Optimal. Leaf size=297 \[ \frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}-\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
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Rubi [A] time = 0.14, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {47, 63, 240, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}-\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{9/4}} \, dx &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 a-x^4}} \, dx,x,\sqrt [4]{a-i a x}\right )}{a}\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {i \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}-\frac {i \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}+\frac {\left (i \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {\left (i \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac {4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {i \sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 0.24 \begin {gather*} \frac {i \sqrt [4]{1+i x} (a-i a x)^{9/4} \, _2F_1\left (\frac {9}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2}-\frac {i x}{2}\right )}{9 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 137, normalized size = 0.46 \begin {gather*} \frac {4 i \sqrt [4]{a-i a x} \left (-5+\frac {a-i a x}{a+i a x}\right )}{5 a \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{-1} \tanh ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {2 (-1)^{3/4} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.49, size = 351, normalized size = 1.18 \begin {gather*} \frac {{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, x - 2 i}\right ) - {\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, x - 2 i}\right ) + {\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, x - 2 i}\right ) - {\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, x - 2 i}\right ) - {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (48 \, x - 32 i\right )}}{10 \, a^{2} x^{2} - 20 i \, a^{2} x - 10 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 490, normalized size = 1.65 \begin {gather*} \frac {8 \left (3 x^{2}+i x +2\right ) \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}}}{5 \left (x -i\right ) \left (i x -1\right ) \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a}-\frac {\left (\RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (\frac {-x^{3}-\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}+i\right )-2 i x^{2}-2 i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{2}+i\right )+i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x +x +i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )+\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )-\sqrt {-x^{4}-2 i x^{3}-2 i x +1}}{\left (i x -1\right )^{2}}\right )+i \RootOf \left (\textit {\_Z}^{2}+i\right ) \ln \left (\frac {-x^{3}-i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}+i\right )-2 i x^{2}+2 \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{2}+i\right )-i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x +x +\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )+i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+i\right )+\sqrt {-x^{4}-2 i x^{3}-2 i x +1}}{\left (i x -1\right )^{2}}\right )\right ) \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (-\left (i x -1\right )^{3} \left (i x +1\right )\right )^{\frac {1}{4}}}{\left (i x -1\right ) \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a} \end {gather*}
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 (-i \, a x + a\right )}^{\frac {5}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}}\,{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 = -0.00 \begin {gather*} \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,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 (- i a \left (x + i\right )\right )^{\frac {5}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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