Optimal. Leaf size=99 \[ \frac {a \sqrt {c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 99, 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 = {457, 79, 65,
214} \begin {gather*} \frac {a \sqrt {c+d x^8}}{8 b \left (a+b x^8\right ) (b c-a d)}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {x}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=\frac {a \sqrt {c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}+\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{16 b (b c-a d)}\\ &=\frac {a \sqrt {c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}+\frac {(2 b c-a d) \text {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 d (b c-a d)}\\ &=\frac {a \sqrt {c+d x^8}}{8 b (b c-a d) \left (a+b x^8\right )}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 100, normalized size = 1.01 \begin {gather*} \frac {\frac {a \sqrt {b} \sqrt {c+d x^8}}{(b c-a d) \left (a+b x^8\right )}-\frac {(2 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{8 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {x^{15}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.83, size = 348, normalized size = 3.52 \begin {gather*} \left [\frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{8} + 2 \, a b c - a^{2} d\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 \, \sqrt {d x^{8} + c} {\left (a b^{2} c - a^{2} b d\right )}}{16 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{8} + a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )}}, \frac {{\left ({\left (2 \, b^{2} c - a b d\right )} x^{8} + 2 \, a b c - a^{2} d\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 ) + \sqrt {d x^{8} + c} {\left (a b^{2} c - a^{2} b d\right )}}{8 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{8} + a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 0.74, size = 116, normalized size = 1.17 \begin {gather*} \frac {\frac {\sqrt {d x^{8} + c} a d^{2}}{{\left (b^{2} c - a b d\right )} {\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.84, size = 95, normalized size = 0.96 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^8+c}}{\sqrt {a\,d-b\,c}}\right )\,\left (a\,d-2\,b\,c\right )}{8\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {a\,d\,\sqrt {d\,x^8+c}}{2\,b\,\left (a\,d-b\,c\right )\,\left (4\,b\,\left (d\,x^8+c\right )+4\,a\,d-4\,b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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