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

Optimal. Leaf size=1225 \[ -\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}{531441 \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}{1594323 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}{3188646 a c^{26/3}}+\frac {2 \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}{\sqrt {3} 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}{3 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}{3 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 (4649045868 a^2 x^2 c^{13}-2324522934 b c^{13}[phone] a^3 x^3 c^9-3482001432 a b x c^9-911118780 b^2 c^5-1963217256 a b^2 x 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 (4649045868 a x c^{13}[phone] a^2 x^2 c^9-1160667144 b c^9-1963217256 b^2 c\right )+\left (-7231849128 a^2 x^2 c^{12}[phone] b c^{12}[phone] a^3 x^3 c^8+558667785312 a b x c^8+1032601284 b^2 c^4+6544057520 a b^2 x\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-6973568802 a x c^{14}-5223002148 a^2 x^2 c^{10}[phone] b c^{10}[phone] 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 (20920706406 a x c^{15}[phone] a^2 x^2 c^{11}-1506635235 b c^{11}-1204701498 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}}}\right )+\left (-7231849128 a^3 x^3 c^{12}[phone] a b x c^{12}[phone] a^4 x^4 c^8-319355415288 b^2 c^8+470457082368 a^2 b x^2 c^8+1032601284 a b^2 x c^4-3272028760 b^3+6544057520 a^2 b^2 x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}+\left (-6973568802 a^2 x^2 c^{14}[phone] b c^{14}-5223002148 a^3 x^3 c^{10}[phone] a b x c^{10}[phone] b^2 c^6+1472412942 a b^2 x 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 (20920706406 a^2 x^2 c^{15}-10460353203 b c^{15}[phone] a^3 x^3 c^{11}-4519905705 a b x c^{11}-748701954 b^2 c^7-1204701498 a b^2 x 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}}}}{161719622064 a c^8 \left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}} \]

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Rubi [F]  time = 0.95, 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]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, 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))^(1/3))/(a*x + Sqrt[-b + a^2*x^2])^(1/4),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \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))^(1/3))/(a*x + Sqrt[-b + a^2*x^2])^(1/4),x
]

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 4.09, size = 1225, normalized size = 1.00 \begin {gather*} \frac {\left (-3272028760 b^3-319355415288 b^2 c^8+1032601284 a b^2 c^4 x+5423886846 a b c^{12} x+6544057520 a^2 b^2 x^2+470457082368 a^2 b c^8 x^2-7231849128 a^3 c^{12} x^3+176421405888 a^4 c^8 x^4\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-748701954 b^2 c^7-10460353203 b c^{15}-1204701498 a b^2 c^3 x-4519905705 a b c^{11} x+20920706406 a^2 c^{15} x^2+6026540940 a^3 c^{11} 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 (820006902 b^2 c^6+3486784401 b c^{14}[phone] a b^2 c^2 x+3917251611 a b c^{10} x-6973568802 a^2 c^{14} x^2-5223002148 a^3 c^{10} 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 (-911118780 b^2 c^5-2324522934 b c^{13}-1963217256 a b^2 c x-3482001432 a b c^9 x+4649045868 a^2 c^{13} x^2+4642668576 a^3 c^9 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 (1032601284 b^2 c^4+1807962282 b c^{12}[phone] a b^2 x+558667785312 a b c^8 x-7231849128 a^2 c^{12} x^2+176421405888 a^3 c^8 x^3\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (-1204701498 b^2 c^3-1506635235 b c^{11}[phone] a c^{15} x+6026540940 a^2 c^{11} 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 (1472412942 b^2 c^2+1305750537 b c^{10}-6973568802 a c^{14} x-5223002148 a^2 c^{10} 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 (-1963217256 b^2 c-1160667144 b c^9+4649045868 a c^{13} x+4642668576 a^2 c^9 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 )}{161719622064 a c^8 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-\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 )}{531441 \sqrt {3} a c^{26/3}}+\frac {2 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 )}{\sqrt {3} 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 )}{1594323 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 )}{3 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 )}{3188646 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 )}{3 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))^(1/3))/(a*x + Sqrt[-b + a^2*x^2]
)^(1/4),x]

[Out]

((-3272028760*b^3 - 319355415288*b^2*c^8 + 1032601284*a*b^2*c^4*x + 5423886846*a*b*c^12*x + 6544057520*a^2*b^2
*x^2 + 470457082368*a^2*b*c^8*x^2 - 7231849128*a^3*c^12*x^3 + 176421405888*a^4*c^8*x^4)*(c + (a*x + Sqrt[-b +
a^2*x^2])^(1/4))^(1/3) + (-748701954*b^2*c^7 - 10460353203*b*c^15 - 1204701498*a*b^2*c^3*x - 4519905705*a*b*c^
11*x + 20920706406*a^2*c^15*x^2 + 6026540940*a^3*c^11*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) + (820006902*b^2*c^6 + 3486784401*b*c^14 + 1472412942*a*b^2*c^2*x + 3917251611*a*b*
c^10*x - 6973568802*a^2*c^14*x^2 - 5223002148*a^3*c^10*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b
+ a^2*x^2])^(1/4))^(1/3) + (-911118780*b^2*c^5 - 2324522934*b*c^13 - 1963217256*a*b^2*c*x - 3482001432*a*b*c^
9*x + 4649045868*a^2*c^13*x^2 + 4642668576*a^3*c^9*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]*((1032601284*b^2*c^4 + 1807962282*b*c^12 + 6544057520*a*b^2*x + 5
58667785312*a*b*c^8*x - 7231849128*a^2*c^12*x^2 + 176421405888*a^3*c^8*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1
/4))^(1/3) + (-1204701498*b^2*c^3 - 1506635235*b*c^11 + 20920706406*a*c^15*x + 6026540940*a^2*c^11*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) + (1472412942*b^2*c^2 + 1305750537*b*c
^10 - 6973568802*a*c^14*x - 5223002148*a^2*c^10*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*
x^2])^(1/4))^(1/3) + (-1963217256*b^2*c - 1160667144*b*c^9 + 4649045868*a*c^13*x + 4642668576*a^2*c^9*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)))/(161719622064*a*c^8*(a*x + Sqrt[-
b + a^2*x^2])^(9/4)) - (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))])/(531441*Sqrt[3]*a*c^(26/3)) + (2*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1
/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[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)])/(1594323*a*c^(26/3)) - (2*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(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 + Sqrt[-b + a^
2*x^2])^(1/4))^(2/3)])/(3188646*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)])/(3*a*c^(2/3))

