3.32.21 \(\int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)

Optimal. Leaf size=699 \[ -\frac {70 b \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{243 a c^{13/3}}+\frac {35 b \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{243 a c^{13/3}}-\frac {70 b \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3} a c^{13/3}}+\frac {\sqrt {\sqrt {a^2 x^2-b}+a x} \left (11550 b c-10935 a c^5 x\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3} \left (9720 a c^4 x-15400 b\right )+\left (8910 b c^3-19683 a c^7 x\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (13122 a c^6 x-9900 b c^2\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt {a^2 x^2-b} \left (-19683 c^7 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+13122 c^6 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}-10935 c^5 \sqrt {\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+9720 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}\right )}{17820 a c^4 \sqrt {a^2 x^2-b}+17820 a^2 c^4 x} \]

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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 \frac {1}{\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 \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.76, size = 575, normalized size = 0.82 \begin {gather*} \frac {6 \left (-\frac {35 b \log \left (1-\frac {\sqrt [3]{c}}{\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}\right )}{729 c^{13/3}}+\frac {35 b \log \left (\frac {c^{2/3}}{\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}+\frac {\sqrt [3]{c}}{\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}+1\right )}{1458 c^{13/3}}+\frac {35 b \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{c}}{\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}+1}{\sqrt {3}}\right )}{243 \sqrt {3} c^{13/3}}+\frac {b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{11/3}}{12 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )}-\frac {37 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{8/3}}{108 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}+\frac {44 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/3}}{81 c^4 \sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {104 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}{243 c^4 \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}-\frac {1}{2} c^3 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\frac {3}{5} c^2 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/3}+\frac {1}{11} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{11/3}-\frac {3}{8} c \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{8/3}\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*(-1/2*(c^3*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)) - (104*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2
/3))/(243*c^4*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) + (3*c^2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(5/3))/5 + (44
*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(5/3))/(81*c^4*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - (3*c*(c + (a*x + Sq
rt[-b + a^2*x^2])^(1/4))^(8/3))/8 - (37*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(8/3))/(108*c^4*(a*x + Sqrt[-
b + a^2*x^2])^(3/4)) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(11/3)/11 + (b*(c + (a*x + Sqrt[-b + a^2*x^2])^(
1/4))^(11/3))/(12*c^4*(a*x + Sqrt[-b + a^2*x^2])) + (35*b*ArcTan[(1 + (2*c^(1/3))/(c + (a*x + Sqrt[-b + a^2*x^
2])^(1/4))^(1/3))/Sqrt[3]])/(243*Sqrt[3]*c^(13/3)) - (35*b*Log[1 - c^(1/3)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/
4))^(1/3)])/(729*c^(13/3)) + (35*b*Log[1 + c^(2/3)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + c^(1/3)/(c +
(a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(1458*c^(13/3))))/a

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IntegrateAlgebraic [A]  time = 1.16, size = 699, normalized size = 1.00 \begin {gather*} \frac {\left (8910 b c^3-19683 a c^7 x\right ) \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-9900 b c^2+13122 a c^6 x\right ) \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}+\left (11550 b c-10935 a c^5 x\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-15400 b+9720 a c^4 x\right ) \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}+\sqrt {-b+a^2 x^2} \left (-19683 c^7 \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+13122 c^6 \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}-10935 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+9720 c^4 \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}\right )}{17820 a^2 c^4 x+17820 a c^4 \sqrt {-b+a^2 x^2}}-\frac {70 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^{13/3}}-\frac {70 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{243 a c^{13/3}}+\frac {35 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^{13/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8910*b*c^3 - 19683*a*c^7*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-9900*b*c^2 + 13122*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))^(2/3) + (11550*b*c - 10935*a*c^5*x)*Sqrt[
a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-15400*b + 9720*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))^(2/3) + Sqrt[-b + a^2*x^2]*(-19683*c^7*(c + (a*x
+ Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 13122*c^6*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2]
)^(1/4))^(2/3) - 10935*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 9720*
c^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)))/(17820*a^2*c^4*x + 17820*a
*c^4*Sqrt[-b + a^2*x^2]) - (70*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^(13/3)) - (70*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(243*
a*c^(13/3)) + (35*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^(13/3))

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fricas [A]  time = 0.74, size = 1039, normalized size = 1.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/53460*(23100*sqrt(1/3)*b*c*sqrt((-c)^(1/3)/c)*log(-6*sqrt(1/3)*(a*(-c)^(2/3)*x - sqrt(a^2*x^2 - b)*(-c)^(2/
3))*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*sqrt((-c)^(1/3)/c) - 3*(a*(-c)
^(2/3)*x - sqrt(1/3)*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt((-c)^(1/3)/c) - sqrt(a^2*x^2 - b)*(-c)^(2/3))*(a*x + s
qrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + 3*(a*c*x - sqrt(1/3)*(a*(-c)^(1/3)*c*x -
sqrt(a^2*x^2 - b)*(-c)^(1/3)*c)*sqrt((-c)^(1/3)/c) - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 2
*b) + 7700*b*(-c)^(2/3)*log((-c)^(2/3) - (-c)^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + (c + (a*x +
sqrt(a^2*x^2 - b))^(1/4))^(2/3)) - 15400*b*(-c)^(2/3)*log((-c)^(1/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(
1/3)) - 3*(19683*c^8 - 8910*a*c^4*x + 8910*sqrt(a^2*x^2 - b)*c^4 - 40*(243*c^5 - 385*a*c*x + 385*sqrt(a^2*x^2
- b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 15*(729*c^6 - 770*a*c^2*x + 770*sqrt(a^2*x^2 - b)*c^2)*sqrt(a*x + sq
rt(a^2*x^2 - b)) - 18*(729*c^7 - 550*a*c^3*x + 550*sqrt(a^2*x^2 - b)*c^3)*(a*x + sqrt(a^2*x^2 - b))^(1/4))*(c
+ (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3))/(a*c^5), -1/53460*(46200*sqrt(1/3)*b*c*sqrt(-(-c)^(1/3)/c)*arctan(-s
qrt(1/3)*(-c)^(1/3)*sqrt(-(-c)^(1/3)/c) + 2*sqrt(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*sqrt(-(-c)^(
1/3)/c)) - 7700*b*(-c)^(2/3)*log((-c)^(2/3) - (-c)^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + (c + (a
*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)) + 15400*b*(-c)^(2/3)*log((-c)^(1/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/
4))^(1/3)) + 3*(19683*c^8 - 8910*a*c^4*x + 8910*sqrt(a^2*x^2 - b)*c^4 - 40*(243*c^5 - 385*a*c*x + 385*sqrt(a^2
*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 15*(729*c^6 - 770*a*c^2*x + 770*sqrt(a^2*x^2 - b)*c^2)*sqrt(a*x
+ sqrt(a^2*x^2 - b)) - 18*(729*c^7 - 550*a*c^3*x + 550*sqrt(a^2*x^2 - b)*c^3)*(a*x + sqrt(a^2*x^2 - b))^(1/4)
)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3))/(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(1/(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 \[\int \frac {1}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

int(1/(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 \frac {1}{\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(1/(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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