Optimal. Leaf size=175 \[ -\frac{\sqrt{c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}-\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}+\frac{a x^8 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.173387, antiderivative size = 175, normalized size of antiderivative = 1., 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, 98, 147, 63, 208} \[ -\frac{\sqrt{c+d x^4} \left (-15 a^2 d^2-b d x^4 (2 b c-5 a d)+8 a b c d+4 b^2 c^2\right )}{12 b^3 d^2 (b c-a d)}-\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}+\frac{a x^8 \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 446
Rule 98
Rule 147
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{15}}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2 \sqrt{c+d x}} \, dx,x,x^4\right )\\ &=\frac{a x^8 \sqrt{c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{x \left (2 a c+\frac{1}{2} (-2 b c+5 a d) x\right )}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{4 b (b c-a d)}\\ &=\frac{a x^8 \sqrt{c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac{\sqrt{c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}+\frac{\left (a^2 (6 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{8 b^3 (b c-a d)}\\ &=\frac{a x^8 \sqrt{c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac{\sqrt{c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}+\frac{\left (a^2 (6 b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^4}\right )}{4 b^3 d (b c-a d)}\\ &=\frac{a x^8 \sqrt{c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac{\sqrt{c+d x^4} \left (4 b^2 c^2+8 a b c d-15 a^2 d^2-b d (2 b c-5 a d) x^4\right )}{12 b^3 d^2 (b c-a d)}-\frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.190138, size = 175, normalized size = 1. \[ \frac{a^2 (5 a d-6 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} \left (2 a^2 b d \left (4 c-5 d x^4\right )-15 a^3 d^2+2 a b^2 \left (2 c^2+3 c d x^4+d^2 x^8\right )+2 b^3 c x^4 \left (2 c-d x^4\right )\right )}{12 b^3 d^2 \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 923, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63743, size = 1270, normalized size = 7.26 \begin{align*} \left [\frac{3 \,{\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} +{\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \sqrt{b^{2} c - a b d} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \,{\left (2 \,{\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \,{\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{24 \,{\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} +{\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}, \frac{3 \,{\left (6 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3} +{\left (6 \, a^{2} b^{2} c d^{2} - 5 \, a^{3} b d^{3}\right )} x^{4}\right )} \sqrt{-b^{2} c + a b d} \arctan \left (\frac{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}{b d x^{4} + b c}\right ) +{\left (2 \,{\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{8} - 4 \, a b^{4} c^{3} - 4 \, a^{2} b^{3} c^{2} d + 23 \, a^{3} b^{2} c d^{2} - 15 \, a^{4} b d^{3} - 2 \,{\left (2 \, b^{5} c^{3} + a b^{4} c^{2} d - 8 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{12 \,{\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} +{\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.02784, size = 243, normalized size = 1.39 \begin{align*} \frac{\sqrt{d x^{4} + c} a^{3} d}{4 \,{\left (b^{4} c - a b^{3} d\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac{{\left (6 \, a^{2} b c - 5 \, a^{3} d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{4} c - a b^{3} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{{\left (d x^{4} + c\right )}^{\frac{3}{2}} b^{4} d^{4} - 3 \, \sqrt{d x^{4} + c} b^{4} c d^{4} - 6 \, \sqrt{d x^{4} + c} a b^{3} d^{5}}{6 \, b^{6} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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