Optimal. Leaf size=646 \[ -\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{24 a^3}-\frac {11 (1-a x)^{7/8} \sqrt [8]{a x+1}}{32 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.70, antiderivative size = 646, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6126, 90, 80, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ -\frac {x (1-a x)^{7/8} (a x+1)^{9/8}}{3 a^2}-\frac {(1-a x)^{7/8} (a x+1)^{9/8}}{24 a^3}-\frac {11 (1-a x)^{7/8} \sqrt [8]{a x+1}}{32 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+1\right )}{256 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{a x+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 90
Rule 204
Rule 299
Rule 331
Rule 618
Rule 628
Rule 634
Rule 1122
Rule 1169
Rule 6126
Rubi steps
\begin {align*} \int e^{\frac {1}{4} \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx\\ &=-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {\int \frac {\left (-1-\frac {a x}{4}\right ) \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx}{3 a^2}\\ &=-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac {11 \int \frac {\sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx}{32 a^2}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac {11 \int \frac {1}{\sqrt [8]{1-a x} (1+a x)^{7/8}} \, dx}{128 a^2}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {11 \operatorname {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-a x}\right )}{16 a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {11 \operatorname {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{16 a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {11 \operatorname {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt {2} a^3}+\frac {11 \operatorname {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt {2} a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac {11 \operatorname {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt {2} a^3}-\frac {11 \operatorname {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{32 \sqrt {2} a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {11 \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt {2 \left (2-\sqrt {2}\right )} a^3}-\frac {11 \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt {2 \left (2-\sqrt {2}\right )} a^3}+\frac {11 \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt {2 \left (2+\sqrt {2}\right )} a^3}+\frac {11 \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 \sqrt {2 \left (2+\sqrt {2}\right )} a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {\left (11 \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}-\frac {\left (11 \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}-\frac {\left (11 \sqrt {2-\sqrt {2}}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {\left (11 \sqrt {2-\sqrt {2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac {\left (11 \sqrt {2+\sqrt {2}}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {\left (11 \sqrt {2+\sqrt {2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac {\left (11 \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}-\frac {\left (11 \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{128 a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}-\frac {11 \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {\left (11 \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}+\frac {\left (11 \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}+\frac {\left (11 \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}+\frac {\left (11 \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{64 a^3}\\ &=-\frac {11 (1-a x)^{7/8} \sqrt [8]{1+a x}}{32 a^3}-\frac {(1-a x)^{7/8} (1+a x)^{9/8}}{24 a^3}-\frac {x (1-a x)^{7/8} (1+a x)^{9/8}}{3 a^2}+\frac {11 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {11 \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}-\frac {11 \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}+\frac {11 \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-a x}}{\sqrt [8]{1+a x}}\right )}{256 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 70, normalized size = 0.11 \[ -\frac {(1-a x)^{7/8} \left (7 \sqrt [8]{a x+1} \left (8 a^2 x^2+9 a x+1\right )+66 \sqrt [8]{2} \, _2F_1\left (-\frac {1}{8},\frac {7}{8};\frac {15}{8};\frac {1}{2} (1-a x)\right )\right )}{168 a^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.75, size = 2486, normalized size = 3.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {1}{4}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt [4]{\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________