\(\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx\) [2966]

Optimal result

Integrand size = 21, antiderivative size = 434 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=-\frac {\sqrt {a+b \sqrt {c x^3}}}{4 x^4}+\frac {21 b^2 c \sqrt {a+b \sqrt {c x^3}}}{160 a^2 x}-\frac {3 b c^3 x^5 \sqrt {a+b \sqrt {c x^3}}}{40 a \left (c x^3\right )^{5/2}}+\frac {7\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{8/3} c^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right ),-7-4 \sqrt {3}\right )}{160 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}} \]

[Out]

7/160*3^(3/4)*b^(8/3)*c^(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)*(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/(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)-1/4*(a+b*(c*x^3)^(1/2))^(1/2)
/x^4+21/160*b^2*c*(a+b*(c*x^3)^(1/2))^(1/2)/a^2/x-3/40*b*c^3*x^5*(a+b*(c*x^3)^(1/2))^(1/2)/a/(c*x^3)^(5/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {376, 348, 283, 331, 224} \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\frac {21 b^2 c \sqrt {a+b \sqrt {c x^3}}}{160 a^2 x}+\frac {7\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{8/3} c^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{160 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}-\frac {3 b c^3 x^5 \sqrt {a+b \sqrt {c x^3}}}{40 a \left (c x^3\right )^{5/2}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{4 x^4} \]

[In]

Int[Sqrt[a + b*Sqrt[c*x^3]]/x^5,x]

[Out]

-1/4*Sqrt[a + b*Sqrt[c*x^3]]/x^4 + (21*b^2*c*Sqrt[a + b*Sqrt[c*x^3]])/(160*a^2*x) - (3*b*c^3*x^5*Sqrt[a + b*Sq
rt[c*x^3]])/(40*a*(c*x^3)^(5/2)) + (7*3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(8/3)*c^(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]])/(160*a^2*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 224

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

Rule 283

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 331

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 348

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 376

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x^5} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (2 \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^3}}{x^9} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^3}}}{4 x^4}+\text {Subst}\left (\frac {1}{8} \left (3 b \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{x^6 \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}}}{4 x^4}-\frac {3 b c^3 x^5 \sqrt {a+b \sqrt {c x^3}}}{40 a \left (c x^3\right )^{5/2}}-\text {Subst}\left (\frac {\left (21 b^2 c\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{80 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^3}}}{4 x^4}+\frac {21 b^2 c \sqrt {a+b \sqrt {c x^3}}}{160 a^2 x}-\frac {3 b c^3 x^5 \sqrt {a+b \sqrt {c x^3}}}{40 a \left (c x^3\right )^{5/2}}+\text {Subst}\left (\frac {\left (21 b^3 c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{320 a^2},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^3}}}{4 x^4}+\frac {21 b^2 c \sqrt {a+b \sqrt {c x^3}}}{160 a^2 x}-\frac {3 b c^3 x^5 \sqrt {a+b \sqrt {c x^3}}}{40 a \left (c x^3\right )^{5/2}}+\frac {7\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{8/3} c^{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 )}{160 a^2 \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*}

Mathematica [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx \]

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^5,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^5, x]

Maple [A] (verified)

Time = 5.08 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {7 i b^{2} \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}-2 b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}-2 b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) c \,x^{4}-42 \sqrt {c \,x^{3}}\, b^{3} c \,x^{3}-18 a \,b^{2} c \,x^{3}+104 \sqrt {c \,x^{3}}\, a^{2} b +80 a^{3}}{320 x^{4} a^{2} \sqrt {a +b \sqrt {c \,x^{3}}}}\)	\(346\)

[In]

int((a+b*(c*x^3)^(1/2))^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/320*(7*I*b^2*3^(1/2)*(-a*b^2*c)^(1/3)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)-2*b*(c*x^3)^(1/2)-x*(-a*b^2
*c)^(1/3))*3^(1/2)/x/(-a*b^2*c)^(1/3))^(1/2)*((b*(c*x^3)^(1/2)-x*(-a*b^2*c)^(1/3))/x/(-a*b^2*c)^(1/3)/(I*3^(1/
2)-3))^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^(1/2)+x*(-a*b^2*c)^(1/3))*3^(1/2)/x/(-a*b^2*c)^(1/3
))^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)-2*b*(c*x^3)^(1/2)-x*(-a*b^2*c)^(1/3))
*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))*c*x^4-42*(c*x^3)^(1/2)*b^3*c*x^3-1
8*a*b^2*c*x^3+104*(c*x^3)^(1/2)*a^2*b+80*a^3)/x^4/a^2/(a+b*(c*x^3)^(1/2))^(1/2)

Fricas [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{5}} \,d x } \]

[In]

integrate((a+b*(c*x^3)^(1/2))^(1/2)/x^5,x, algorithm="fricas")

[Out]

integral(sqrt(sqrt(c*x^3)*b + a)/x^5, x)

Sympy [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{3}}}}{x^{5}}\, dx \]

[In]

integrate((a+b*(c*x**3)**(1/2))**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**3))/x**5, x)

Maxima [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{5}} \,d x } \]

[In]

integrate((a+b*(c*x^3)^(1/2))^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)/x^5, x)

Giac [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{5}} \,d x } \]

[In]

integrate((a+b*(c*x^3)^(1/2))^(1/2)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^5} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x^5} \,d x \]

[In]

int((a + b*(c*x^3)^(1/2))^(1/2)/x^5,x)

[Out]

int((a + b*(c*x^3)^(1/2))^(1/2)/x^5, x)