3.2968 \(\int \sqrt {a+b \sqrt {c x^3}} \, dx\)

Optimal. Leaf size=770 \[ \frac {4 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} E\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {12 a \sqrt {a+b \sqrt {c x^3}}}{7 b^{2/3} \sqrt [3]{c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}+\frac {4}{7} x \sqrt {a+b \sqrt {c x^3}} \]

[Out]

4/7*3^(3/4)*a^(4/3)*EllipticF((a^(1/3)*(1-3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(
1/3)*c^(2/3)*x^2/(c*x^3)^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))*((a^(2/3)+b
^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(
1/2))^2)^(1/2)/b^(2/3)/c^(1/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/
3)*c^(2/3)*x^2/(c*x^3)^(1/2))^2)^(1/2)/(a+b*(c*x^3)^(1/2))^(1/2)-6/7*3^(1/4)*a^(4/3)*EllipticE((a^(1/3)*(1-3^(
1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2)),I*3^(1/2)+2*I
)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^
(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))^2)^(1/2)/b^(2/3)/c^(1
/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2
))^2)^(1/2)/(a+b*(c*x^3)^(1/2))^(1/2)+4/7*x*(a+b*(c*x^3)^(1/2))^(1/2)+12/7*a*(a+b*(c*x^3)^(1/2))^(1/2)/b^(2/3)
/c^(1/3)/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))

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Rubi [A]  time = 0.39, antiderivative size = 770, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {255, 243, 279, 303, 218, 1877} \[ \frac {4 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} E\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {12 a \sqrt {a+b \sqrt {c x^3}}}{7 b^{2/3} \sqrt [3]{c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}+\frac {4}{7} x \sqrt {a+b \sqrt {c x^3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(4*x*Sqrt[a + b*Sqrt[c*x^3]])/7 + (12*a*Sqrt[a + b*Sqrt[c*x^3]])/(7*b^(2/3)*c^(1/3)*((1 + Sqrt[3])*a^(1/3) + (
b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])) - (6*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3
) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt
[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*b^(2/3)*c^(1/3)*Sqr
t[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[
c*x^3])^2]*Sqrt[a + b*Sqrt[c*x^3]]) + (4*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])
*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1
/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/
((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*b^(2/3)*c^(1/3)*Sqrt[(a^(1/3
)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2
]*Sqrt[a + b*Sqrt[c*x^3]])

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 243

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*
x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, p}, x] && FractionQ[n]

Rule 255

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Subst[Int[(a + b*c^n
*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, p, q}, x] && Fraction
Q[n]

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \int \sqrt {a+b \sqrt {c x^3}} \, dx &=\operatorname {Subst}\left (\int \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int x \sqrt {a+b \sqrt {c} x^3} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{7} x \sqrt {a+b \sqrt {c x^3}}+\operatorname {Subst}\left (\frac {1}{7} (6 a) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{7} x \sqrt {a+b \sqrt {c x^3}}+\operatorname {Subst}\left (\frac {(6 a) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{c} x}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{7 \sqrt [3]{b} \sqrt [6]{c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )+\operatorname {Subst}\left (\frac {\left (6 \sqrt {2 \left (2-\sqrt {3}\right )} a^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{7 \sqrt [3]{b} \sqrt [6]{c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{7} x \sqrt {a+b \sqrt {c x^3}}+\frac {12 a \sqrt {a+b \sqrt {c x^3}}}{7 b^{2/3} \sqrt [3]{c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {4 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}\\ \end {align*}

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Mathematica [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \sqrt {a+b \sqrt {c x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^3]], x]

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fricas [F]  time = 5.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sqrt {c x^{3}} b + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(sqrt(c*x^3)*b + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {c x^{3}} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a), x)

