Optimal. Leaf size=119 \[ -\frac {16 a q^3 x^q \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {1}{q}\right );\frac {1}{2} \left (4-\frac {1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt {d x}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_3\left (a x^q\right )}{d \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{d \sqrt {d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6591, 2455, 20, 364} \[ -\frac {4 q \text {PolyLog}\left (2,a x^q\right )}{d \sqrt {d x}}-\frac {2 \text {PolyLog}\left (3,a x^q\right )}{d \sqrt {d x}}-\frac {16 a q^3 x^q \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {1}{q}\right );\frac {1}{2} \left (4-\frac {1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{d \sqrt {d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^q\right )}{(d x)^{3/2}} \, dx &=-\frac {2 \text {Li}_3\left (a x^q\right )}{d \sqrt {d x}}+(2 q) \int \frac {\text {Li}_2\left (a x^q\right )}{(d x)^{3/2}} \, dx\\ &=-\frac {4 q \text {Li}_2\left (a x^q\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_3\left (a x^q\right )}{d \sqrt {d x}}-\left (4 q^2\right ) \int \frac {\log \left (1-a x^q\right )}{(d x)^{3/2}} \, dx\\ &=\frac {8 q^2 \log \left (1-a x^q\right )}{d \sqrt {d x}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_3\left (a x^q\right )}{d \sqrt {d x}}+\frac {\left (8 a q^3\right ) \int \frac {x^{-1+q}}{\sqrt {d x} \left (1-a x^q\right )} \, dx}{d}\\ &=\frac {8 q^2 \log \left (1-a x^q\right )}{d \sqrt {d x}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_3\left (a x^q\right )}{d \sqrt {d x}}+\frac {\left (8 a q^3 \sqrt {x}\right ) \int \frac {x^{-\frac {3}{2}+q}}{1-a x^q} \, dx}{d \sqrt {d x}}\\ &=-\frac {16 a q^3 x^q \, _2F_1\left (1,\frac {1}{2} \left (2-\frac {1}{q}\right );\frac {1}{2} \left (4-\frac {1}{q}\right );a x^q\right )}{d (1-2 q) \sqrt {d x}}+\frac {8 q^2 \log \left (1-a x^q\right )}{d \sqrt {d x}}-\frac {4 q \text {Li}_2\left (a x^q\right )}{d \sqrt {d x}}-\frac {2 \text {Li}_3\left (a x^q\right )}{d \sqrt {d x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 50, normalized size = 0.42 \[ -\frac {x G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1+\frac {1}{2 q} \\ 1,0,0,0,\frac {1}{2 q} \\\end {array}\right )}{q (d x)^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\rm polylog}\left (3, a x^{q}\right )}{d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{q})}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.28, size = 145, normalized size = 1.22 \[ -\frac {x^{\frac {3}{2}} \left (-a \right )^{\frac {1}{2 q}} \left (-\frac {8 q^{3} \left (-a \right )^{-\frac {1}{2 q}} \ln \left (1-a \,x^{q}\right )}{\sqrt {x}}+\frac {4 q^{2} \left (-a \right )^{-\frac {1}{2 q}} \polylog \left (2, a \,x^{q}\right )}{\sqrt {x}}-\frac {2 q \left (-a \right )^{-\frac {1}{2 q}} \left (1-2 q \right ) \polylog \left (3, a \,x^{q}\right )}{\left (2 q -1\right ) \sqrt {x}}-8 q^{3} x^{q -\frac {1}{2}} a \left (-a \right )^{-\frac {1}{2 q}} \Phi \left (a \,x^{q}, 1, \frac {2 q -1}{2 q}\right )\right )}{\left (d x \right )^{\frac {3}{2}} q} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 16 \, q^{4} \int \frac {1}{{\left (a^{2} d^{\frac {3}{2}} {\left (2 \, q + 1\right )} x^{2 \, q} - 2 \, a d^{\frac {3}{2}} {\left (2 \, q + 1\right )} x^{q} + d^{\frac {3}{2}} {\left (2 \, q + 1\right )}\right )} x^{\frac {3}{2}}}\,{d x} - \frac {2 \, {\left (\frac {2 \, {\left ({\left (2 \, q^{2} + q\right )} a x x^{q} - {\left (2 \, q^{2} + q\right )} x\right )} {\rm Li}_2\left (a x^{q}\right )}{x^{\frac {3}{2}}} - \frac {4 \, {\left ({\left (2 \, q^{3} + q^{2}\right )} a x x^{q} - {\left (2 \, q^{3} + q^{2}\right )} x\right )} \log \left (-a x^{q} + 1\right )}{x^{\frac {3}{2}}} + \frac {{\left (a {\left (2 \, q + 1\right )} x x^{q} - {\left (2 \, q + 1\right )} x\right )} {\rm Li}_{3}(a x^{q})}{x^{\frac {3}{2}}} + \frac {8 \, {\left (2 \, q^{4} x - {\left (2 \, q^{4} + q^{3}\right )} a x x^{q}\right )}}{x^{\frac {3}{2}}}\right )}}{a d^{\frac {3}{2}} {\left (2 \, q + 1\right )} x^{q} - d^{\frac {3}{2}} {\left (2 \, q + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (3,a\,x^q\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{q}\right )}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________