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

Optimal. Leaf size=963 \[ \frac {308 \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}{243 \sqrt {3} a c^{17/3}}-\frac {308 \log \left (\sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}-\sqrt [3]{c}\right ) b^2}{729 a c^{17/3}}+\frac {154 \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}{729 a c^{17/3}}+\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 (774840978 a x c^{13}[phone] a^2 x^2 c^9-386889048 b c^9+1109188080 a^3 x^3 c^5-3287950380 a b x c^5+1089560472 b^2 c\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 (774840978 c^{13}[phone] a x c^9+1109188080 a^2 x^2 c^5-2733356340 b c^5\right )+\left (-1205308188 a x c^{12}-1400169888 a^2 x^2 c^8+350042472 b c^8-1815934120 b^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-1162261467 c^{14}-870500358 a x c^{10}-1188415800 a^2 x^2 c^6+3981192930 b c^6\right ) \sqrt {a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (3486784401 c^{15}[phone] a x c^{11}[phone] a^2 x^2 c^7-11373139206 b c^7\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 (-1205308188 a^2 x^2 c^{12}[phone] b c^{12}-1400169888 a^3 x^3 c^8+1050127416 a b x c^8-573080508 b^2 c^4-1815934120 a b^2 x\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-1162261467 a x c^{14}-870500358 a^2 x^2 c^{10}[phone] b c^{10}-1188415800 a^3 x^3 c^6+4575400830 a b x c^6-817170354 b^2 c^2\right ) \sqrt {a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (3486784401 a x c^{15}[phone] a^2 x^2 c^{11}-502211745 b c^{11}[phone] a^3 x^3 c^7-12014883738 a b x c^7+668593926 b^2 c^3\right ) \sqrt [4]{a x+\sqrt {a^2 x^2-b}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}}{2865402540 a c^5 \left (a x+\sqrt {a^2 x^2-b}\right )^{5/4}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.13, 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 (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\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]*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]

[Out]

Defer[Int][(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(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 (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\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 (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\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]*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x
]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.73, size = 963, normalized size = 1.00 \begin {gather*} \frac {\left (-573080508 b^2 c^4+602654094 b c^{12}-1815934120 a b^2 x+1050127416 a b c^8 x-1205308188 a^2 c^{12} x^2-1400169888 a^3 c^8 x^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (668593926 b^2 c^3-502211745 b c^{11}-12014883738 a b c^7 x+3486784401 a c^{15} x+1004423490 a^2 c^{11} x^2+1283489064 a^3 c^7 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 (-817170354 b^2 c^2+435250179 b c^{10}[phone] a b c^6 x-1162261467 a c^{14} x-870500358 a^2 c^{10} x^2-1188415800 a^3 c^6 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 (1089560472 b^2 c-386889048 b c^9-3287950380 a b c^5 x+774840978 a c^{13} x+773778096 a^2 c^9 x^2+1109188080 a^3 c^5 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 (-1815934120 b^2+350042472 b c^8-1205308188 a c^{12} x-1400169888 a^2 c^8 x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-11373139206 b c^7+3486784401 c^{15}[phone] a c^{11} x+1283489064 a^2 c^7 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 (3981192930 b c^6-1162261467 c^{14}-870500358 a c^{10} x-1188415800 a^2 c^6 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 (-2733356340 b c^5+774840978 c^{13}[phone] a c^9 x+1109188080 a^2 c^5 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 )}{2865402540 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {308 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 )}{243 \sqrt {3} a c^{17/3}}-\frac {308 b^2 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{729 a c^{17/3}}+\frac {154 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 )}{729 a c^{17/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-573080508*b^2*c^4 + 602654094*b*c^12 - 1815934120*a*b^2*x + 1050127416*a*b*c^8*x - 1205308188*a^2*c^12*x^2
- 1400169888*a^3*c^8*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (668593926*b^2*c^3 - 502211745*b*c^11
- 12014883738*a*b*c^7*x + 3486784401*a*c^15*x + 1004423490*a^2*c^11*x^2 + 1283489064*a^3*c^7*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) + (-817170354*b^2*c^2 + 435250179*b*c^10 +
4575400830*a*b*c^6*x - 1162261467*a*c^14*x - 870500358*a^2*c^10*x^2 - 1188415800*a^3*c^6*x^3)*Sqrt[a*x + Sqrt[
-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (1089560472*b^2*c - 386889048*b*c^9 - 3287950380
*a*b*c^5*x + 774840978*a*c^13*x + 773778096*a^2*c^9*x^2 + 1109188080*a^3*c^5*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]*((-1815934120*b^2 + 350042472*b*c^8 - 1
205308188*a*c^12*x - 1400169888*a^2*c^8*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-11373139206*b*c^
7 + 3486784401*c^15 + 1004423490*a*c^11*x + 1283489064*a^2*c^7*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x
+ Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (3981192930*b*c^6 - 1162261467*c^14 - 870500358*a*c^10*x - 1188415800*a^
2*c^6*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-2733356340*b*c^5 +
774840978*c^13 + 773778096*a*c^9*x + 1109188080*a^2*c^5*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)))/(2865402540*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (308*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))])/(243*Sqrt[3]*a*c^(17/3)) - (308*b^2*Lo
g[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(729*a*c^(17/3)) + (154*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 + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(729*a*c^(17/3))

