b c-a d) \left (a+b x^8\right )}-\frac {(a (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^8}\right )}{8 b^2 d (b c-a d)}\\ &=\frac {\sqrt {c+d x^8}}{4 b^2 d}-\frac {a^2 \sqrt {c+d x^8}}{8 b^2 (b c-a d) \left (a+b x^8\right )}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 107, normalized size = 0.87 \[ \frac {1}{8} \left (\frac {\sqrt {c+d x^8} \left (\frac {a^2}{\left (a+b x^8\right ) (a d-b c)}+\frac {2}{d}\right )}{b^2}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 475, normalized size = 3.86 \[ \left [\frac {{\left ({\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{8} + 4 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{8} + 2 \, b c - a d + 2 \, \sqrt {d x^{8} + c} \sqrt {b^{2} c - a b d}}{b x^{8} + a}\right ) + 2 \, {\left (2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{8} + 2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} \sqrt {d x^{8} + c}}{16 \, {\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} + {\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{8}\right )}}, -\frac {{\left ({\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{8} + 4 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-b^{2} c + a b d}}{b d x^{8} + b c}\right ) - {\left (2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{8} + 2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} \sqrt {d x^{8} + c}}{8 \, {\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} + {\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{8}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 134, normalized size = 1.09 \[ -\frac {\sqrt {d x^{8} + c} a^{2} d}{8 \, {\left (b^{3} c - a b^{2} d\right )} {\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}} - \frac {{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {\sqrt {d x^{8} + c}}{4 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {x^{23}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.02, size = 144, normalized size = 1.17 \[ \frac {\sqrt {d\,x^8+c}}{4\,b^2\,d}-\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^8+c}\,\left (3\,a\,d-4\,b\,c\right )}{\left (3\,a^2\,d-4\,a\,b\,c\right )\,\sqrt {a\,d-b\,c}}\right )\,\left (3\,a\,d-4\,b\,c\right )}{8\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{3/2}}+\frac {a^2\,d\,\sqrt {d\,x^8+c}}{2\,\left (a\,d-b\,c\right )\,\left (4\,b^3\,\left (d\,x^8+c\right )-4\,b^3\,c+4\,a\,b^2\,d\right )} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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