Optimal. Leaf size=165 \[ -\frac {3 a^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4+b}}{\sqrt {a x^4+b}-\sqrt {a} x^2}\right )}{2 \sqrt {2} b}-\frac {3 a^{7/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^4+b}}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^4+b}}\right )}{2 \sqrt {2} b}+\frac {\left (b-6 a x^4\right ) \left (a x^4+b\right )^{3/4}}{7 b x^7} \]
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Rubi [A] time = 0.29, antiderivative size = 255, normalized size of antiderivative = 1.55, number of steps used = 13, number of rules used = 10, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {580, 583, 12, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {3 a^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt {2} b}-\frac {3 a^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}+1\right )}{2 \sqrt {2} b}+\frac {3 a^{7/4} \log \left (-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1\right )}{4 \sqrt {2} b}-\frac {3 a^{7/4} \log \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1\right )}{4 \sqrt {2} b}+\frac {\left (a x^4+b\right )^{3/4}}{7 x^7}-\frac {6 a \left (a x^4+b\right )^{3/4}}{7 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 211
Rule 377
Rule 580
Rule 583
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8 \left (b+2 a x^4\right )} \, dx &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}+\frac {\int \frac {18 a b^2+15 a^2 b x^4}{x^4 \sqrt [4]{b+a x^4} \left (b+2 a x^4\right )} \, dx}{7 b}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}-\frac {\int \frac {63 a^2 b^3}{\sqrt [4]{b+a x^4} \left (b+2 a x^4\right )} \, dx}{21 b^3}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}-\left (3 a^2\right ) \int \frac {1}{\sqrt [4]{b+a x^4} \left (b+2 a x^4\right )} \, dx\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}-\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}-\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {a} x^2}{b+a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {a} x^2}{b+a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}-\frac {\left (3 a^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {a}}-\frac {\sqrt {2} x}{\sqrt [4]{a}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 b}-\frac {\left (3 a^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {a}}+\frac {\sqrt {2} x}{\sqrt [4]{a}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 b}+\frac {\left (3 a^{7/4}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{a}}+2 x}{-\frac {1}{\sqrt {a}}-\frac {\sqrt {2} x}{\sqrt [4]{a}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {2} b}+\frac {\left (3 a^{7/4}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{a}}-2 x}{-\frac {1}{\sqrt {a}}+\frac {\sqrt {2} x}{\sqrt [4]{a}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {2} b}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}+\frac {3 a^{7/4} \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {b+a x^4}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {2} b}-\frac {3 a^{7/4} \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {b+a x^4}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {2} b}-\frac {\left (3 a^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b}+\frac {\left (3 a^{7/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {6 a \left (b+a x^4\right )^{3/4}}{7 b x^3}+\frac {3 a^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b}-\frac {3 a^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b}+\frac {3 a^{7/4} \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {b+a x^4}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {2} b}-\frac {3 a^{7/4} \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {b+a x^4}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {2} b}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 202, normalized size = 1.22 \begin {gather*} \left (\frac {1}{7 x^7}-\frac {6 a}{7 b x^3}\right ) \left (a x^4+b\right )^{3/4}-\frac {3 a^{7/4} \left (-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}+1\right )-\log \left (-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}+\frac {\sqrt {a} x^2}{\sqrt {a+b x^4}}+1\right )+\log \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}+\frac {\sqrt {a} x^2}{\sqrt {a+b x^4}}+1\right )\right )}{4 \sqrt {2} b} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.52, size = 165, normalized size = 1.00 \begin {gather*} \frac {\left (b-6 a x^4\right ) \left (b+a x^4\right )^{3/4}}{7 b x^7}-\frac {3 a^{7/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{b+a x^4}}{-\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{2 \sqrt {2} b}-\frac {3 a^{7/4} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^4}}{\sqrt {2} \sqrt [4]{a}}}{x \sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 75.87, size = 429, normalized size = 2.60 \begin {gather*} \frac {84 \, \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \arctan \left (\frac {{\left (\left (-\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} a^{5} b^{3} x^{2} + \sqrt {\frac {a^{10}}{b^{2}}} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{4} x^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left ({\left (a^{4} b^{2} x^{4} + a^{3} b^{3}\right )} \sqrt {\frac {a^{10}}{b^{2}}} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} - {\left (a^{9} b x^{4} + a^{8} b^{2}\right )} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}\right )}}{2 \, {\left (a^{11} x^{5} + a^{10} b x\right )}}\right ) - 21 \, \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (-\frac {27 \, {\left (2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} a^{4} x^{3} + 2 \, \sqrt {a x^{4} + b} \sqrt {-\frac {a^{7}}{b^{4}}} a^{2} b x^{2} - a^{5} + 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{2} x\right )}}{2 \, a x^{4} + b}\right ) + 21 \, \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {27 \, {\left (2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} a^{4} x^{3} - 2 \, \sqrt {a x^{4} + b} \sqrt {-\frac {a^{7}}{b^{4}}} a^{2} b x^{2} + a^{5} + 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} \left (-\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{2} x\right )}}{2 \, a x^{4} + b}\right ) - 8 \, {\left (6 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{56 \, b x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )}}{{\left (2 \, a x^{4} + b\right )} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \right )^{\frac {3}{4}}}{x^{8} \left (2 a \,x^{4}+b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )}}{{\left (2 \, a x^{4} + b\right )} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (a\,x^4+b\right )}^{3/4}\,\left (b-a\,x^4\right )}{x^8\,\left (2\,a\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{4} - b\right ) \left (a x^{4} + b\right )^{\frac {3}{4}}}{x^{8} \left (2 a x^{4} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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