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

Optimal. Leaf size=719 \[ \frac {20 b \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{243 a c^{11/3}}-\frac {10 b \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{243 a c^{11/3}}-\frac {20 b \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3} a c^{11/3}}+\frac {\sqrt {\sqrt {a^2 x^2-b}+a x} \left (-4374 a c^5 x-5460 b c\right ) \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c} \left (3402 a c^4 x+9100 b\right )+\left (68040 a^2 c^3 x^2-19683 a c^7 x+2835 b c^3\right ) \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (6561 a c^6 x+4095 b c^2\right ) \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {a^2 x^2-b} \left (6561 c^6 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-4374 c^5 \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+3402 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (68040 a c^3 x-19683 c^7\right ) \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}\right )}{73710 a c^3 \sqrt {a^2 x^2-b}+73710 a^2 c^3 x} \]

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Rubi [F]  time = 0.05, 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 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.86, size = 569, normalized size = 0.79 \begin {gather*} -\frac {6 \left (-\frac {10 b \log \left (\sqrt [3]{c}-\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}\right )}{729 c^{11/3}}+\frac {5 b \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{729 c^{11/3}}+\frac {10 b \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [3]{c}}+1}{\sqrt {3}}\right )}{243 \sqrt {3} c^{11/3}}+\frac {1}{4} c^3 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{4/3}-\frac {5 b \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{243 c^3 \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}-\frac {3}{7} c^2 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{7/3}+\frac {b \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{81 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {1}{13} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{13/3}+\frac {3}{10} c \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{10/3}-\frac {b \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{108 c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}-\frac {b \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{12 \left (\sqrt {a^2 x^2-b}+a x\right )}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-1/12*(b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(a*x + Sqrt[-b + a^2*x^2]) - (b*(c + (a*x + Sqrt[-
b + a^2*x^2])^(1/4))^(1/3))/(108*c*(a*x + Sqrt[-b + a^2*x^2])^(3/4)) + (b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4
))^(1/3))/(81*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - (5*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(243*c^
3*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) + (c^3*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(4/3))/4 - (3*c^2*(c + (a*x
+ Sqrt[-b + a^2*x^2])^(1/4))^(7/3))/7 + (3*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(10/3))/10 - (c + (a*x + S
qrt[-b + a^2*x^2])^(1/4))^(13/3)/13 + (10*b*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/c^(1/
3))/Sqrt[3]])/(243*Sqrt[3]*c^(11/3)) - (10*b*Log[c^(1/3) - (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(729
*c^(11/3)) + (5*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)])/(729*c^(11/3))))/a

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IntegrateAlgebraic [A]  time = 1.24, size = 719, normalized size = 1.00 \begin {gather*} \frac {\left (2835 b c^3-19683 a c^7 x+68040 a^2 c^3 x^2\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (4095 b c^2+6561 a c^6 x\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 (-5460 b c-4374 a c^5 x\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (9100 b+3402 a c^4 x\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 (-19683 c^7+68040 a c^3 x\right ) \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+6561 c^6 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-4374 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+3402 c^4 \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 )}{73710 a^2 c^3 x+73710 a c^3 \sqrt {-b+a^2 x^2}}-\frac {20 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 )}{81 \sqrt {3} a c^{11/3}}+\frac {20 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{243 a c^{11/3}}-\frac {10 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 )}{243 a c^{11/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2835*b*c^3 - 19683*a*c^7*x + 68040*a^2*c^3*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (4095*b*c^2 +
6561*a*c^6*x)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-5460*b*c - 43
74*a*c^5*x)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (9100*b + 3402*a*c^4
*x)*(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]*((-1968
3*c^7 + 68040*a*c^3*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + 6561*c^6*(a*x + Sqrt[-b + a^2*x^2])^(1/4
)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) - 4374*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b +
a^2*x^2])^(1/4))^(1/3) + 3402*c^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/
3)))/(73710*a^2*c^3*x + 73710*a*c^3*Sqrt[-b + a^2*x^2]) - (20*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^
2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(81*Sqrt[3]*a*c^(11/3)) + (20*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b +
a^2*x^2])^(1/4))^(1/3)])/(243*a*c^(11/3)) - (10*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)])/(243*a*c^(11/3))

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fricas [A]  time = 0.74, size = 396, normalized size = 0.55 \begin {gather*} -\frac {18200 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} \sqrt {c^{2}} c + 2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {5}{6}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{3 \, c^{2}}\right ) + 9100 \, b {\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 ) - 18200 \, b {\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 (19683 \, c^{9} - 70875 \, a c^{5} x + 2835 \, \sqrt {a^{2} x^{2} - b} c^{5} - 14 \, {\left (243 \, c^{6} + 650 \, a c^{2} x - 650 \, \sqrt {a^{2} x^{2} - b} c^{2}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 6 \, {\left (729 \, c^{7} + 910 \, a c^{3} x - 910 \, \sqrt {a^{2} x^{2} - b} c^{3}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 9 \, {\left (729 \, c^{8} + 455 \, a c^{4} x - 455 \, \sqrt {a^{2} x^{2} - b} c^{4}\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}}}{221130 \, a c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/221130*(18200*sqrt(3)*b*(c^2)^(1/6)*c*arctan(1/3*(sqrt(3)*sqrt(c^2)*c + 2*sqrt(3)*(c^2)^(5/6)*(c + (a*x + s
qrt(a^2*x^2 - b))^(1/4))^(1/3))/c^2) + 9100*b*(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)) - 18200*b*(c^2)^(2/3)*log((c + (a*x +
sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c - (c^2)^(2/3)) + 3*(19683*c^9 - 70875*a*c^5*x + 2835*sqrt(a^2*x^2 - b)*c^5
- 14*(243*c^6 + 650*a*c^2*x - 650*sqrt(a^2*x^2 - b)*c^2)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 6*(729*c^7 + 910*a*
c^3*x - 910*sqrt(a^2*x^2 - b)*c^3)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 9*(729*c^8 + 455*a*c^4*x - 455*sqrt(a^2*x^2
- b)*c^4)*(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^5)

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

[Out]

Timed out

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

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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