Optimal. Leaf size=349 \[ -\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]
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Rubi [A] time = 0.27, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {413, 385, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 385
Rule 413
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 )^3} \, dx &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}+\frac {\int \frac {a (b c+7 a d)+b (5 b c+3 a d) x^4}{\left (c+d x^4\right )^2} \, dx}{8 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{c+d x^4} \, dx}{32 c^2 d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d^2}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{5/2}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{5/2}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \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}{128 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \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}{128 \sqrt {2} c^{11/4} d^{9/4}}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 319, normalized size = 0.91 \[ \frac {-\sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )+\sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )-2 \sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )-\frac {8 c^{3/4} \sqrt [4]{d} x \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right )}{c+d x^4}+\frac {32 c^{7/4} \sqrt [4]{d} x (b c-a d)^2}{\left (c+d x^4\right )^2}}{256 c^{11/4} d^{9/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.40, size = 1411, normalized size = 4.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 407, normalized size = 1.17 \[ \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac {9 \, b^{2} c^{2} d x^{5} - 2 \, a b c d^{2} x^{5} - 7 \, a^{2} d^{3} x^{5} + 5 \, b^{2} c^{3} x + 6 \, a b c^{2} d x - 11 \, a^{2} c d^{2} x}{32 \, {\left (d x^{4} + c\right )}^{2} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 499, normalized size = 1.43 \[ \frac {21 \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 )}{128 c^{3}}+\frac {21 \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 )}{128 c^{3}}+\frac {21 \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 )}{256 c^{3}}+\frac {3 \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 )}{64 c^{2} d}+\frac {3 \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 )}{64 c^{2} d}+\frac {3 \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 )}{128 c^{2} d}+\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 )}{128 c \,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 )}{128 c \,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 )}{256 c \,d^{2}}+\frac {\frac {\left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{5}}{32 c^{2} d}+\frac {\left (11 a^{2} d^{2}-6 a b c d -5 b^{2} c^{2}\right ) x}{32 c \,d^{2}}}{\left (d \,x^{4}+c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 361, normalized size = 1.03 \[ -\frac {{\left (9 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 7 \, a^{2} d^{3}\right )} x^{5} + {\left (5 \, b^{2} c^{3} + 6 \, a b c^{2} d - 11 \, a^{2} c d^{2}\right )} x}{32 \, {\left (c^{2} d^{4} x^{8} + 2 \, c^{3} d^{3} x^{4} + c^{4} d^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{256 \, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.66, size = 1401, normalized size = 4.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.85, size = 264, normalized size = 0.76 \[ \frac {x^{5} \left (7 a^{2} d^{3} + 2 a b c d^{2} - 9 b^{2} c^{2} d\right ) + x \left (11 a^{2} c d^{2} - 6 a b c^{2} d - 5 b^{2} c^{3}\right )}{32 c^{4} d^{2} + 64 c^{3} d^{3} x^{4} + 32 c^{2} d^{4} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} c^{11} d^{9} + 194481 a^{8} d^{8} + 222264 a^{7} b c d^{7} + 280476 a^{6} b^{2} c^{2} d^{6} + 176904 a^{5} b^{3} c^{3} d^{5} + 112806 a^{4} b^{4} c^{4} d^{4} + 42120 a^{3} b^{5} c^{5} d^{3} + 15900 a^{2} b^{6} c^{6} d^{2} + 3000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left (t \mapsto t \log {\left (\frac {128 t c^{3} d^{2}}{21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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