Optimal. Leaf size=359 \[ -\frac {d x}{c e^2}+\frac {x^3}{3 c e}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (c d^2+a e^2\right )}-\frac {a^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {a^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )} \]
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Rubi [A]
time = 0.23, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1302, 211,
1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {a^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}+\frac {a^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}-\frac {a^{5/4} \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}+\frac {a^{5/4} \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}+\frac {d^{7/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (a e^2+c d^2\right )}-\frac {d x}{c e^2}+\frac {x^3}{3 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1302
Rubi steps
\begin {align*} \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\int \left (-\frac {d}{c e^2}+\frac {x^2}{c e}+\frac {d^4}{e^2 \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {a^2 \left (d-e x^2\right )}{c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {d x}{c e^2}+\frac {x^3}{3 c e}+\frac {a^2 \int \frac {d-e x^2}{a+c x^4} \, dx}{c \left (c d^2+a e^2\right )}+\frac {d^4 \int \frac {1}{d+e x^2} \, dx}{e^2 \left (c d^2+a e^2\right )}\\ &=-\frac {d x}{c e^2}+\frac {x^3}{3 c e}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (c d^2+a e^2\right )}+\frac {\left (a^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c^2 \left (c d^2+a e^2\right )}+\frac {\left (a^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c^2 \left (c d^2+a e^2\right )}\\ &=-\frac {d x}{c e^2}+\frac {x^3}{3 c e}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (c d^2+a e^2\right )}+\frac {\left (a^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\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 \left (c d^2+a e^2\right )}+\frac {\left (a^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\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 \left (c d^2+a e^2\right )}-\frac {\left (a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right )\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} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\left (a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right )\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} c^{7/4} \left (c d^2+a e^2\right )}\\ &=-\frac {d x}{c e^2}+\frac {x^3}{3 c e}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (c d^2+a e^2\right )}-\frac {a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\left (a^{7/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \text {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} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\left (a^{7/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \text {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} c^{7/4} \left (c d^2+a e^2\right )}\\ &=-\frac {d x}{c e^2}+\frac {x^3}{3 c e}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} \left (c d^2+a e^2\right )}-\frac {a^{7/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {a^{7/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {a^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 344, normalized size = 0.96 \begin {gather*} \frac {-24 c^{3/4} d \sqrt {e} \left (c d^2+a e^2\right ) x+8 c^{3/4} e^{3/2} \left (c d^2+a e^2\right ) x^3+24 c^{7/4} d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+6 \sqrt {2} a^{5/4} e^{5/2} \left (-\sqrt {c} d+\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-6 \sqrt {2} a^{5/4} e^{5/2} \left (-\sqrt {c} d+\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-3 \sqrt {2} a e^{5/2} \left (\sqrt [4]{a} \sqrt {c} d+a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+3 \sqrt {2} a e^{5/2} \left (\sqrt [4]{a} \sqrt {c} d+a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{24 c^{7/4} e^{5/2} \left (c d^2+a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 279, normalized size = 0.78
method | result | size |
default | \(-\frac {-\frac {1}{3} e \,x^{3}+d x}{c \,e^{2}}+\frac {d^{4} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {d e}}+\frac {a^{2} \left (\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}-\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) c}\) | \(279\) |
