Optimal. Leaf size=173 \[ \frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac {(b c-a d)^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}-\frac {(b c-a d)^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d} \]
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Rubi [A] time = 0.10, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {408, 240, 212, 206, 203, 377, 208, 205} \[ \frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac {(b c-a d)^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}-\frac {(b c-a d)^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 408
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{c+d x^4} \, dx &=\frac {b \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{d}-\frac {(b c-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 )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac {(b c-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 )}{2 \sqrt {c} d}-\frac {(b c-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 )}{2 \sqrt {c} d}\\ &=\frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac {(b c-a d)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac {(b c-a d)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 161, normalized size = 0.93 \[ \frac {5 a c x \left (a+b x^4\right )^{3/4} F_1\left (\frac {1}{4};-\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (3 b c F_1\left (\frac {5}{4};\frac {1}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )-4 a d F_1\left (\frac {5}{4};-\frac {3}{4},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};-\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.06, size = 844, normalized size = 4.88 \[ \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {c d x \sqrt {\frac {{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} \sqrt {\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b x^{4} + a}}{x^{2}}} \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {1}{4}} - {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {1}{4}}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}\right ) + \left (\frac {b^{3}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} d \left (\frac {b^{3}}{d^{4}}\right )^{\frac {1}{4}} - d x \left (\frac {b^{3}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {b^{3} d^{2} x^{2} \sqrt {\frac {b^{3}}{d^{4}}} + \sqrt {b x^{4} + a} b^{4}}{x^{2}}}}{b^{3} x}\right ) - \frac {1}{4} \, \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c^{2} d^{3} x \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{x}\right ) + \frac {1}{4} \, \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c^{2} d^{3} x \left (\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{c^{3} d^{4}}\right )^{\frac {3}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{x}\right ) + \frac {1}{4} \, \left (\frac {b^{3}}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d^{3} x \left (\frac {b^{3}}{d^{4}}\right )^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}}{x}\right ) - \frac {1}{4} \, \left (\frac {b^{3}}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d^{3} x \left (\frac {b^{3}}{d^{4}}\right )^{\frac {3}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}}{x}\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{d x^{4} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{d \,x^{4}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{d x^{4} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/4}}{d\,x^4+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{4}\right )^{\frac {3}{4}}}{c + d x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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