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

Optimal. Leaf size=697 \[ \frac {182 b \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{729 a c^{16/3}}-\frac {91 b \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{729 a c^{16/3}}+\frac {182 b \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{243 \sqrt {3} a c^{16/3}}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (3645 a c^5 x-2730 b c\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3} \left (3640 a b x+1944 b c^4\right )+\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (6561 a c^7 x-2106 b c^3\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt {\sqrt {a^2 x^2-b}+a x} \left (2340 b c^2-4374 a c^6 x\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt {a^2 x^2-b} \left (6561 c^7 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}-4374 c^6 \sqrt {\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+3645 c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+3640 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}\right )}{4860 a c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \]

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Rubi [F]  time = 0.36, 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 [4]{a x+\sqrt {-b+a^2 x^2}} \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[1/((a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]

[Out]

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

Rubi steps

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 1.52, size = 697, normalized size = 1.00 \begin {gather*} \frac {\left (1944 b c^4+3640 a b x\right ) \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-2106 b c^3+6561 a c^7 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 (2340 b c^2-4374 a c^6 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 (-2730 b c+3645 a c^5 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 (3640 b \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+6561 c^7 \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}-4374 c^6 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+3645 c^5 \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 )}{4860 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {182 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 )}{243 \sqrt {3} a c^{16/3}}+\frac {182 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{729 a c^{16/3}}-\frac {91 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 )}{729 a c^{16/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1944*b*c^4 + 3640*a*b*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-2106*b*c^3 + 6561*a*c^7*x)*(a*x +
Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (2340*b*c^2 - 4374*a*c^6*x)*Sqrt[a*x
+ Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-2730*b*c + 3645*a*c^5*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]*(3640*b*(c + (a*x + Sqrt[
-b + a^2*x^2])^(1/4))^(2/3) + 6561*c^7*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))
^(2/3) - 4374*c^6*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 3645*c^5*(a*x
+ Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)))/(4860*a*c^5*(a*x + Sqrt[-b + a^2*x^
2])^(5/4)) + (182*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(2
43*Sqrt[3]*a*c^(16/3)) + (182*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(729*a*c^(16/3))
- (91*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*a*c^(16/3))

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fricas [A]  time = 0.87, size = 1036, normalized size = 1.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/14580*(5460*sqrt(1/3)*b^2*c*sqrt(-1/c^(2/3))*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(-1/c^(2/3)) - 3*(a*c^(2/3)*x + sq
rt(1/3)*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt(-1/c^(2/3)) - 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))^(1/3) + 3*(a*c*x - sqrt(1/3)*(a*c^(4/3)*x - sqrt(a^2*x^2 - b)*c^(4
/3))*sqrt(-1/c^(2/3)) - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 2*b) - 1820*b^2*c^(2/3)*log((c
+ (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 36
40*b^2*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) - c^(1/3)) + 3*(6561*b*c^8 - 2106*a*b*c^4*x + 2
106*sqrt(a^2*x^2 - b)*b*c^4 + 8*(486*a^2*c^5*x^2 - 243*b*c^5 + 455*a*b*c*x - (486*a*c^5*x + 455*b*c)*sqrt(a^2*
x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 15*(243*b*c^6 - 182*a*b*c^2*x + 182*sqrt(a^2*x^2 - b)*b*c^2)*sqrt(
a*x + sqrt(a^2*x^2 - b)) - 18*(243*b*c^7 - 130*a*b*c^3*x + 130*sqrt(a^2*x^2 - b)*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*b*c^6), 1/14580*(10920*sqrt(1/3)*b^2*c^(2/3)*arctan
(sqrt(1/3) + 2*sqrt(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)/c^(1/3)) - 1820*b^2*c^(2/3)*log((c + (a*x
+ sqrt(a^2*x^2 - b))^(1/4))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*c^(1/3) + c^(2/3)) + 3640*b^2
*c^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) - c^(1/3)) + 3*(6561*b*c^8 - 2106*a*b*c^4*x + 2106*sq
rt(a^2*x^2 - b)*b*c^4 + 8*(486*a^2*c^5*x^2 - 243*b*c^5 + 455*a*b*c*x - (486*a*c^5*x + 455*b*c)*sqrt(a^2*x^2 -
b))*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 15*(243*b*c^6 - 182*a*b*c^2*x + 182*sqrt(a^2*x^2 - b)*b*c^2)*sqrt(a*x +
sqrt(a^2*x^2 - b)) - 18*(243*b*c^7 - 130*a*b*c^3*x + 130*sqrt(a^2*x^2 - b)*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*b*c^6)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

Integral(1/((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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