Optimal. Leaf size=87 \[ -\frac {\sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 87, 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 = {455, 44, 65,
214} \begin {gather*} \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{3/2}}-\frac {\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 44
Rule 65
Rule 214
Rule 455
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=-\frac {\sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}-\frac {d \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{16 (b c-a d)}\\ &=-\frac {\sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}-\frac {\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 c-a d)}\\ &=-\frac {\sqrt {c+d x^8}}{8 (b c-a d) \left (a+b x^8\right )}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 86, normalized size = 0.99 \begin {gather*} \frac {1}{8} \left (-\frac {\sqrt {c+d x^8}}{(b c-a d) \left (a+b x^8\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{3/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {x^{7}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (71) = 142\).
time = 2.19, size = 302, normalized size = 3.47 \begin {gather*} \left [-\frac {{\left (b d x^{8} + a 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 (b^{2} c - a b d\right )}}{16 \, {\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{8} + a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}}, -\frac {{\left (b d x^{8} + a 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 (b^{2} c - a b d\right )}}{8 \, {\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{8} + a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b 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 = 1.19, size = 93, normalized size = 1.07 \begin {gather*} -\frac {d \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, \sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x^{8} + c} d}{8 \, {\left ({\left (d x^{8} + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.80, size = 84, normalized size = 0.97 \begin {gather*} \frac {d\,\sqrt {d\,x^8+c}}{2\,\left (a\,d-b\,c\right )\,\left (4\,b\,\left (d\,x^8+c\right )+4\,a\,d-4\,b\,c\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^8+c}}{\sqrt {a\,d-b\,c}}\right )}{8\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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