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

Optimal. Leaf size=445 \[ -\frac {585 b \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{2048 a c^{17/4}}+\frac {585 b \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{2048 a c^{17/4}}-\frac {585 b \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{1024 a c^4 \sqrt [3]{\sqrt {a^2 x^2-b}+a x}}+\frac {117 b \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{256 a c^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{2/3}}-\frac {13 b \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{32 a c^2 \left (\sqrt {a^2 x^2-b}+a x\right )}+\frac {3 b \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{8 a c \left (\sqrt {a^2 x^2-b}+a x\right )^{4/3}}-\frac {8 c \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{7 a}+\frac {6 \sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}}{7 a} \]

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Rubi [F]  time = 0.38, 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]{a x+\sqrt {-b+a^2 x^2}} \sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 1.38, size = 445, normalized size = 1.00 \begin {gather*} -\frac {8 c \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{7 a}+\frac {3 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{8 a c \left (a x+\sqrt {-b+a^2 x^2}\right )^{4/3}}-\frac {13 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{32 a c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )}+\frac {117 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{256 a c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}}-\frac {585 b \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{1024 a c^4 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}+\frac {6 \sqrt [3]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}}{7 a}-\frac {585 b \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{2048 a c^{17/4}}+\frac {585 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{2048 a c^{17/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(7*a) + (3*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))
/(8*a*c*(a*x + Sqrt[-b + a^2*x^2])^(4/3)) - (13*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(32*a*c^2*(a*x
+ Sqrt[-b + a^2*x^2])) + (117*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(256*a*c^3*(a*x + Sqrt[-b + a^2
*x^2])^(2/3)) - (585*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(1024*a*c^4*(a*x + Sqrt[-b + a^2*x^2])^(1
/3)) + (6*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(7*a) - (585*b*ArcTan
[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2048*a*c^(17/4)) + (585*b*ArcTanh[(c + (a*x + Sqrt[-b
+ a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2048*a*c^(17/4))

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fricas [A]  time = 1.06, size = 469, normalized size = 1.05 \begin {gather*} \frac {16380 \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}} c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} - \sqrt {a^{2} b^{4} c^{9} \sqrt {\frac {b^{4}}{a^{4} c^{17}}} + b^{6} \sqrt {c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}}} a c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}}}{b^{4}}\right ) + 4095 \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \log \left (200201625 \, a^{3} c^{13} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {3}{4}} + 200201625 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4095 \, a b c^{4} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {1}{4}} \log \left (-200201625 \, a^{3} c^{13} \left (\frac {b^{4}}{a^{4} c^{17}}\right )^{\frac {3}{4}} + 200201625 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (8192 \, b c^{5} + 2912 \, a b c^{2} x - 2912 \, \sqrt {a^{2} x^{2} - b} b c^{2} - 21 \, {\left (256 \, a^{2} c^{3} x^{2} - 128 \, b c^{3} - 195 \, a b x - {\left (256 \, a c^{3} x - 195 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 12 \, {\left (512 \, b c^{4} + 273 \, a b c x - 273 \, \sqrt {a^{2} x^{2} - b} b c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {3}{4}}}{28672 \, a b c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28672*(16380*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*arctan(-(a*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)*c^4*(
b^4/(a^4*c^17))^(1/4) - sqrt(a^2*b^4*c^9*sqrt(b^4/(a^4*c^17)) + b^6*sqrt(c + (a*x + sqrt(a^2*x^2 - b))^(1/3)))
*a*c^4*(b^4/(a^4*c^17))^(1/4))/b^4) + 4095*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*log(200201625*a^3*c^13*(b^4/(a^4*c^1
7))^(3/4) + 200201625*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 4095*a*b*c^4*(b^4/(a^4*c^17))^(1/4)*l
og(-200201625*a^3*c^13*(b^4/(a^4*c^17))^(3/4) + 200201625*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 4
*(8192*b*c^5 + 2912*a*b*c^2*x - 2912*sqrt(a^2*x^2 - b)*b*c^2 - 21*(256*a^2*c^3*x^2 - 128*b*c^3 - 195*a*b*x - (
256*a*c^3*x - 195*b)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3) - 12*(512*b*c^4 + 273*a*b*c*x - 273*sq
rt(a^2*x^2 - b)*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/3))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(3/4))/(a*b*c^4)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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