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maple [A]  time = 0.35, size = 854, normalized size = 1.11 \[ \frac {4 a \,b^{2} c x +4 \sqrt {c \,x^{3}}\, b^{3} c x +3 i \sqrt {\frac {\sqrt {c \,x^{3}}\, b -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right ) x}}\, \sqrt {-\frac {i \left (2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \left (-a \,b^{2} c \right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {\frac {i \left (2 \sqrt {c \,x^{3}}\, b -i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {2}\, a \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (2 \sqrt {c \,x^{3}}\, b -i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )+3 \sqrt {\frac {\sqrt {c \,x^{3}}\, b -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right ) x}}\, \sqrt {-\frac {i \left (2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \left (-a \,b^{2} c \right )^{\frac {2}{3}} \sqrt {\frac {i \left (2 \sqrt {c \,x^{3}}\, b -i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {2}\, a \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (2 \sqrt {c \,x^{3}}\, b -i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )-2 i \sqrt {\frac {\sqrt {c \,x^{3}}\, b -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right ) x}}\, \sqrt {-\frac {i \left (2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \left (-a \,b^{2} c \right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {\frac {i \left (2 \sqrt {c \,x^{3}}\, b -i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {2}\, a \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (2 \sqrt {c \,x^{3}}\, b -i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )}{7 \sqrt {a +\sqrt {c \,x^{3}}\, b}\, b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7/c*(3*I*(((c*x^3)^(1/2)*b-(-a*b^2*c)^(1/3)*x)/(-a*b^2*c)^(1/3)/(I*3^(1/2)-3)/x)^(1/2)*(-I*(2*(c*x^3)^(1/2)*
b+I*3^(1/2)*(-a*b^2*c)^(1/3)*x+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2)*EllipticE(1/6*3^(1/2)*2^(
1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*(c*x^3)^(1/2)*b+(-a*b^2*c)^(1/3)*x)*3^(1/2)/x/(-a*b^2*c)^(1/3))^(1/2)
,2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-a*b^2*c)^(2/3)*3^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*(c*x^3)
^(1/2)*b+(-a*b^2*c)^(1/3)*x)*3^(1/2)/x/(-a*b^2*c)^(1/3))^(1/2)*2^(1/2)*a-2*I*(((c*x^3)^(1/2)*b-(-a*b^2*c)^(1/3
)*x)/(-a*b^2*c)^(1/3)/(I*3^(1/2)-3)/x)^(1/2)*(-I*(2*(c*x^3)^(1/2)*b+I*3^(1/2)*(-a*b^2*c)^(1/3)*x+(-a*b^2*c)^(1
/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*(c*
x^3)^(1/2)*b+(-a*b^2*c)^(1/3)*x)*3^(1/2)/x/(-a*b^2*c)^(1/3))^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-
a*b^2*c)^(2/3)*3^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*(c*x^3)^(1/2)*b+(-a*b^2*c)^(1/3)*x)*3^(1/2)/x/(-a*b
^2*c)^(1/3))^(1/2)*2^(1/2)*a+3*(((c*x^3)^(1/2)*b-(-a*b^2*c)^(1/3)*x)/(-a*b^2*c)^(1/3)/(I*3^(1/2)-3)/x)^(1/2)*(
-I*(2*(c*x^3)^(1/2)*b+I*3^(1/2)*(-a*b^2*c)^(1/3)*x+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2)*Ellip
ticE(1/6*3^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*(c*x^3)^(1/2)*b+(-a*b^2*c)^(1/3)*x)*3^(1/2)/x/(-a
*b^2*c)^(1/3))^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-a*b^2*c)^(2/3)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/
3)+2*(c*x^3)^(1/2)*b+(-a*b^2*c)^(1/3)*x)*3^(1/2)/x/(-a*b^2*c)^(1/3))^(1/2)*2^(1/2)*a+4*(c*x^3)^(1/2)*x*b^3*c+4
*x*a*b^2*c)/b^2/(a+(c*x^3)^(1/2)*b)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {c x^{3}} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+b\,\sqrt {c\,x^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^3)^(1/2))^(1/2),x)

[Out]

int((a + b*(c*x^3)^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sqrt {c x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(c*x**3)), x)

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