3.2969 \(\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx\)

Optimal. Leaf size=810 \[ -\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{2/3} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right ) \sqrt {\frac {-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) b^{4/3}}{8 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {3^{3/4} c^{2/3} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right ) \sqrt {\frac {-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) b^{4/3}}{2 \sqrt {2} a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {3 c^{2/3} \sqrt {a+b \sqrt {c x^3}} b^{4/3}}{4 a \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )}-\frac {3 c x \sqrt {a+b \sqrt {c x^3}} b}{4 a \sqrt {c x^3}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2} \]

[Out]

1/4*3^(3/4)*b^(4/3)*c^(2/3)*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)*(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)/a^(2/3)*2^(1/2)/(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)-3/8*3^(1/4)*b^(4/3)*c^(2/3)*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)/a^(2
/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)-1/2*(a+b*(c*x^3)^(1/2))^(1/2)/x^2+3/4*b^(4/3)*c^(2/3)*(a+b*(c*x^3)^(1/2)
)^(1/2)/a/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))-3/4*b*c*x*(a+b*(c*x^3)^(1/2))^(1/2)/a/(c*x^3
)^(1/2)

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Rubi [A]  time = 0.46, antiderivative size = 810, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {369, 341, 277, 325, 303, 218, 1877} \[ -\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{2/3} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right ) \sqrt {\frac {-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) b^{4/3}}{8 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {3^{3/4} c^{2/3} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right ) \sqrt {\frac {-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) b^{4/3}}{2 \sqrt {2} a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {3 c^{2/3} \sqrt {a+b \sqrt {c x^3}} b^{4/3}}{4 a \left (\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )}-\frac {3 c x \sqrt {a+b \sqrt {c x^3}} b}{4 a \sqrt {c x^3}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c*x^3]]/x^3,x]

[Out]

-Sqrt[a + b*Sqrt[c*x^3]]/(2*x^2) - (3*b*c*x*Sqrt[a + b*Sqrt[c*x^3]])/(4*a*Sqrt[c*x^3]) + (3*b^(4/3)*c^(2/3)*Sq
rt[a + b*Sqrt[c*x^3]])/(4*a*((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])) - (3*3^(1/4)*Sqrt[2 -
Sqrt[3]]*b^(4/3)*c^(2/3)*(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]*Ellipt
icE[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]])/(8*a^(2/3)*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]]) + (3^(3/4)*b^(4/3)*c
^(2/3)*(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 - Sq
rt[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]])/(2*Sqrt[2]*a^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqr
t[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 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 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 325

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

Rule 341

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

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(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, d, m, p, q}, x] && FractionQ[n]

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 \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x^3} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^3}}{x^5} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2}+\operatorname {Subst}\left (\frac {1}{4} \left (3 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2}-\frac {3 b c x \sqrt {a+b \sqrt {c x^3}}}{4 a \sqrt {c x^3}}+\operatorname {Subst}\left (\frac {\left (3 b^2 c\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{8 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2}-\frac {3 b c x \sqrt {a+b \sqrt {c x^3}}}{4 a \sqrt {c x^3}}+\operatorname {Subst}\left (\frac {\left (3 b^{5/3} c^{5/6}\right ) \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 )}{8 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )+\operatorname {Subst}\left (\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} b^{5/3} c^{5/6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{4 a^{2/3}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2}-\frac {3 b c x \sqrt {a+b \sqrt {c x^3}}}{4 a \sqrt {c x^3}}+\frac {3 b^{4/3} c^{2/3} \sqrt {a+b \sqrt {c x^3}}}{4 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} c^{2/3} \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 )}{8 a^{2/3} \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 {3^{3/4} b^{4/3} c^{2/3} \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 )}{2 \sqrt {2} a^{2/3} \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 = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^3,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^3, x]

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fricas [F]  time = 6.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(sqrt(c*x^3)*b + a)/x^3, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(x)]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time: 22.11-2*ab
s(c)/c^2/c*16*c^3/64/sqrt(c*x)/sqrt(c*x)*sqrt(a*c^2/(c*x)^2+b*c/sqrt(c*x))+integrate(2*abs(c)/c^2/c*24*c^4*b/6
4/2*c*c/(sqrt(c*x)*sqrt(b*c^2*x*sqrt(c*x)+a*c^2)*c*x*abs(c)*sign(x)),x)

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maple [A]  time = 0.30, size = 863, normalized size = 1.07 \[ \frac {-12 b^{2} c \,x^{3}+3 i \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}}\, \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}\, x^{2} \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 {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}}\, \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}} x^{2} \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 {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}}\, \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}\, x^{2} \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 )-8 a^{2}-20 \sqrt {c \,x^{3}}\, a b}{16 \sqrt {a +\sqrt {c \,x^{3}}\, b}\, a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*I*2^(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)*(((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*(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
),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-a*b^2*c)^(2/3)*3^(1/2)*x^2-2*I*2^(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)*(((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*(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),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(
-a*b^2*c)^(2/3)*3^(1/2)*x^2+3*2^(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)*(((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)*Ell
ipticE(1/6*3^(1/2)*2^(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),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-a*b^2*c)^(2/3)*x^2-12*c*x^3*b^2-20*(c*x^3)^(1
/2)*a*b-8*a^2)/x^2/a/(a+(c*x^3)^(1/2)*b)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(c*x**3))/x**3, x)

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