Optimal. Leaf size=106 \[ -\frac {2 a \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 106, 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 = {50, 42, 229, 227, 196} \[ -\frac {2 a \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 42
Rule 50
Rule 196
Rule 227
Rule 229
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+a \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\\ &=-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {\left (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 {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {\left (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 {2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (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 {2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {2 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*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 70, normalized size = 0.66 \[ \frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{7/4} \, _2F_1\left (\frac {1}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2}-\frac {i x}{2}\right )}{7 a \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ \frac {3 \, a x {\rm integral}\left (\frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{a x^{4} + a x^{2}}, x\right ) - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (i \, x - 3\right )}}{3 \, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.07, size = 94, normalized size = 0.89 \[ \frac {\left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} a x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{\left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}}-\frac {2 i \left (x -i\right ) \left (x +i\right ) a}{3 \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\sqrt [4]{i a \left (x - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________