Optimal. Leaf size=332 \[ -\frac {7 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.29, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {288, 290, 329, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac {7 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 288
Rule 290
Rule 297
Rule 298
Rule 301
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{13/2}}{\left (a+c x^4\right )^3} \, dx &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 \int \frac {x^{5/2}}{\left (a+c x^4\right )^2} \, dx}{16 c}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \int \frac {x^{5/2}}{a+c x^4} \, dx}{128 a c}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \operatorname {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a c}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a c^{3/2}}+\frac {7 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a c^{3/2}}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 a c^{7/4}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 a c^{7/4}}-\frac {7 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 a c^{7/4}}+\frac {7 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 a c^{7/4}}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{512 a c^2}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{512 a c^2}-\frac {7 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}\\ &=-\frac {x^{7/2}}{8 c \left (a+c x^4\right )^2}+\frac {7 x^{7/2}}{64 a c \left (a+c x^4\right )}+\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{9/8} c^{15/8}}-\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{9/8} c^{15/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{9/8} c^{15/8}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 45, normalized size = 0.14 \[ \frac {2 x^{7/2} \left (\frac {\, _2F_1\left (\frac {7}{8},3;\frac {15}{8};-\frac {c x^4}{a}\right )}{a^2}-\frac {1}{\left (a+c x^4\right )^2}\right )}{9 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 688, normalized size = 2.07 \[ -\frac {28 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} a^{8} c^{13} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} - a^{7} c^{11} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {3}{4}} + x} a c^{2} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} - \sqrt {2} a c^{2} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} + 1\right ) + 28 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {-\sqrt {2} a^{8} c^{13} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} - a^{7} c^{11} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {3}{4}} + x} a c^{2} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} - \sqrt {2} a c^{2} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} - 1\right ) - 7 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{8} c^{13} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} - a^{7} c^{11} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {3}{4}} + x\right ) + 7 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{8} c^{13} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} - a^{7} c^{11} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {3}{4}} + x\right ) + 56 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {-a^{7} c^{11} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {3}{4}} + x} a c^{2} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} - a c^{2} \sqrt {x} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}}\right ) - 14 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 14 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {1}{8}} \log \left (-a^{8} c^{13} \left (-\frac {1}{a^{9} c^{15}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 16 \, {\left (7 \, c x^{7} - a x^{3}\right )} \sqrt {x}}{1024 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.88, size = 499, normalized size = 1.50 \[ \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, c x^{\frac {15}{2}} - a x^{\frac {7}{2}}}{64 \, {\left (c x^{4} + a\right )}^{2} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 61, normalized size = 0.18 \[ \frac {7 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 a \,c^{2} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )}+\frac {\frac {7 x^{\frac {15}{2}}}{64 a}-\frac {x^{\frac {7}{2}}}{64 c}}{\left (c \,x^{4}+a \right )^{2}} \]
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 \[ \frac {7 \, c x^{\frac {15}{2}} - a x^{\frac {7}{2}}}{64 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} + 7 \, \int \frac {x^{\frac {5}{2}}}{128 \, {\left (a c^{2} x^{4} + a^{2} c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 156, normalized size = 0.47 \[ \frac {\frac {7\,x^{15/2}}{64\,a}-\frac {x^{7/2}}{64\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {7\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{9/8}\,c^{15/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,7{}\mathrm {i}}{256\,{\left (-a\right )}^{9/8}\,c^{15/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {7}{512}+\frac {7}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{9/8}\,c^{15/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {7}{512}-\frac {7}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{9/8}\,c^{15/8}} \]
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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