Optimal. Leaf size=137 \[ \frac {14 a \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac {14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
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Rubi [A] time = 0.03, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {47, 50, 42, 229, 227, 196} \[ \frac {14 a \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac {14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 47
Rule 50
Rule 196
Rule 227
Rule 229
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx &=\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}-7 \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-(7 a) \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (7 a \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (7 a \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {\left (7 a \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {14 a \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 0.51 \[ \frac {i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac {5}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2}-\frac {i x}{2}\right )}{11 a^2 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ \frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (2 i \, x^{2} - 16 \, x + 42 i\right )} + {\left (3 \, a x^{2} - 3 i \, a x\right )} {\rm integral}\left (-\frac {14 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{a x^{4} + a x^{2}}, x\right )}{3 \, a x^{2} - 3 i \, a x} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 96, normalized size = 0.70 \[ -\frac {7 \left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} a x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{\left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}}+\frac {2 i \left (x^{2}-12 i x +13\right ) a}{3 \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}} \]
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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {5}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \]
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 \[ \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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