Optimal. Leaf size=332 \[ \frac {15 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{13/8} c^{11/8}}+\frac {15 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.30, 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, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac {15 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{13/8} c^{11/8}}+\frac {15 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 208
Rule 288
Rule 290
Rule 297
Rule 298
Rule 300
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{9/2}}{\left (a+c x^4\right )^3} \, dx &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx}{16 c}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}+\frac {15 \int \frac {\sqrt {x}}{a+c x^4} \, dx}{128 a c}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}+\frac {15 \operatorname {Subst}\left (\int \frac {x^2}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a c}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}+\frac {15 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{3/2} c}+\frac {15 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{3/2} c}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}+\frac {15 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{3/2} c^{5/4}}-\frac {15 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{3/2} c^{5/4}}-\frac {15 \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)^{3/2} c^{5/4}}+\frac {15 \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)^{3/2} c^{5/4}}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \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)^{3/2} c^{3/2}}+\frac {15 \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)^{3/2} c^{3/2}}+\frac {15 \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)^{13/8} c^{11/8}}+\frac {15 \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)^{13/8} c^{11/8}}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}+\frac {15 \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)^{13/8} c^{11/8}}-\frac {15 \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)^{13/8} c^{11/8}}\\ &=-\frac {x^{3/2}}{8 c \left (a+c x^4\right )^2}+\frac {3 x^{3/2}}{64 a c \left (a+c x^4\right )}-\frac {15 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{13/8} c^{11/8}}+\frac {15 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{13/8} c^{11/8}}+\frac {15 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}-\frac {15 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{13/8} c^{11/8}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 45, normalized size = 0.14 \[ \frac {2 x^{3/2} \left (\frac {\, _2F_1\left (\frac {3}{8},3;\frac {11}{8};-\frac {c x^4}{a}\right )}{a^2}-\frac {1}{\left (a+c x^4\right )^2}\right )}{13 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 693, normalized size = 2.09 \[ -\frac {60 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{10} c^{8} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a^{5} c^{4} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{8}} + x} a^{8} c^{7} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} a^{8} c^{7} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {5}{8}} + 1\right ) + 60 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{10} c^{8} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a^{5} c^{4} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{8}} + x} a^{8} c^{7} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} a^{8} c^{7} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {5}{8}} - 1\right ) - 15 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{10} c^{8} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a^{5} c^{4} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{8}} + x\right ) + 15 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{10} c^{8} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a^{5} c^{4} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{8}} + x\right ) - 120 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{10} c^{8} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{4}} + x} a^{8} c^{7} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {5}{8}} - a^{8} c^{7} \sqrt {x} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {5}{8}}\right ) + 30 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \log \left (a^{5} c^{4} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - 30 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {1}{8}} \log \left (-a^{5} c^{4} \left (-\frac {1}{a^{13} c^{11}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - 16 \, {\left (3 \, c x^{5} - 5 \, a x\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 = 0.86, size = 499, normalized size = 1.50 \[ -\frac {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {15 \, \left (\frac {a}{c}\right )^{\frac {3}{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 {3 \, c x^{\frac {11}{2}} - 5 \, a x^{\frac {3}{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 \[ \frac {15 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 a \,c^{2} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{5}}+\frac {\frac {3 x^{\frac {11}{2}}}{64 a}-\frac {5 x^{\frac {3}{2}}}{64 c}}{\left (c \,x^{4}+a \right )^{2}} \]
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 \[ \frac {3 \, c x^{\frac {11}{2}} - 5 \, a x^{\frac {3}{2}}}{64 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} + 15 \, \int \frac {\sqrt {x}}{128 \, {\left (a c^{2} x^{4} + a^{2} c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 156, normalized size = 0.47 \[ \frac {\frac {3\,x^{11/2}}{64\,a}-\frac {5\,x^{3/2}}{64\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {15\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{13/8}\,c^{11/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,15{}\mathrm {i}}{256\,{\left (-a\right )}^{13/8}\,c^{11/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {15}{512}-\frac {15}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{13/8}\,c^{11/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {15}{512}+\frac {15}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{13/8}\,c^{11/8}} \]
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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