3.32.41 \(\int \frac {\sqrt {-b+a^2 x^2}}{(c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}})^{2/3}} \, dx\)

Optimal. Leaf size=1186 \[ -\frac {21505 \tan ^{-1}\left (\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right ) b^2}{19683 \sqrt {3} a c^{26/3}}+\frac {21505 \log \left (\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}-\sqrt [3]{c}\right ) b^2}{59049 a c^{26/3}}-\frac {21505 \log \left (c^{2/3}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \sqrt [3]{c}+\left (c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\right )^{2/3}\right ) b^2}{118098 a c^{26/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right ) b}{a c^{2/3}}-\frac {2 \log \left (\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}-\sqrt [3]{c}\right ) b}{a c^{2/3}}+\frac {\log \left (c^{2/3}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \sqrt [3]{c}+\left (c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}\right )^{2/3}\right ) b}{a c^{2/3}}+\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \left (2812385772 a^2 x^2 c^{12}-1406192886 b c^{12}[phone] a^3 x^3 c^8-2450297304 a b x c^8+1032601284 b^2 c^4+3272028760 a b^2 x\right )+\sqrt {a^2 x^2-b} \left (\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}} \left (2812385772 a x c^{12}[phone] a^2 x^2 c^8-816765768 b c^8+3272028760 b^2\right )+\left (-16271660538 a x c^{15}-4687309620 a^2 x^2 c^{11}[phone] b c^{11}-1204701498 b^2 c^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-3615924564 a x c^{13}-3610964448 a^2 x^2 c^9+902741112 b c^9-1963217256 b^2 c\right ) \sqrt {a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (5423886846 a x c^{14}[phone] a^2 x^2 c^{10}-1015583751 b c^{10}[phone] b^2 c^2\right ) \sqrt [4]{a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}\right )+\left (-16271660538 a^2 x^2 c^{15}[phone] b c^{15}-4687309620 a^3 x^3 c^{11}[phone] a b x c^{11}-748701954 b^2 c^7-1204701498 a b^2 x c^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-3615924564 a^2 x^2 c^{13}[phone] b c^{13}-3610964448 a^3 x^3 c^9+2708223336 a b x c^9-911118780 b^2 c^5-1963217256 a b^2 x c\right ) \sqrt {a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (5423886846 a^2 x^2 c^{14}-2711943423 b c^{14}[phone] a^3 x^3 c^{10}-3046751253 a b x c^{10}[phone] b^2 c^6+1472412942 a b^2 x c^2\right ) \sqrt [4]{a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{5989615632 a \left (2 a^2 x^2-b\right ) c^8+11979231264 a^2 x \sqrt {a^2 x^2-b} c^8} \]

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Rubi [F]  time = 0.30, 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}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]

[Out]

Defer[Int][Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx &=\int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]

[Out]

Integrate[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3), x]

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IntegrateAlgebraic [A]  time = 3.16, size = 1186, normalized size = 1.00 \begin {gather*} \frac {\left (-748701954 b^2 c^7+8135830269 b c^{15}-1204701498 a b^2 c^3 x+3515482215 a b c^{11} x-16271660538 a^2 c^{15} x^2-4687309620 a^3 c^{11} x^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (820006902 b^2 c^6-2711943423 b c^{14}[phone] a b^2 c^2 x-3046751253 a b c^{10} x+5423886846 a^2 c^{14} x^2+4062335004 a^3 c^{10} x^3\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-911118780 b^2 c^5+1807962282 b c^{13}-1963217256 a b^2 c x+2708223336 a b c^9 x-3615924564 a^2 c^{13} x^2-3610964448 a^3 c^9 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (1032601284 b^2 c^4-1406192886 b c^{12}[phone] a b^2 x-2450297304 a b c^8 x+2812385772 a^2 c^{12} x^2+3267063072 a^3 c^8 x^3\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-1204701498 b^2 c^3+1171827405 b c^{11}-16271660538 a c^{15} x-4687309620 a^2 c^{11} x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (1472412942 b^2 c^2-1015583751 b c^{10}[phone] a c^{14} x+4062335004 a^2 c^{10} x^2\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-1963217256 b^2 c+902741112 b c^9-3615924564 a c^{13} x-3610964448 a^2 c^9 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (3272028760 b^2-816765768 b c^8+2812385772 a c^{12} x+3267063072 a^2 c^8 x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{11979231264 a^2 c^8 x \sqrt {-b+a^2 x^2}[phone] a c^8 \left (-b+2 a^2 x^2\right )}-\frac {21505 b^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{19683 \sqrt {3} a c^{26/3}}+\frac {2 \sqrt {3} b \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{a c^{2/3}}+\frac {21505 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{59049 a c^{26/3}}-\frac {2 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{a c^{2/3}}-\frac {21505 b^2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{118098 a c^{26/3}}+\frac {b \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{a c^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]

