Optimal. Leaf size=115 \[ -\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {47, 48, 42, 197, 196} \[ -\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 42
Rule 47
Rule 48
Rule 196
Rule 197
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx &=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{5/4}} \, dx\\ &=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {1}{5} (3 a) \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx\\ &=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {\left (3 a \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {\left (3 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 70, normalized size = 0.61 \[ \frac {i \sqrt [4]{1+i x} (a-i a x)^{7/4} \, _2F_1\left (\frac {7}{4},\frac {9}{4};\frac {11}{4};\frac {1}{2}-\frac {i x}{2}\right )}{7 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (5 i \, x + 3\right )} - 5 \, {\left (a^{3} x^{3} - 2 i \, a^{3} x^{2} - a^{3} x\right )} {\rm integral}\left (\frac {6 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{5 \, {\left (a^{3} x^{4} + a^{3} x^{2}\right )}}, x\right )}{5 \, {\left (a^{3} x^{3} - 2 i \, a^{3} x^{2} - a^{3} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.04, size = 107, normalized size = 0.93 \[ \frac {3 \left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{5 \left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a}-\frac {2 \left (3 x^{2}+2 i x +1\right )}{5 \left (x -i\right ) \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________