Optimal. Leaf size=370 \[ -\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )-\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )-\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )+\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )+\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} c^{7/4}}+\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c} \]
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Rubi [A] time = 0.50, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1171, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )-\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )-\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )+\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (\sqrt {c} d \left (c d^2-3 a e^2\right )+\sqrt {a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} c^{7/4}}+\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1171
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3}{a+c x^4} \, dx &=\int \left (\frac {3 d e^2}{c}+\frac {e^3 x^2}{c}+\frac {c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x^2}{c \left (a+c x^4\right )}\right ) \, dx\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c}+\frac {\int \frac {c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x^2}{a+c x^4} \, dx}{c}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c}-\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c^2}+\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c^2}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c}+\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c^2}+\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c^2}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c}+\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{7/4}}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^3}{3 c}-\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (3 c d^2 e-a e^3+\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{7/4}}+\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}-\frac {\left (3 c d^2 e-a e^3-\frac {\sqrt {c} d \left (c d^2-3 a e^2\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 360, normalized size = 0.97 \begin {gather*} \frac {-3 \sqrt {2} \left (a^{3/2} e^3-3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+3 \sqrt {2} \left (a^{3/2} e^3-3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+6 \sqrt {2} \left (a^{3/2} e^3-3 \sqrt {a} c d^2 e+3 a \sqrt {c} d e^2-c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt {2} \left (-a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+72 a^{3/4} c^{3/4} d e^2 x+8 a^{3/4} c^{3/4} e^3 x^3}{24 a^{3/4} c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^3}{a+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 2.98, size = 2133, normalized size = 5.76
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 = 405, normalized size = 1.09 \begin {gather*} \frac {c^{2} x^{3} e^{3} + 9 \, c^{2} d x e^{2}}{3 \, c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 572, normalized size = 1.55 \begin {gather*} \frac {e^{3} x^{3}}{3 c}-\frac {\sqrt {2}\, a \,e^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, a \,e^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, a \,e^{3} \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 a}+\frac {3 \sqrt {2}\, d^{2} e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {3 \sqrt {2}\, d^{2} e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {3 \sqrt {2}\, d^{2} e \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {3 d \,e^{2} x}{c}-\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 c}-\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 c}-\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \,e^{2} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.48, size = 342, normalized size = 0.92 \begin {gather*} \frac {e^{3} x^{3} + 9 \, d e^{2} x}{3 \, c} + \frac {\frac {2 \, \sqrt {2} {\left (c^{\frac {3}{2}} d^{3} + 3 \, \sqrt {a} c d^{2} e - 3 \, a \sqrt {c} d e^{2} - a^{\frac {3}{2}} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (c^{\frac {3}{2}} d^{3} + 3 \, \sqrt {a} c d^{2} e - 3 \, a \sqrt {c} d e^{2} - a^{\frac {3}{2}} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (c^{\frac {3}{2}} d^{3} - 3 \, \sqrt {a} c d^{2} e - 3 \, a \sqrt {c} d e^{2} + a^{\frac {3}{2}} e^{3}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (c^{\frac {3}{2}} d^{3} - 3 \, \sqrt {a} c d^{2} e - 3 \, a \sqrt {c} d e^{2} + a^{\frac {3}{2}} e^{3}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.88, size = 2712, normalized size = 7.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.27, size = 350, normalized size = 0.95 \begin {gather*} \operatorname {RootSum} {\left (256 t^{4} a^{3} c^{7} + t^{2} \left (192 a^{4} c^{4} d e^{5} - 640 a^{3} c^{5} d^{3} e^{3} + 192 a^{2} c^{6} d^{5} e\right ) + a^{6} e^{12} + 6 a^{5} c d^{2} e^{10} + 15 a^{4} c^{2} d^{4} e^{8} + 20 a^{3} c^{3} d^{6} e^{6} + 15 a^{2} c^{4} d^{8} e^{4} + 6 a c^{5} d^{10} e^{2} + c^{6} d^{12}, \left (t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} c^{5} e^{3} + 192 t^{3} a^{3} c^{6} d^{2} e - 36 t a^{5} c^{2} d e^{8} + 336 t a^{4} c^{3} d^{3} e^{6} - 504 t a^{3} c^{4} d^{5} e^{4} + 144 t a^{2} c^{5} d^{7} e^{2} - 4 t a c^{6} d^{9}}{a^{6} e^{12} - 12 a^{5} c d^{2} e^{10} - 27 a^{4} c^{2} d^{4} e^{8} + 27 a^{2} c^{4} d^{8} e^{4} + 12 a c^{5} d^{10} e^{2} - c^{6} d^{12}} \right )} \right )\right )} + \frac {3 d e^{2} x}{c} + \frac {e^{3} x^{3}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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