[Out]

((-748701954*b^2*c^7 + 8135830269*b*c^15 - 1204701498*a*b^2*c^3*x + 3515482215*a*b*c^11*x - 16271660538*a^2*c^
15*x^2 - 4687309620*a^3*c^11*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (820006902*b^2*c^6 - 27119434
23*b*c^14 + 1472412942*a*b^2*c^2*x - 3046751253*a*b*c^10*x + 5423886846*a^2*c^14*x^2 + 4062335004*a^3*c^10*x^3
)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-911118780*b^2*c^5 + 180796
2282*b*c^13 - 1963217256*a*b^2*c*x + 2708223336*a*b*c^9*x - 3615924564*a^2*c^13*x^2 - 3610964448*a^3*c^9*x^3)*
Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (1032601284*b^2*c^4 - 1406192886
*b*c^12 + 3272028760*a*b^2*x - 2450297304*a*b*c^8*x + 2812385772*a^2*c^12*x^2 + 3267063072*a^3*c^8*x^3)*(a*x +
Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + Sqrt[-b + a^2*x^2]*((-1204701498*b^2
*c^3 + 1171827405*b*c^11 - 16271660538*a*c^15*x - 4687309620*a^2*c^11*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/
4))^(1/3) + (1472412942*b^2*c^2 - 1015583751*b*c^10 + 5423886846*a*c^14*x + 4062335004*a^2*c^10*x^2)*(a*x + Sq
rt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-1963217256*b^2*c + 902741112*b*c^9 -
3615924564*a*c^13*x - 3610964448*a^2*c^9*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(
1/4))^(1/3) + (3272028760*b^2 - 816765768*b*c^8 + 2812385772*a*c^12*x + 3267063072*a^2*c^8*x^2)*(a*x + Sqrt[-b
+ a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)))/(11979231264*a^2*c^8*x*Sqrt[-b + a^2*x^2] +
5989615632*a*c^8*(-b + 2*a^2*x^2)) - (21505*b^2*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(
1/3))/(Sqrt[3]*c^(1/3))])/(19683*Sqrt[3]*a*c^(26/3)) + (2*Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b
+ a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(a*c^(2/3)) + (21505*b^2*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^
2*x^2])^(1/4))^(1/3)])/(59049*a*c^(26/3)) - (2*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])
/(a*c^(2/3)) - (21505*b^2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqr
t[-b + a^2*x^2])^(1/4))^(2/3)])/(118098*a*c^(26/3)) + (b*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])
^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(a*c^(2/3))