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fricas [A]  time = 0.71, size = 719, normalized size = 0.59 \begin {gather*} \frac {304304 \, \sqrt {3} {\left (1062882 \, b^{2} c^{9} - 21505 \, b^{3} 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 (1062882 \, b^{2} c^{8} - 21505 \, b^{3}\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 (1062882 \, b^{2} c^{8} - 21505 \, b^{3}\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 (10460353203 \, b c^{17} - 1497403908 \, a^{2} b c^{9} x^{2} + 748701954 \, b^{2} c^{9} + 567 \, {\left (2657205 \, a b c^{13} - 2124694 \, a b^{2} c^{5}\right )} x - 2 \, {\left (35937693792 \, a^{3} c^{10} x^{3} + 903981141 \, b c^{14} - 1032601284 \, a^{2} b c^{6} x^{2} + 516300642 \, b^{2} c^{6} - 6916 \, {\left (28874961 \, a b c^{10} + 236555 \, a b^{2} c^{2}\right )} x - 988 \, {\left (36374184 \, a^{2} c^{10} x^{2} - 161617113 \, b c^{10} - 1045143 \, a b c^{6} x - 1655885 \, b^{2} 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 (2657205 \, b c^{13} + 2640924 \, a b c^{9} x + 2124694 \, b^{2} c^{5}\right )} \sqrt {a^{2} x^{2} - b} + 6 \, {\left (387420489 \, b c^{15} - 303706260 \, a^{2} b c^{7} x^{2} + 151853130 \, b^{2} c^{7} + 364 \, {\left (531441 \, a b c^{11} - 898909 \, a b^{2} c^{3}\right )} x + 52 \, {\left (3720087 \, b c^{11} + 5840505 \, a b c^{7} x + 6292363 \, b^{2} c^{3}\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (387420489 \, b c^{16} - 182223756 \, a^{2} b c^{8} x^{2} + 91111878 \, b^{2} c^{8} + 91 \, {\left (1594323 \, a b c^{12} - 1797818 \, a b^{2} c^{4}\right )} x + 13 \, {\left (11160261 \, b c^{12} + 14017212 \, a b c^{8} x + 12584726 \, b^{2} 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}}}{485158866192 \, a b 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))^(1/3)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="fricas")

[Out]

1/485158866192*(304304*sqrt(3)*(1062882*b^2*c^9 - 21505*b^3*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*(1062882*b^2*c^8 - 21505*b^3)*(-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*(1062882*b^2*c^8 -
21505*b^3)*(-c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - (-c^2)^(2/3)) + 3*(10460353203*b*
c^17 - 1497403908*a^2*b*c^9*x^2 + 748701954*b^2*c^9 + 567*(2657205*a*b*c^13 - 2124694*a*b^2*c^5)*x - 2*(359376
93792*a^3*c^10*x^3 + 903981141*b*c^14 - 1032601284*a^2*b*c^6*x^2 + 516300642*b^2*c^6 - 6916*(28874961*a*b*c^10
+ 236555*a*b^2*c^2)*x - 988*(36374184*a^2*c^10*x^2 - 161617113*b*c^10 - 1045143*a*b*c^6*x - 1655885*b^2*c^2)*
sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 567*(2657205*b*c^13 + 2640924*a*b*c^9*x + 2124694*b^2*c^5
)*sqrt(a^2*x^2 - b) + 6*(387420489*b*c^15 - 303706260*a^2*b*c^7*x^2 + 151853130*b^2*c^7 + 364*(531441*a*b*c^11
- 898909*a*b^2*c^3)*x + 52*(3720087*b*c^11 + 5840505*a*b*c^7*x + 6292363*b^2*c^3)*sqrt(a^2*x^2 - b))*sqrt(a*x
+ sqrt(a^2*x^2 - b)) - 9*(387420489*b*c^16 - 182223756*a^2*b*c^8*x^2 + 91111878*b^2*c^8 + 91*(1594323*a*b*c^1
2 - 1797818*a*b^2*c^4)*x + 13*(11160261*b*c^12 + 14017212*a*b*c^8*x + 12584726*b^2*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*b*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))^(1/3)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="giac")

[Out]

Timed out

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

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

[Out]

int((a^2*x^2-b)^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),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 {1}{3}}}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}\,{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))^(1/3)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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