Optimal. Leaf size=149 \[ -\frac{\sqrt{c+d x^4} (3 b c-2 a d)}{4 a^2 c x^2 (b c-a d)}-\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}+\frac{b \sqrt{c+d x^4}}{4 a x^2 \left (a+b x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.198745, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 472, 583, 12, 377, 205} \[ -\frac{\sqrt{c+d x^4} (3 b c-2 a d)}{4 a^2 c x^2 (b c-a d)}-\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}+\frac{b \sqrt{c+d x^4}}{4 a x^2 \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 465
Rule 472
Rule 583
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^2\right )\\ &=\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 b c+2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{4 a (b c-a d)}\\ &=-\frac{(3 b c-2 a d) \sqrt{c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{b c (3 b c-4 a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{4 a^2 c (b c-a d)}\\ &=-\frac{(3 b c-2 a d) \sqrt{c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac{(b (3 b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{4 a^2 (b c-a d)}\\ &=-\frac{(3 b c-2 a d) \sqrt{c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac{(b (3 b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^2}{\sqrt{c+d x^4}}\right )}{4 a^2 (b c-a d)}\\ &=-\frac{(3 b c-2 a d) \sqrt{c+d x^4}}{4 a^2 c (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^2 \left (a+b x^4\right )}-\frac{b (3 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{5/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [C] time = 1.56581, size = 869, normalized size = 5.83 \[ -\frac{\sqrt{d x^4+c} \left (120 d^2 \sin ^{-1}\left (\sqrt{\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}}\right ) x^8+96 d^2 \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{5/2} \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right ) x^8+32 d^2 \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{5/2} \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}} \text{HypergeometricPFQ}\left (\{2,2,3\},\left \{1,\frac{7}{2}\right \},\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right ) x^8-120 d^2 \sqrt{\frac{a (b c-a d) x^4 \left (d x^4+c\right )}{c^2 \left (b x^4+a\right )^2}} x^8+180 c d \sin ^{-1}\left (\sqrt{\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}}\right ) x^4+160 c d \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{5/2} \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right ) x^4+64 c d \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{5/2} \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}} \text{HypergeometricPFQ}\left (\{2,2,3\},\left \{1,\frac{7}{2}\right \},\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right ) x^4-180 c d \sqrt{\frac{a (b c-a d) x^4 \left (d x^4+c\right )}{c^2 \left (b x^4+a\right )^2}} x^4+45 c^2 \sin ^{-1}\left (\sqrt{\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}}\right )+64 c^2 \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{5/2} \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}} \, _2F_1\left (2,3;\frac{7}{2};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+32 c^2 \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{5/2} \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}} \text{HypergeometricPFQ}\left (\{2,2,3\},\left \{1,\frac{7}{2}\right \},\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-45 c^2 \sqrt{\frac{a (b c-a d) x^4 \left (d x^4+c\right )}{c^2 \left (b x^4+a\right )^2}}\right )}{60 c^3 x^2 \left (\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )^{3/2} \left (b x^4+a\right )^2 \sqrt{\frac{a \left (d x^4+c\right )}{c \left (b x^4+a\right )}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.015, size = 885, normalized size = 5.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84819, size = 1256, normalized size = 8.43 \begin{align*} \left [-\frac{{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{6} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \,{\left (2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2} +{\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{16 \,{\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{6} +{\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{2}\right )}}, -\frac{{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{6} +{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \arctan \left (\frac{{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d}}{2 \,{\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} +{\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) + 2 \,{\left (2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2} +{\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{8 \,{\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{6} +{\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{2}\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 = 1.60352, size = 181, normalized size = 1.21 \begin{align*} -\frac{b^{2} c \sqrt{d + \frac{c}{x^{4}}}}{4 \,{\left (a^{2} b c - a^{3} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}} + \frac{{\left (3 \, b^{2} c - 4 \, a b d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \,{\left (a^{2} b c - a^{3} d\right )} \sqrt{a b c - a^{2} d}} - \frac{\sqrt{d + \frac{c}{x^{4}}}}{2 \, a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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