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fricas [A]  time = 0.83, size = 644, normalized size = 0.54 \begin {gather*} \frac {304304 \, \sqrt {3} {\left (118098 \, b c^{9} - 21505 \, b^{2} c\right )} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} c \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} - 2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}}}{3 \, c^{2}}\right ) + 152152 \, {\left (118098 \, b c^{8} - 21505 \, b^{2}\right )} \left (-c^{2}\right )^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c - \left (-c^{2}\right )^{\frac {1}{3}} c + \left (-c^{2}\right )^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 304304 \, {\left (118098 \, b c^{8} - 21505 \, b^{2}\right )} \left (-c^{2}\right )^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - \left (-c^{2}\right )^{\frac {2}{3}}\right ) - 3 \, {\left (8135830269 \, c^{17} + 1497403908 \, a^{2} c^{9} x^{2} - 748701954 \, b c^{9} + 567 \, {\left (2066715 \, a c^{13} + 2124694 \, a b c^{5}\right )} x - 2 \, {\left (703096443 \, c^{14} + 1032601284 \, a^{2} c^{6} x^{2} - 516300642 \, b c^{6} + 6916 \, {\left (59049 \, a c^{10} + 236555 \, a b c^{2}\right )} x + 988 \, {\left (413343 \, c^{10} - 1045143 \, a c^{6} x - 1655885 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 567 \, {\left (2066715 \, c^{13} - 2640924 \, a c^{9} x - 2124694 \, b c^{5}\right )} \sqrt {a^{2} x^{2} - b} + 6 \, {\left (301327047 \, c^{15} + 303706260 \, a^{2} c^{7} x^{2} - 151853130 \, b c^{7} + 364 \, {\left (413343 \, a c^{11} + 898909 \, a b c^{3}\right )} x + 52 \, {\left (2893401 \, c^{11} - 5840505 \, a c^{7} x - 6292363 \, b c^{3}\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (301327047 \, c^{16} + 182223756 \, a^{2} c^{8} x^{2} - 91111878 \, b c^{8} + 91 \, {\left (1240029 \, a c^{12} + 1797818 \, a b c^{4}\right )} x + 13 \, {\left (8680203 \, c^{12} - 14017212 \, a c^{8} x - 12584726 \, b c^{4}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{17968846896 \, a c^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3),x, algorithm="fricas")

[Out]

1/17968846896*(304304*sqrt(3)*(118098*b*c^9 - 21505*b^2*c)*sqrt(-(-c^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-c^2)^(1/
3)*c*sqrt(-(-c^2)^(1/3)) - 2*sqrt(3)*(-c^2)^(2/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*sqrt(-(-c^2)^(1/
3)))/c^2) + 152152*(118098*b*c^8 - 21505*b^2)*(-c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*c -
(-c^2)^(1/3)*c + (-c^2)^(2/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 304304*(118098*b*c^8 - 21505*b^2
)*(-c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - (-c^2)^(2/3)) - 3*(8135830269*c^17 + 149740
3908*a^2*c^9*x^2 - 748701954*b*c^9 + 567*(2066715*a*c^13 + 2124694*a*b*c^5)*x - 2*(703096443*c^14 + 1032601284
*a^2*c^6*x^2 - 516300642*b*c^6 + 6916*(59049*a*c^10 + 236555*a*b*c^2)*x + 988*(413343*c^10 - 1045143*a*c^6*x -
1655885*b*c^2)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 567*(2066715*c^13 - 2640924*a*c^9*x - 212
4694*b*c^5)*sqrt(a^2*x^2 - b) + 6*(301327047*c^15 + 303706260*a^2*c^7*x^2 - 151853130*b*c^7 + 364*(413343*a*c^
11 + 898909*a*b*c^3)*x + 52*(2893401*c^11 - 5840505*a*c^7*x - 6292363*b*c^3)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqr
t(a^2*x^2 - b)) - 9*(301327047*c^16 + 182223756*a^2*c^8*x^2 - 91111878*b*c^8 + 91*(1240029*a*c^12 + 1797818*a*
b*c^4)*x + 13*(8680203*c^12 - 14017212*a*c^8*x - 12584726*b*c^4)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^
(1/4))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))/(a*c^10)

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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]

integrate((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3),x)

[Out]

int((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b}}{{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(2/3),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*x^2 - b)/(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a^2\,x^2-b}}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2 - b)^(1/2)/(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(2/3),x)

[Out]

int((a^2*x^2 - b)^(1/2)/(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(2/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b}}{\left (c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**2-b)**(1/2)/(c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(2/3),x)

[Out]

Integral(sqrt(a**2*x**2 - b)/(c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(2/3), x)

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