Optimal. Leaf size=291 \[ \frac {(b c-a d) (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}+\frac {x (b c-a d)^2}{4 c d^2 \left (c+d x^4\right )}+\frac {b^2 x}{d^2} \]
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Rubi [A] time = 0.37, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {390, 385, 211, 1165, 628, 1162, 617, 204} \[ \frac {(b c-a d) (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}+\frac {x (b c-a d)^2}{4 c d^2 \left (c+d x^4\right )}+\frac {b^2 x}{d^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 385
Rule 390
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^2} \, dx &=\int \left (\frac {b^2}{d^2}-\frac {b^2 c^2-a^2 d^2+2 b d (b c-a d) x^4}{d^2 \left (c+d x^4\right )^2}\right ) \, dx\\ &=\frac {b^2 x}{d^2}-\frac {\int \frac {b^2 c^2-a^2 d^2+2 b d (b c-a d) x^4}{\left (c+d x^4\right )^2} \, dx}{d^2}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 x}{4 c d^2 \left (c+d x^4\right )}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {1}{c+d x^4} \, dx}{4 c d^2}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 x}{4 c d^2 \left (c+d x^4\right )}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{8 c^{3/2} d^2}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{8 c^{3/2} d^2}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 x}{4 c d^2 \left (c+d x^4\right )}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{16 c^{3/2} d^{5/2}}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{16 c^{3/2} d^{5/2}}+\frac {((b c-a d) (5 b c+3 a d)) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{16 \sqrt {2} c^{7/4} d^{9/4}}+\frac {((b c-a d) (5 b c+3 a d)) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{16 \sqrt {2} c^{7/4} d^{9/4}}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 x}{4 c d^2 \left (c+d x^4\right )}+\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}-\frac {((b c-a d) (5 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}+\frac {((b c-a d) (5 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}\\ &=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 x}{4 c d^2 \left (c+d x^4\right )}+\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{16 \sqrt {2} c^{7/4} d^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 298, normalized size = 1.02 \[ \frac {\frac {\sqrt {2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{c^{7/4}}-\frac {\sqrt {2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{c^{7/4}}+\frac {2 \sqrt {2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac {2 \sqrt {2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{7/4}}+\frac {8 \sqrt [4]{d} x (b c-a d)^2}{c \left (c+d x^4\right )}+32 b^2 \sqrt [4]{d} x}{32 d^{9/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.27, size = 1335, normalized size = 4.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 376, normalized size = 1.29 \[ \frac {b^{2} x}{d^{2}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{16 \, c^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{16 \, c^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{32 \, c^{2} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{32 \, c^{2} d^{3}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{4 \, {\left (d x^{4} + c\right )} c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 475, normalized size = 1.63 \[ \frac {a^{2} x}{4 \left (d \,x^{4}+c \right ) c}-\frac {a b x}{2 \left (d \,x^{4}+c \right ) d}+\frac {b^{2} c x}{4 \left (d \,x^{4}+c \right ) d^{2}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{16 c^{2}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{16 c^{2}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{32 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 c d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 c d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{16 c d}+\frac {b^{2} x}{d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{16 d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{16 d^{2}}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{32 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 319, normalized size = 1.10 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{4 \, {\left (c d^{3} x^{4} + c^{2} d^{2}\right )}} + \frac {b^{2} x}{d^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{32 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 1254, normalized size = 4.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 219, normalized size = 0.75 \[ \frac {b^{2} x}{d^{2}} + \frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{4 c^{2} d^{2} + 4 c d^{3} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} c^{7} d^{9} + 81 a^{8} d^{8} + 216 a^{7} b c d^{7} - 324 a^{6} b^{2} c^{2} d^{6} - 984 a^{5} b^{3} c^{3} d^{5} + 646 a^{4} b^{4} c^{4} d^{4} + 1640 a^{3} b^{5} c^{5} d^{3} - 900 a^{2} b^{6} c^{6} d^{2} - 1000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left (t \mapsto t \log {\left (\frac {16 t c^{2} d^{2}}{3 a^{2} d^{2} + 2 a b c d - 5 b^{2} c^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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