Optimal. Leaf size=112 \[ \frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)}+\frac {a d+b c}{2 a^2 c^2 x^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)}-\frac {1}{6 a c x^6} \]
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Rubi [A] time = 0.22, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {465, 480, 583, 522, 205} \[ \frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)}+\frac {a d+b c}{2 a^2 c^2 x^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)}-\frac {1}{6 a c x^6} \]
Antiderivative was successfully verified.
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Rule 205
Rule 465
Rule 480
Rule 522
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{6 a c x^6}+\frac {\operatorname {Subst}\left (\int \frac {-3 (b c+a d)-3 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac {1}{6 a c x^6}+\frac {b c+a d}{2 a^2 c^2 x^2}-\frac {\operatorname {Subst}\left (\int \frac {-3 \left (b^2 c^2+a b c d+a^2 d^2\right )-3 b d (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 a^2 c^2}\\ &=-\frac {1}{6 a c x^6}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 a^2 (b c-a d)}-\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{c+d x^2} \, dx,x,x^2\right )}{2 c^2 (b c-a d)}\\ &=-\frac {1}{6 a c x^6}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 193, normalized size = 1.72 \[ \frac {\frac {3 b^{5/2} x^6 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/2}}+\frac {3 b^{5/2} x^6 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{5/2}}-\frac {3 b^2 x^4}{a^2}+\frac {b}{a}-\frac {3 d^{5/2} x^6 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/2}}-\frac {3 d^{5/2} x^6 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{5/2}}+\frac {3 d^2 x^4}{c^2}-\frac {d}{c}}{6 x^6 (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.46, size = 592, normalized size = 5.29 \[ \left [-\frac {3 \, b^{2} c^{2} x^{6} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) - 6 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, \frac {6 \, a^{2} d^{2} x^{6} \sqrt {\frac {d}{c}} \arctan \left (\frac {c \sqrt {\frac {d}{c}}}{d x^{2}}\right ) - 3 \, b^{2} c^{2} x^{6} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + 6 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} - 2 \, a b c^{2} + 2 \, a^{2} c d}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac {6 \, b^{2} c^{2} x^{6} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b x^{2}}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) - 6 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac {3 \, b^{2} c^{2} x^{6} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b x^{2}}\right ) - 3 \, a^{2} d^{2} x^{6} \sqrt {\frac {d}{c}} \arctan \left (\frac {c \sqrt {\frac {d}{c}}}{d x^{2}}\right ) - 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + a b c^{2} - a^{2} c d}{6 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 103, normalized size = 0.92 \[ \frac {b^{3} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b}} - \frac {d^{3} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )} \sqrt {c d}} + \frac {3 \, b c x^{4} + 3 \, a d x^{4} - a c}{6 \, a^{2} c^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 105, normalized size = 0.94 \[ -\frac {b^{3} \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -b c \right ) \sqrt {a b}\, a^{2}}+\frac {d^{3} \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -b c \right ) \sqrt {c d}\, c^{2}}+\frac {d}{2 a \,c^{2} x^{2}}+\frac {b}{2 a^{2} c \,x^{2}}-\frac {1}{6 a c \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 101, normalized size = 0.90 \[ \frac {b^{3} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b}} - \frac {d^{3} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )} \sqrt {c d}} + \frac {3 \, {\left (b c + a d\right )} x^{4} - a c}{6 \, a^{2} c^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.53, size = 535, normalized size = 4.78 \[ \frac {\ln \left (c^{10}\,{\left (-a^5\,b^5\right )}^{5/2}+a^{20}\,d^{10}\,\sqrt {-a^5\,b^5}-a^{12}\,b^{13}\,c^{10}\,x^2-a^{22}\,b^3\,d^{10}\,x^2+2\,a^{10}\,c^5\,d^5\,{\left (-a^5\,b^5\right )}^{3/2}+2\,a^{17}\,b^8\,c^5\,d^5\,x^2\right )\,\sqrt {-a^5\,b^5}}{4\,a^6\,d-4\,a^5\,b\,c}-\frac {\ln \left (c^{10}\,{\left (-a^5\,b^5\right )}^{5/2}+a^{20}\,d^{10}\,\sqrt {-a^5\,b^5}+a^{12}\,b^{13}\,c^{10}\,x^2+a^{22}\,b^3\,d^{10}\,x^2+2\,a^{10}\,c^5\,d^5\,{\left (-a^5\,b^5\right )}^{3/2}-2\,a^{17}\,b^8\,c^5\,d^5\,x^2\right )\,\sqrt {-a^5\,b^5}}{4\,\left (a^6\,d-a^5\,b\,c\right )}-\frac {\frac {1}{6\,a\,c}-\frac {x^4\,\left (a\,d+b\,c\right )}{2\,a^2\,c^2}}{x^6}-\frac {\ln \left (a^{10}\,{\left (-c^5\,d^5\right )}^{5/2}+b^{10}\,c^{20}\,\sqrt {-c^5\,d^5}+a^{10}\,c^{12}\,d^{13}\,x^2+b^{10}\,c^{22}\,d^3\,x^2+2\,a^5\,b^5\,c^{10}\,{\left (-c^5\,d^5\right )}^{3/2}-2\,a^5\,b^5\,c^{17}\,d^8\,x^2\right )\,\sqrt {-c^5\,d^5}}{4\,\left (b\,c^6-a\,c^5\,d\right )}+\frac {\ln \left (a^{10}\,{\left (-c^5\,d^5\right )}^{5/2}+b^{10}\,c^{20}\,\sqrt {-c^5\,d^5}-a^{10}\,c^{12}\,d^{13}\,x^2-b^{10}\,c^{22}\,d^3\,x^2+2\,a^5\,b^5\,c^{10}\,{\left (-c^5\,d^5\right )}^{3/2}+2\,a^5\,b^5\,c^{17}\,d^8\,x^2\right )\,\sqrt {-c^5\,d^5}}{4\,b\,c^6-4\,a\,c^5\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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