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

Optimal. Leaf size=674 \[ -\frac {1989 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {1989 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {\sqrt {a^2 x^2-b} \left (\left (6055526400 a^2 c^7 x^2-4026531840 a c^{10} x+2409402996 b^2 c-1513881600 b c^7\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-5752750080 a^2 c^6 x^2+3523215360 a c^9 x-3011753745 b^2+1438187520 b c^6\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5513052160 a^2 c^5 x^2-3229614080 a c^8 x-26186997760 b c^5+2684354560 c^{11}\right )\right )+\left (6055526400 a^3 c^7 x^3-4026531840 a^2 c^{10} x^2+2409402996 a b^2 c x-4541644800 a b c^7 x-1860655104 b^2 c^4+2013265920 b c^{10}\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\sqrt [3]{\sqrt {a^2 x^2-b}+a x} \left (-5752750080 a^3 c^6 x^3+3523215360 a^2 c^9 x^2-3011753745 a b^2 x+4314562560 a b c^6 x+1976946048 b^2 c^3-1761607680 b c^9\right ) \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{2/3} \left (\sqrt [3]{\sqrt {a^2 x^2-b}+a x}+c\right )^{3/4} \left (5513052160 a^3 c^5 x^3-3229614080 a^2 c^8 x^2-28943523840 a b c^5 x+2684354560 a c^{11} x-2141691552 b^2 c^2+1614807040 b c^8\right )}{12404367360 a c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/3}} \]

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Rubi [F]  time = 1.05, 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 {\sqrt {-b+a^2 x^2} \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[(Sqrt[-b + a^2*x^2]*(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][(Sqrt[-b + a^2*x^2]*(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 {-b+a^2 x^2} \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 {-b+a^2 x^2} \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 = 157.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \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[(Sqrt[-b + a^2*x^2]*(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[(Sqrt[-b + a^2*x^2]*(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 = 1.89, size = 674, normalized size = 1.00 \begin {gather*} \frac {\left (-1860655104 b^2 c^4+2013265920 b c^{10}+2409402996 a b^2 c x-4541644800 a b c^7 x-4026531840 a^2 c^{10} x^2+6055526400 a^3 c^7 x^3\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (1976946048 b^2 c^3-1761607680 b c^9-3011753745 a b^2 x+4314562560 a b c^6 x+3523215360 a^2 c^9 x^2-5752750080 a^3 c^6 x^3\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 (-2141691552 b^2 c^2+1614807040 b c^8-28943523840 a b c^5 x+2684354560 a c^{11} x-3229614080 a^2 c^8 x^2+5513052160 a^3 c^5 x^3\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 (\left (2409402996 b^2 c-1513881600 b c^7-4026531840 a c^{10} x+6055526400 a^2 c^7 x^2\right ) \left (c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}\right )^{3/4}+\left (-3011753745 b^2+1438187520 b c^6+3523215360 a c^9 x-5752750080 a^2 c^6 x^2\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 (-26186997760 b c^5+2684354560 c^{11}-3229614080 a c^8 x+5513052160 a^2 c^5 x^2\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}\right )}{12404367360 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/3}}-\frac {1989 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}}+\frac {1989 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c+\sqrt [3]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [4]{c}}\right )}{16384 a c^{21/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(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]

((-1860655104*b^2*c^4 + 2013265920*b*c^10 + 2409402996*a*b^2*c*x - 4541644800*a*b*c^7*x - 4026531840*a^2*c^10*
x^2 + 6055526400*a^3*c^7*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (1976946048*b^2*c^3 - 1761607680*
b*c^9 - 3011753745*a*b^2*x + 4314562560*a*b*c^6*x + 3523215360*a^2*c^9*x^2 - 5752750080*a^3*c^6*x^3)*(a*x + Sq
rt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-2141691552*b^2*c^2 + 1614807040*b*c^8
 - 28943523840*a*b*c^5*x + 2684354560*a*c^11*x - 3229614080*a^2*c^8*x^2 + 5513052160*a^3*c^5*x^3)*(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]*((2409402996*b^2*c - 15
13881600*b*c^7 - 4026531840*a*c^10*x + 6055526400*a^2*c^7*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) +
(-3011753745*b^2 + 1438187520*b*c^6 + 3523215360*a*c^9*x - 5752750080*a^2*c^6*x^2)*(a*x + Sqrt[-b + a^2*x^2])^
(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-26186997760*b*c^5 + 2684354560*c^11 - 3229614080*a*c^8*
x + 5513052160*a^2*c^5*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(1
2404367360*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/3)) - (1989*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(
1/4)/c^(1/4)])/(16384*a*c^(21/4)) + (1989*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(
16384*a*c^(21/4))

