\(\int x^2 \sqrt {a+b (c x^3)^{3/2}} \, dx\) [2974]

Optimal result

Integrand size = 21, antiderivative size = 642 \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {4}{21} x^3 \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {4 a \sqrt {a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}-\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}}+\frac {4 \sqrt {2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}\right ),-7-4 \sqrt {3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}} \]

[Out]

4/21*x^3*(a+b*(c*x^3)^(3/2))^(1/2)+4/7*a*(a+b*(c*x^3)^(3/2))^(1/2)/b^(2/3)/c/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x
^3)^(1/2))+4/21*a^(4/3)*EllipticF((a^(1/3)*(1-3^(1/2))+b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*
x^3)^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*(a^(1/3)+b^(1/3)*(c*x^3)^(1/2))*((a^(2/3)+b^(2/3)*c*x^3-a^(1/3)*b^(1/3)*(c*
x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2))^2)^(1/2)*3^(3/4)/b^(2/3)/c/(a+b*(c*x^3)^(3/2))^(1/2)/(
a^(1/3)*(a^(1/3)+b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2))^2)^(1/2)-2/7*3^(1/4)*a^(4/
3)*EllipticE((a^(1/3)*(1-3^(1/2))+b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2)),I*3^(1/2)
+2*I)*(a^(1/3)+b^(1/3)*(c*x^3)^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((a^(2/3)+b^(2/3)*c*x^3-a^(1/3)*b^(1/3)*(c*x^3
)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2))^2)^(1/2)/b^(2/3)/c/(a+b*(c*x^3)^(3/2))^(1/2)/(a^(1/3)*(a^
(1/3)+b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {376, 348, 281, 285, 309, 224, 1891} \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {4 \sqrt {2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^3}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^3}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}}-\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^3}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^3}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}}+\frac {4 a \sqrt {a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}+\frac {4}{21} x^3 \sqrt {a+b \left (c x^3\right )^{3/2}} \]

[In]

Int[x^2*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

(4*x^3*Sqrt[a + b*(c*x^3)^(3/2)])/21 + (4*a*Sqrt[a + b*(c*x^3)^(3/2)])/(7*b^(2/3)*c*((1 + Sqrt[3])*a^(1/3) + b
^(1/3)*Sqrt[c*x^3])) - (2*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^
(2/3)*c*x^3 - a^(1/3)*b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*EllipticE[ArcSin[(
(1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/
(7*b^(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*S
qrt[a + b*(c*x^3)^(3/2)]) + (4*Sqrt[2]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c*x^3 -
a^(1/3)*b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])
*a^(1/3) + b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*3^(1/4)*b^
(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*Sqrt[a
+ b*(c*x^3)^(3/2)])

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 285

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 309

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

Rule 1891

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], 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*} \text {integral}& = \text {Subst}\left (\int x^2 \sqrt {a+b c^{3/2} x^{9/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (2 \text {Subst}\left (\int x^5 \sqrt {a+b c^{3/2} x^9} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (\frac {2}{3} \text {Subst}\left (\int x \sqrt {a+b c^{3/2} x^3} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {4}{21} x^3 \sqrt {a+b \left (c x^3\right )^{3/2}}+\text {Subst}\left (\frac {1}{7} (2 a) \text {Subst}\left (\int \frac {x}{\sqrt {a+b c^{3/2} x^3}} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {4}{21} x^3 \sqrt {a+b \left (c x^3\right )^{3/2}}+\text {Subst}\left (\frac {(2 a) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c} x}{\sqrt {a+b c^{3/2} x^3}} \, dx,x,x^{3/2}\right )}{7 \sqrt [3]{b} \sqrt {c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )-\text {Subst}\left (\frac {\left (2 \left (1-\sqrt {3}\right ) a^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b c^{3/2} x^3}} \, dx,x,x^{3/2}\right )}{7 \sqrt [3]{b} \sqrt {c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {4}{21} x^3 \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {4 a \sqrt {a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}-\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}}+\frac {4 \sqrt {2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}\right )|-7-4 \sqrt {3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.11 \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {x^3 \sqrt {a+b \left (c x^3\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{3 \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}}} \]

[In]

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

[Out]

(x^3*Sqrt[a + b*(c*x^3)^(3/2)]*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*(c*x^3)^(3/2))/a)])/(3*Sqrt[1 + (b*(c*x^
3)^(3/2))/a])

Maple [A] (verified)

Time = 3.96 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {\frac {4 c \,x^{3} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{7}-\frac {4 i a \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\sqrt {c \,x^{3}}-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) E\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{7 b \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}}{3 c}\)	\(495\)
default	\(\frac {\frac {4 c \,x^{3} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{7}-\frac {4 i a \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\sqrt {c \,x^{3}}-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) E\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{7 b \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}}{3 c}\)	\(495\)

[In]

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

[Out]

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

Fricas [F]

\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} x^{2} \,d x } \]

[In]

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

[Out]

integral(sqrt(sqrt(c*x^3)*b*c*x^3 + a)*x^2, x)

Sympy [F]

\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int x^{2} \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(x**2*sqrt(a + b*(c*x**3)**(3/2)), x)

Maxima [F]

\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} x^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.51 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.08 \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {x^3\,\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {2}{3};\ \frac {5}{3};\ -\frac {b\,{\left (c\,x^3\right )}^{3/2}}{a}\right )}{3\,\sqrt {\frac {b\,{\left (c\,x^3\right )}^{3/2}}{a}+1}} \]

[In]

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

[Out]

(x^3*(a + b*(c*x^3)^(3/2))^(1/2)*hypergeom([-1/2, 2/3], 5/3, -(b*(c*x^3)^(3/2))/a))/(3*((b*(c*x^3)^(3/2))/a +
1)^(1/2))