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 617, normalized size = 0.64 \begin {gather*} \frac {3631868240 \, \sqrt {3} b^{2} c \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 ) + 1815934120 \, \left (-c^{2}\right )^{\frac {2}{3}} b^{2} \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 ) - 3631868240 \, \left (-c^{2}\right )^{\frac {2}{3}} b^{2} \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 (3486784401 \, c^{17} + 641744532 \, a^{2} c^{9} x^{2} - 11373139206 \, b c^{9} + 567 \, {\left (885735 \, a c^{13} + 1179178 \, a b c^{5}\right )} x - 2 \, {\left (301327047 \, c^{14} + 573080508 \, a^{2} c^{6} x^{2} - 286540254 \, b c^{6} + 988 \, {\left (177147 \, a c^{10} + 918995 \, a b c^{2}\right )} x + 988 \, {\left (177147 \, c^{10} - 580041 \, a c^{6} x - 918995 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 81 \, {\left (6200145 \, c^{13} + 7922772 \, a c^{9} x - 8254246 \, b c^{5}\right )} \sqrt {a^{2} x^{2} - b} + 6 \, {\left (129140163 \, c^{15} + 92432340 \, a^{2} c^{7} x^{2} - 455559390 \, b c^{7} + 364 \, {\left (177147 \, a c^{11} + 498883 \, a b c^{3}\right )} x + 364 \, {\left (177147 \, c^{11} + 253935 \, a c^{7} x - 498883 \, b c^{3}\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (129140163 \, c^{16} + 66023100 \, a^{2} c^{8} x^{2} - 442354770 \, b c^{8} + 91 \, {\left (531441 \, a c^{12} + 997766 \, a b c^{4}\right )} x + 13 \, {\left (3720087 \, c^{12} + 5078700 \, a c^{8} x - 6984362 \, 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}}}{8596207620 \, a c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8596207620*(3631868240*sqrt(3)*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) + 18159341
20*(-c^2)^(2/3)*b^2*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)) - 3631868240*(-c^2)^(2/3)*b^2*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(
1/3)*c - (-c^2)^(2/3)) + 3*(3486784401*c^17 + 641744532*a^2*c^9*x^2 - 11373139206*b*c^9 + 567*(885735*a*c^13 +
1179178*a*b*c^5)*x - 2*(301327047*c^14 + 573080508*a^2*c^6*x^2 - 286540254*b*c^6 + 988*(177147*a*c^10 + 91899
5*a*b*c^2)*x + 988*(177147*c^10 - 580041*a*c^6*x - 918995*b*c^2)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^
(3/4) + 81*(6200145*c^13 + 7922772*a*c^9*x - 8254246*b*c^5)*sqrt(a^2*x^2 - b) + 6*(129140163*c^15 + 92432340*a
^2*c^7*x^2 - 455559390*b*c^7 + 364*(177147*a*c^11 + 498883*a*b*c^3)*x + 364*(177147*c^11 + 253935*a*c^7*x - 49
8883*b*c^3)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)) - 9*(129140163*c^16 + 66023100*a^2*c^8*x^2 - 4423
54770*b*c^8 + 91*(531441*a*c^12 + 997766*a*b*c^4)*x + 13*(3720087*c^12 + 5078700*a*c^8*x - 6984362*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^7)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}}}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________