Optimal. Leaf size=100 \[ -\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac {16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37}
\begin {gather*} -\frac {16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}+\frac {4 \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx}{11 a}\\ &=-\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}+\frac {8 \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{77 a^2}\\ &=-\frac {2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac {8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac {16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 52, normalized size = 0.52 \begin {gather*} \frac {2 (a+i a x)^{3/4} \left (41 i+28 x-8 i x^2\right )}{231 a^4 (i+x)^2 (a-i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 50, normalized size = 0.50
method | result | size |
risch | \(\frac {\frac {16}{231} x^{3}+\frac {40}{231} i x^{2}-\frac {26}{231} x +\frac {82}{231} i}{a^{3} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x +i\right )^{2}}\) | \(50\) |
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 [A]
time = 0.30, size = 57, normalized size = 0.57 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} + 28 i \, x - 41\right )}}{231 \, {\left (a^{5} x^{3} + 3 i \, a^{5} x^{2} - 3 \, a^{5} x - i \, a^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [B]
time = 0.75, size = 51, normalized size = 0.51 \begin {gather*} \frac {{\left (x-\mathrm {i}\right )}^4\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (8\,x^2+x\,28{}\mathrm {i}-41\right )\,2{}\mathrm {i}}{231\,a^4\,{\left (x^2+1\right )}^3\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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