risch | \(\frac {x^{3}}{3 c e}-\frac {d x}{c \,e^{2}}+\frac {\sqrt {-d e}\, d^{3} \ln \left (\left (-16 \left (-d e \right )^{\frac {5}{2}} a \,c^{7} d^{12} e^{2}+16 \left (-d e \right )^{\frac {5}{2}} c^{8} d^{14}+14 a^{4} c^{4} d^{7} e^{9} \left (-d e \right )^{\frac {3}{2}}-4 a^{3} c^{5} d^{9} e^{7} \left (-d e \right )^{\frac {3}{2}}-2 a^{2} c^{6} d^{11} e^{5} \left (-d e \right )^{\frac {3}{2}}+16 c^{8} d^{15} \left (-d e \right )^{\frac {3}{2}} e -\sqrt {-d e}\, a^{8} e^{18}-2 \sqrt {-d e}\, a^{7} c \,d^{2} e^{16}-\sqrt {-d e}\, a^{6} c^{2} d^{4} e^{14}-2 \sqrt {-d e}\, a^{4} c^{4} d^{8} e^{10}-4 \sqrt {-d e}\, a^{3} c^{5} d^{10} e^{8}-2 \sqrt {-d e}\, a^{2} c^{6} d^{12} e^{6}\right ) x +a^{8} d \,e^{18}+2 a^{7} c \,d^{3} e^{16}+a^{6} c^{2} d^{5} e^{14}+16 a^{4} c^{4} d^{9} e^{10}+16 a \,c^{7} d^{15} e^{4}\right )}{2 e^{3} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {\sqrt {-d e}\, d^{3} \ln \left (\left (16 \left (-d e \right )^{\frac {5}{2}} a \,c^{7} d^{12} e^{2}-16 \left (-d e \right )^{\frac {5}{2}} c^{8} d^{14}-14 a^{4} c^{4} d^{7} e^{9} \left (-d e \right )^{\frac {3}{2}}+4 a^{3} c^{5} d^{9} e^{7} \left (-d e \right )^{\frac {3}{2}}+2 a^{2} c^{6} d^{11} e^{5} \left (-d e \right )^{\frac {3}{2}}-16 c^{8} d^{15} \left (-d e \right )^{\frac {3}{2}} e +\sqrt {-d e}\, a^{8} e^{18}+2 \sqrt {-d e}\, a^{7} c \,d^{2} e^{16}+\sqrt {-d e}\, a^{6} c^{2} d^{4} e^{14}+2 \sqrt {-d e}\, a^{4} c^{4} d^{8} e^{10}+4 \sqrt {-d e}\, a^{3} c^{5} d^{10} e^{8}+2 \sqrt {-d e}\, a^{2} c^{6} d^{12} e^{6}\right ) x +a^{8} d \,e^{18}+2 a^{7} c \,d^{3} e^{16}+a^{6} c^{2} d^{5} e^{14}+16 a^{4} c^{4} d^{9} e^{10}+16 a \,c^{7} d^{15} e^{4}\right )}{2 e^{3} \left (a \,e^{2}+c \,d^{2}\right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (\left (a^{2} c^{3} e^{4}+2 a \,c^{4} d^{2} e^{2}+c^{5} d^{4}\right ) \textit {\_Z}^{4}-4 a^{3} c^{2} d \,e^{5} \textit {\_Z}^{2}+a^{5} e^{8}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} c^{3} e^{7}-2 a^{2} c^{4} d^{2} e^{5}+2 a \,c^{5} d^{4} e^{3}+2 c^{6} d^{6} e \right ) \textit {\_R}^{5}+\left (7 a^{4} c^{2} d \,e^{8}-2 a^{3} c^{3} d^{3} e^{6}-a^{2} c^{4} d^{5} e^{4}+8 c^{6} d^{9}\right ) \textit {\_R}^{3}+\left (-2 a^{6} e^{11}-4 a^{2} c^{4} d^{8} e^{3}\right ) \textit {\_R} \right ) x +\left (3 a^{3} c^{3} d^{2} e^{6}+6 a^{2} c^{4} d^{4} e^{4}+3 a \,c^{5} d^{6} e^{2}\right ) \textit {\_R}^{4}+\left (-a^{5} c d \,e^{9}-a^{4} c^{2} d^{3} e^{7}-16 a^{3} c^{3} d^{5} e^{5}+16 a^{2} c^{4} d^{7} e^{3}-4 a \,c^{5} d^{9} e \right ) \textit {\_R}^{2}+4 a^{5} c \,d^{4} e^{8}-4 a^{4} c^{2} d^{6} e^{6}\right )}{4 e^{2} c}\) | \(984\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 291, normalized size = 0.81 \begin {gather*} \frac {d^{\frac {7}{2}} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{c d^{2} e^{2} + a e^{4}} + \frac {a^{2} {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\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 (\sqrt {c} d - \sqrt {a} e\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 (\sqrt {c} d + \sqrt {a} e\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 (\sqrt {c} d + \sqrt {a} e\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}}}\right )}}{8 \, {\left (c^{2} d^{2} + a c e^{2}\right )}} + \frac {{\left (x^{3} e - 3 \, d x\right )} e^{\left (-2\right )}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2061 vs.
\(2 (263) = 526\).
time = 5.53, size = 4154, normalized size = 11.57 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.00, size = 363, normalized size = 1.01 \begin {gather*} \frac {d^{\frac {7}{2}} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{c d^{2} e^{2} + a e^{4}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} a e\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 )}{2 \, {\left (\sqrt {2} c^{5} d^{2} + \sqrt {2} a c^{4} e^{2}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} a e\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 )}{2 \, {\left (\sqrt {2} c^{5} d^{2} + \sqrt {2} a c^{4} e^{2}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} a e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} c^{5} d^{2} + \sqrt {2} a c^{4} e^{2}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} a e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} c^{5} d^{2} + \sqrt {2} a c^{4} e^{2}\right )}} + \frac {{\left (c^{2} x^{3} e^{2} - 3 \, c^{2} d x e\right )} e^{\left (-3\right )}}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.06, size = 2500, normalized size = 6.96 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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