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fricas [A]  time = 0.65, size = 562, normalized size = 0.83 \begin {gather*} \frac {12047014980 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \arctan \left (-\frac {a b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}} c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} - \sqrt {a^{2} b^{8} c^{11} \sqrt {\frac {b^{8}}{a^{4} c^{21}}} + b^{12} \sqrt {c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}}} a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}}}{b^{8}}\right ) + 3011753745 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (7868724669 \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) - 3011753745 \, a c^{5} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {1}{4}} \log \left (-7868724669 \, a^{3} c^{16} \left (\frac {b^{8}}{a^{4} c^{21}}\right )^{\frac {3}{4}} + 7868724669 \, b^{6} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{3}}\right )}^{\frac {1}{4}}\right ) + 4 \, {\left (2684354560 \, c^{11} + 2756526080 \, a^{2} c^{5} x^{2} - 26186997760 \, b c^{5} - 2464 \, {\left (655360 \, a c^{8} + 869193 \, a b c^{2}\right )} x + 21 \, {\left (83886080 \, c^{9} + 188280576 \, a^{2} c^{3} x^{2} - 94140288 \, b c^{3} - 1045 \, {\left (65536 \, a c^{6} + 137241 \, a b\right )} x - 209 \, {\left (327680 \, c^{6} + 900864 \, a c^{3} x - 686205 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {2}{3}} - 2464 \, {\left (655360 \, c^{8} - 1118720 \, a c^{5} x - 869193 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - 12 \, {\left (167772160 \, c^{10} + 310109184 \, a^{2} c^{4} x^{2} - 155054592 \, b c^{4} - 77 \, {\left (1638400 \, a c^{7} + 2607579 \, a b c\right )} x - 77 \, {\left (1638400 \, c^{7} + 4027392 \, a c^{4} x - 2607579 \, b c\right )} \sqrt {a^{2} x^{2} - b}\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}}}{49617469440 \, a c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2-b)^(1/2)*(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]

1/49617469440*(12047014980*a*c^5*(b^8/(a^4*c^21))^(1/4)*arctan(-(a*b^6*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(
1/4)*c^5*(b^8/(a^4*c^21))^(1/4) - sqrt(a^2*b^8*c^11*sqrt(b^8/(a^4*c^21)) + b^12*sqrt(c + (a*x + sqrt(a^2*x^2 -
 b))^(1/3)))*a*c^5*(b^8/(a^4*c^21))^(1/4))/b^8) + 3011753745*a*c^5*(b^8/(a^4*c^21))^(1/4)*log(7868724669*a^3*c
^16*(b^8/(a^4*c^21))^(3/4) + 7868724669*b^6*(c + (a*x + sqrt(a^2*x^2 - b))^(1/3))^(1/4)) - 3011753745*a*c^5*(b
^8/(a^4*c^21))^(1/4)*log(-7868724669*a^3*c^16*(b^8/(a^4*c^21))^(3/4) + 7868724669*b^6*(c + (a*x + sqrt(a^2*x^2
 - b))^(1/3))^(1/4)) + 4*(2684354560*c^11 + 2756526080*a^2*c^5*x^2 - 26186997760*b*c^5 - 2464*(655360*a*c^8 +
869193*a*b*c^2)*x + 21*(83886080*c^9 + 188280576*a^2*c^3*x^2 - 94140288*b*c^3 - 1045*(65536*a*c^6 + 137241*a*b
)*x - 209*(327680*c^6 + 900864*a*c^3*x - 686205*b)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(2/3) - 2464*(
655360*c^8 - 1118720*a*c^5*x - 869193*b*c^2)*sqrt(a^2*x^2 - b) - 12*(167772160*c^10 + 310109184*a^2*c^4*x^2 -
155054592*b*c^4 - 77*(1638400*a*c^7 + 2607579*a*b*c)*x - 77*(1638400*c^7 + 4027392*a*c^4*x - 2607579*b*c)*sqrt
(a^2*x^2 - b))*(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^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((a^2*x^2-b)^(1/2)*(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 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{3}}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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