Optimal. Leaf size=144 \[ \frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}-\frac {14 a^2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rubi [A]
time = 0.03, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 42, 235,
233, 202} \begin {gather*} -\frac {14 a^2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 42
Rule 52
Rule 202
Rule 233
Rule 235
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}+\frac {1}{5} (7 a) \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx\\ &=-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}+\frac {1}{5} \left (7 a^2\right ) \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\\ &=-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}+\frac {\left (7 a^2 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}+\frac {\left (7 a^2 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}-\frac {\left (7 a^2 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}-\frac {14 a^2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 70, normalized size = 0.49 \begin {gather*} \frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac {1}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2}-\frac {i x}{2}\right )}{11 a \sqrt [4]{a+i a x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.22, size = 104, normalized size = 0.72
method | result | size |
risch | \(-\frac {2 \left (10 i+3 x \right ) \left (x -i\right ) \left (x +i\right ) a^{2}}{15 \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {7 x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) a^{2} \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(104\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {7}{4}}}{\sqrt [4]{i a \left (x - i\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \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-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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