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

Optimal. Leaf size=470 \[ -\frac {15 b \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{32 a c^{9/4}}+\frac {15 b \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{32 a c^{9/4}}+\frac {\left (3072 a c^4 x+4620 b c\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-2688 a c^3 x-5775 b\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (2464 a c^2 x-4096 c^5\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt {a^2 x^2-b} \left (3072 c^4 \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}-2688 c^3 \sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+2464 c^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}\right )}{6160 a c^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{2/3}} \]

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Rubi [F]  time = 0.31, 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 [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[(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][(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 {\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 {\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 = 174.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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]

Integrate[(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]

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

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IntegrateAlgebraic [A]  time = 0.96, size = 470, normalized size = 1.00 \begin {gather*} \frac {\left (4620 b c+3072 a c^4 x\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-5775 b-2688 a c^3 x\right ) \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}+\left (-4096 c^5+2464 a c^2 x\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\sqrt {-b+a^2 x^2} \left (3072 c^4 \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}-2688 c^3 \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}+2464 c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3} \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}\right )}{6160 a c^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{2/3}}-\frac {15 b \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{32 a c^{9/4}}+\frac {15 b \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{32 a c^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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]

((4620*b*c + 3072*a*c^4*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-5775*b - 2688*a*c^3*x)*(a*x + Sqrt
[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-4096*c^5 + 2464*a*c^2*x)*(a*x + Sqrt[-b
+ a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*(3072*c^4*(c + (a*x + Sqr
t[-b + a^2*x^2])^(1/3))^(3/4) - 2688*c^3*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3
))^(3/4) + 2464*c^2*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(6160*a*c^
2*(a*x + Sqrt[-b + a^2*x^2])^(2/3)) - (15*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(32*
a*c^(9/4)) + (15*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(32*a*c^(9/4))

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fricas [A]  time = 0.67, size = 433, normalized size = 0.92 \begin {gather*} \frac {23100 \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\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^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} - \sqrt {a^{2} b^{4} c^{5} \sqrt {\frac {b^{4}}{a^{4} c^{9}}} + b^{6} \sqrt {c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}}} a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}}}{b^{4}}\right ) + 5775 \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} \log \left (3375 \, a^{3} c^{7} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {3}{4}} + 3375 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 5775 \, a c^{2} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {1}{4}} \log \left (-3375 \, a^{3} c^{7} \left (\frac {b^{4}}{a^{4} c^{9}}\right )^{\frac {3}{4}} + 3375 \, b^{3} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (4096 \, c^{5} - 2464 \, a c^{2} x - 2464 \, \sqrt {a^{2} x^{2} - b} c^{2} + 21 \, {\left (128 \, c^{3} + 275 \, a x - 275 \, \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 12 \, {\left (256 \, c^{4} + 385 \, a c x - 385 \, \sqrt {a^{2} x^{2} - 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}}}{24640 \, a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

1/24640*(23100*a*c^2*(b^4/(a^4*c^9))^(1/4)*arctan(-(a*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)*c^2*(b^4
/(a^4*c^9))^(1/4) - sqrt(a^2*b^4*c^5*sqrt(b^4/(a^4*c^9)) + b^6*sqrt(c + (a*x + sqrt(a^2*x^2 - b))^(1/3)))*a*c^
2*(b^4/(a^4*c^9))^(1/4))/b^4) + 5775*a*c^2*(b^4/(a^4*c^9))^(1/4)*log(3375*a^3*c^7*(b^4/(a^4*c^9))^(3/4) + 3375
*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 5775*a*c^2*(b^4/(a^4*c^9))^(1/4)*log(-3375*a^3*c^7*(b^4/(a
^4*c^9))^(3/4) + 3375*b^3*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 4*(4096*c^5 - 2464*a*c^2*x - 2464*sqr
t(a^2*x^2 - b)*c^2 + 21*(128*c^3 + 275*a*x - 275*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3) - 12*(256*
c^4 + 385*a*c*x - 385*sqrt(a^2*x^2 - 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*c^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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