Optimal. Leaf size=230 \[ \frac {b^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac {b^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac {(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
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Rubi [A] time = 0.18, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {413, 530, 240, 212, 206, 203, 377, 208, 205} \[ \frac {b^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac {b^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac {(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac {x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 413
Rule 530
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{7/4}}{\left (c+d x^4\right )^2} \, dx &=-\frac {(b c-a d) x \left (a+b x^4\right )^{3/4}}{4 c d \left (c+d x^4\right )}+\frac {\int \frac {a (b c+3 a d)+4 b^2 c x^4}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )^{3/4}}{4 c d \left (c+d x^4\right )}+\frac {b^2 \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{d^2}-\frac {((b c-a d) (4 b c+3 a d)) \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )^{3/4}}{4 c d \left (c+d x^4\right )}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{d^2}-\frac {((b c-a d) (4 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 c d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )^{3/4}}{4 c d \left (c+d x^4\right )}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {((b c-a d) (4 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c}-\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} d^2}-\frac {((b c-a d) (4 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c}+\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )^{3/4}}{4 c d \left (c+d x^4\right )}+\frac {b^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {(b c-a d)^{3/4} (4 b c+3 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}+\frac {b^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac {(b c-a d)^{3/4} (4 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}\\ \end {align*}
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Mathematica [C] time = 0.64, size = 358, normalized size = 1.56 \[ \frac {\frac {15 a^2 \left (-\log \left (\sqrt [4]{c}-\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{\sqrt [4]{b c-a d}}+\frac {16 b^2 c^{3/4} x^5 \sqrt [4]{\frac {b x^4}{a}+1} F_1\left (\frac {5}{4};\frac {1}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{d \sqrt [4]{a+b x^4}}-\frac {20 c^{3/4} x \left (a+b x^4\right )^{3/4} (b c-a d)}{d \left (c+d x^4\right )}+\frac {5 a b c \left (-\log \left (\sqrt [4]{c}-\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{d \sqrt [4]{b c-a d}}}{80 c^{7/4}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 3.26, size = 1667, normalized size = 7.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}}}{\left (d \,x^{4}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^4+a\right )}^{7/4}}{{\left (d\,x^4+c\right )}^2} \,d x \]
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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