Optimal. Leaf size=674 \[ -\frac {1989 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {1989 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {\sqrt {a^2 x^2-b} \left (\left (6055526400 a^2 c^7 x^2-4026531840 a c^{10} x+2409402996 b^2 c-1513881600 b c^7\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-5752750080 a^2 c^6 x^2+3523215360 a c^9 x-3011753745 b^2+1438187520 b c^6\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5513052160 a^2 c^5 x^2-3229614080 a c^8 x-26186997760 b c^5+2684354560 c^{11}\right )\right )+\left (6055526400 a^3 c^7 x^3-4026531840 a^2 c^{10} x^2+2409402996 a b^2 c x-4541644800 a b c^7 x-1860655104 b^2 c^4+2013265920 b c^{10}\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-5752750080 a^3 c^6 x^3+3523215360 a^2 c^9 x^2-3011753745 a b^2 x+4314562560 a b c^6 x+1976946048 b^2 c^3-1761607680 b c^9\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5513052160 a^3 c^5 x^3-3229614080 a^2 c^8 x^2-28943523840 a b c^5 x+2684354560 a c^{11} x-2141691552 b^2 c^2+1614807040 b c^8\right )}{12404367360 a c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/3}} \]
________________________________________________________________________________________
Rubi [F] time = 1.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 157.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.94, size = 674, normalized size = 1.00 \begin {gather*} -\frac {1989 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {1989 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {\sqrt {a^2 x^2-b} \left (\left (6055526400 a^2 c^7 x^2-4026531840 a c^{10} x+2409402996 b^2 c-1513881600 b c^7\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-5752750080 a^2 c^6 x^2+3523215360 a c^9 x-3011753745 b^2+1438187520 b c^6\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5513052160 a^2 c^5 x^2-3229614080 a c^8 x-26186997760 b c^5+2684354560 c^{11}\right )\right )+\left (6055526400 a^3 c^7 x^3-4026531840 a^2 c^{10} x^2+2409402996 a b^2 c x-4541644800 a b c^7 x-1860655104 b^2 c^4+2013265920 b c^{10}\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-5752750080 a^3 c^6 x^3+3523215360 a^2 c^9 x^2-3011753745 a b^2 x+4314562560 a b c^6 x+1976946048 b^2 c^3-1761607680 b c^9\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5513052160 a^3 c^5 x^3-3229614080 a^2 c^8 x^2-28943523840 a b c^5 x+2684354560 a c^{11} x-2141691552 b^2 c^2+1614807040 b c^8\right )}{12404367360 a c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 562, normalized size = 0.83 \begin {gather*} \frac {12047014980 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}} c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} - \sqrt {a^{2} b^{8} c^{11} \sqrt {\frac {b^{8}}{a^{4} c^{21}}} + b^{12} \sqrt {c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}}} a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}}}{b^{8}}\right ) + 3011753745 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (7868724669 \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 3011753745 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (-7868724669 \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + 4 \, {\left (2684354560 \, c^{11} + 2756526080 \, a^{2} c^{5} x^{2} - 26186997760 \, b c^{5} - 2464 \, {\left (655360 \, a c^{8} + 869193 \, a b c^{2}\right )} x + 21 \, {\left (83886080 \, c^{9} + 188280576 \, a^{2} c^{3} x^{2} - 94140288 \, b c^{3} - 1045 \, {\left (65536 \, a c^{6} + 137241 \, a b\right )} x - 209 \, {\left (327680 \, c^{6} + 900864 \, a c^{3} x - 686205 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 2464 \, {\left (655360 \, c^{8} - 1118720 \, a c^{5} x - 869193 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - 12 \, {\left (167772160 \, c^{10} + 310109184 \, a^{2} c^{4} x^{2} - 155054592 \, b c^{4} - 77 \, {\left (1638400 \, a c^{7} + 2607579 \, a b c\right )} x - 77 \, {\left (1638400 \, c^{7} + 4027392 \, a c^{4} x - 2607579 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}}{49617469440 \, a c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}}{{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\,\sqrt {a^2\,x^2-b}}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/3}\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}{\sqrt [4]{c + \sqrt [3]{a x + \sqrt {a^{2} x^{2} - b}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________