3.2.5 \(\int x^{3/2} \sqrt {a x+b x^3+c x^5} \, dx\) [105]

Optimal. Leaf size=380 \[ -\frac {2 \left (b^2-3 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (b+3 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{15 c}+\frac {2 \sqrt [4]{a} \left (b^2-3 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a x+b x^3+c x^5}} \]

[Out]

-2/15*(-3*a*c+b^2)*x^(3/2)*(c*x^4+b*x^2+a)/c^(3/2)/(a^(1/2)+x^2*c^(1/2))/(c*x^5+b*x^3+a*x)^(1/2)+1/15*(3*c*x^2
+b)*x^(1/2)*(c*x^5+b*x^3+a*x)^(1/2)/c+2/15*a^(1/4)*(-3*a*c+b^2)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos
(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(
1/2)+x^2*c^(1/2))*x^(1/2)*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(7/4)/(c*x^5+b*x^3+a*x)^(1/2)-1/30
*a^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^
(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*(2*b^2-6*a*c+b*a^(1/2)*c^(1/2))*x^(1/
2)*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(7/4)/(c*x^5+b*x^3+a*x)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1933, 1967, 1211, 1117, 1209} \begin {gather*} -\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {a} b \sqrt {c}-6 a c+2 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a x+b x^3+c x^5}}+\frac {2 \sqrt [4]{a} \sqrt {x} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a x+b x^3+c x^5}}-\frac {2 x^{3/2} \left (b^2-3 a c\right ) \left (a+b x^2+c x^4\right )}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (b+3 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{15 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^2 - 3*a*c)*x^(3/2)*(a + b*x^2 + c*x^4))/(15*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[a*x + b*x^3 + c*x^5])
+ (Sqrt[x]*(b + 3*c*x^2)*Sqrt[a*x + b*x^3 + c*x^5])/(15*c) + (2*a^(1/4)*(b^2 - 3*a*c)*Sqrt[x]*(Sqrt[a] + Sqrt[
c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(S
qrt[a]*Sqrt[c]))/4])/(15*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5]) - (a^(1/4)*(2*b^2 + Sqrt[a]*b*Sqrt[c] - 6*a*c)*Sqr
t[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x
)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(30*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5])

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]
/; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1933

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol] :> Simp[x^(m - n + q +
1)*(b*(n - q)*p + c*(m + p*q + (n - q)*(2*p - 1) + 1)*x^(n - q))*((a*x^q + b*x^n + c*x^(2*n - q))^p/(c*(m + p
*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1))), x] + Dist[(n - q)*(p/(c*(m + p*(2*n - q) + 1)*(m + p*q +
(n - q)*(2*p - 1) + 1))), Int[x^(m - (n - 2*q))*Simp[(-a)*b*(m + p*q - n + q + 1) + (2*a*c*(m + p*q + (n - q)*
(2*p - 1) + 1) - b^2*(m + p*q + (n - q)*(p - 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x
], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ
[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, n - q] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p*
q + (n - q)*(2*p - 1) + 1, 0]

Rule 1967

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(j_.)))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Sym
bol] :> Dist[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)]), Int[x^(m -
q/2)*((A + B*x^(n - q))/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, A, B, m, n, q}, x
] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q
, 1]

Rubi steps

\begin {align*} \int x^{3/2} \sqrt {a x+b x^3+c x^5} \, dx &=\frac {\sqrt {x} \left (b+3 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{15 c}+\frac {\int \frac {\sqrt {x} \left (-a b-2 \left (b^2-3 a c\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{15 c}\\ &=\frac {\sqrt {x} \left (b+3 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{15 c}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {-a b-2 \left (b^2-3 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {x} \left (b+3 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{15 c}+\frac {\left (2 \sqrt {a} \left (b^2-3 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^{3/2} \sqrt {a x+b x^3+c x^5}}+\frac {\left (\sqrt {a} \left (-\sqrt {a} b \sqrt {c}-2 \left (b^2-3 a c\right )\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^{3/2} \sqrt {a x+b x^3+c x^5}}\\ &=-\frac {2 \left (b^2-3 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (b+3 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{15 c}+\frac {2 \sqrt [4]{a} \left (b^2-3 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.32, size = 486, normalized size = 1.28 \begin {gather*} \frac {\sqrt {x} \left (2 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{30 c^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x \left (a+b x^2+c x^4\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b + 3*c*x^2)*(a + b*x^2 + c*x^4) - I*(b^2 - 3*a*c)*(-b + Sqrt
[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c]
+ 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt
[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*(-b^3 + 4*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 3*a*c*Sqrt[b^2 - 4*a*c])
*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b
- Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])
/(b - Sqrt[b^2 - 4*a*c])]))/(30*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1041\) vs. \(2(364)=728\).
time = 0.07, size = 1042, normalized size = 2.74

method	result	size
risch	\(\frac {x^{\frac {3}{2}} \left (3 c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}+a \right )}{15 c \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}-\frac {\left (\frac {\left (6 a c -2 b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {x}}{15 c \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\)	\(455\)
default	\(-\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (-6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b \,c^{2} x^{7}-6 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, c^{2} x^{7}-8 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} c \,x^{5}-8 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b c \,x^{5}-6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b c \,x^{3}-6 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a c \,x^{3}-2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{3} x^{3}-2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} x^{3}+12 \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a^{2} c -3 b^{2} a \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right )+b a \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) \sqrt {-4 a c +b^{2}}-12 \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a^{2} c +4 \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a \,b^{2}-2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a \,b^{2} x -2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b x \right )}{30 \sqrt {x}\, \left (c \,x^{4}+b \,x^{2}+a \right ) c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(1042\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(x*(c*x^4+b*x^2+a))^(1/2)*(-6*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*b*c^2*x^7-6*(-4*a*c+b^2)^(1/2)*((-b+(-4*
a*c+b^2)^(1/2))/a)^(1/2)*c^2*x^7-8*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*b^2*c*x^5-8*(-4*a*c+b^2)^(1/2)*((-b+(-4*a
*c+b^2)^(1/2))/a)^(1/2)*b*c*x^5-6*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*a*b*c*x^3-6*(-4*a*c+b^2)^(1/2)*((-b+(-4*a*
c+b^2)^(1/2))/a)^(1/2)*a*c*x^3-2*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*b^3*x^3-2*(-4*a*c+b^2)^(1/2)*((-b+(-4*a*c+b
^2)^(1/2))/a)^(1/2)*b^2*x^3+12*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+
2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a
*c+b^2)/a/c)^(1/2))*a^2*c-3*b^2*a*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x
^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-
2*a*c+b^2)/a/c)^(1/2))+b*a*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)
/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b
^2)/a/c)^(1/2))*(-4*a*c+b^2)^(1/2)-12*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)
+b*x^2+2*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1
/2)-2*a*c+b^2)/a/c)^(1/2))*a^2*c+4*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*
x^2+2*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)
-2*a*c+b^2)/a/c)^(1/2))*a*b^2-2*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*a*b^2*x-2*(-4*a*c+b^2)^(1/2)*((-b+(-4*a*c+b^
2)^(1/2))/a)^(1/2)*a*b*x)/x^(1/2)/(c*x^4+b*x^2+a)/c/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{\frac {3}{2}} \sqrt {x \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**(3/2)*sqrt(x*(a + b*x**2 + c*x**4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{3/2}\,\sqrt {c\,x^5+b\,x^3+a\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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