Optimal. Leaf size=87 \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]
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Rubi [A] time = 0.07, 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 = {444, 51, 63, 208} \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 \sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 444
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {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 \operatorname {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 {\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 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 = 0.10, size = 85, normalized size = 0.98 \[ \frac {1}{8} \left (\frac {\sqrt {c+d x^8}}{\left (a+b x^8\right ) (a d-b c)}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {a d-b c}}\right )}{\sqrt {b} (a d-b c)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 302, normalized size = 3.47 \[ \left [-\frac {{\left (b d x^{8} + a d\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{8} + 2 \, b c - a d - 2 \, \sqrt {d x^{8} + c} \sqrt {b^{2} c - a b d}}{b x^{8} + a}\right ) + 2 \, \sqrt {d x^{8} + c} {\left (b^{2} c - a b d\right )}}{16 \, {\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{8} + a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}}, -\frac {{\left (b d x^{8} + a d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-b^{2} c + a b d}}{b d x^{8} + b c}\right ) + \sqrt {d x^{8} + c} {\left (b^{2} c - a b d\right )}}{8 \, {\left ({\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{8} + a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 93, normalized size = 1.07 \[ -\frac {d \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, \sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x^{8} + c} d}{8 \, {\left ({\left (d x^{8} + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \]
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 \[ \int \frac {x^{7}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \frac {d\,\sqrt {d\,x^8+c}}{2\,\left (a\,d-b\,c\right )\,\left (4\,b\,\left (d\,x^8+c\right )+4\,a\,d-4\,b\,c\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^8+c}}{\sqrt {a\,d-b\,c}}\right )}{8\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}} \]
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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