3.2952 \(\int x \sqrt{a+b (c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=642 \[ \frac{2 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}}\right ),-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{a+b \left (c x^2\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}+\frac{2}{7} x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \]

[Out]

(2*x^2*Sqrt[a + b*(c*x^2)^(3/2)])/7 + (6*a*Sqrt[a + b*(c*x^2)^(3/2)])/(7*b^(2/3)*c*((1 + Sqrt[3])*a^(1/3) + b^
(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(
2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticE[ArcSin[((
1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/(
7*b^(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sq
rt[a + b*(c*x^2)^(3/2)]) + (2*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*
c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - S
qrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/(7*b^(
2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a
+ b*(c*x^2)^(3/2)])

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Rubi [A]  time = 0.361392, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {368, 279, 303, 218, 1877} \[ \frac{2 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{a+b \left (c x^2\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}+\frac{2}{7} x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

(2*x^2*Sqrt[a + b*(c*x^2)^(3/2)])/7 + (6*a*Sqrt[a + b*(c*x^2)^(3/2)])/(7*b^(2/3)*c*((1 + Sqrt[3])*a^(1/3) + b^
(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(
2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticE[ArcSin[((
1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/(
7*b^(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sq
rt[a + b*(c*x^2)^(3/2)]) + (2*Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*
c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - S
qrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])], -7 - 4*Sqrt[3]])/(7*b^(
2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a
+ b*(c*x^2)^(3/2)])

Rule 368

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

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

Mathematica [C]  time = 0.0181169, size = 69, normalized size = 0.11 \[ \frac{x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )}{2 \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 495, normalized size = 0.8 \begin{align*}{\frac{1}{2\,c} \left ({\frac{4\,c{x}^{2}}{7}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}-{\frac{{\frac{4\,i}{7}}a\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}\sqrt{{ \left ( \sqrt{c{x}^{2}}-{\frac{1}{b}\sqrt [3]{-{b}^{2}a}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}} \left ( \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a} \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}} \right ) +{\frac{1}{b}\sqrt [3]{-{b}^{2}a}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a} \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}} \right ) } \right ){\frac{1}{\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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