\(\int \sqrt {x} (a x+b x^3+c x^5)^{3/2} \, dx\) [110]

Optimal result

Integrand size = 24, antiderivative size = 487 \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {x} \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{11/4} \sqrt {a x+b x^3+c x^5}} \]

[Out]

1/63*(7*c*x^2+3*b)*(c*x^5+b*x^3+a*x)^(3/2)/c/x^(1/2)+1/315*(84*a^2*c^2-57*a*b^2*c+8*b^4)*x^(3/2)*(c*x^4+b*x^2+
a)/c^(5/2)/(a^(1/2)+x^2*c^(1/2))/(c*x^5+b*x^3+a*x)^(1/2)-1/315*(b*(-9*a*c+4*b^2)+6*c*(-7*a*c+2*b^2)*x^2)*x^(1/
2)*(c*x^5+b*x^3+a*x)^(1/2)/c^2-1/315*a^(1/4)*(84*a^2*c^2-57*a*b^2*c+8*b^4)*(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^(11/4)/(c*x^5+b*x^3+a*x
)^(1/2)+1/630*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(si
n(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))*(8*b^4-57*a*b^2*c+84*a^2
*c^2+4*b*(-6*a*c+b^2)*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^(11/4)/(c*x^5
+b*x^3+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1933, 1959, 1967, 1211, 1117, 1209} \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\sqrt [4]{a} \sqrt {x} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{11/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} \sqrt {x} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a x+b x^3+c x^5}}+\frac {x^{3/2} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (a+b x^2+c x^4\right )}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}} \]

[In]

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

[Out]

((8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*x^(3/2)*(a + b*x^2 + c*x^4))/(315*c^(5/2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[a*x
+ b*x^3 + c*x^5]) - (Sqrt[x]*(b*(4*b^2 - 9*a*c) + 6*c*(2*b^2 - 7*a*c)*x^2)*Sqrt[a*x + b*x^3 + c*x^5])/(315*c^2
) + ((3*b + 7*c*x^2)*(a*x + b*x^3 + c*x^5)^(3/2))/(63*c*Sqrt[x]) - (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*
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/(Sqrt[a]*Sqrt[c]))/4])/(315*c^(11/4)*Sqrt[a*x + b*x^3 + c*x^5]) + (a^(1/4)*(8*b^4 - 57*a
*b^2*c + 84*a^2*c^2 + 4*Sqrt[a]*b*Sqrt[c]*(b^2 - 6*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]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(
11/4)*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 1959

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> Simp[x^(m + 1)*(b*B*(n - q)*p + A*c*(m + p*q + (n - q)*(2*p + 1) + 1) + B*c*(m + p*q + 2*(n - q)*p + 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 + q)*Simp[2*a*A*c*
(m + p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(m + p*q + 1) + (2*a*B*c*(m + p*q + 2*(n - q)*p + 1) + A*b*c*(m + p*
q + (n - q)*(2*p + 1) + 1) - b^2*B*(m + p*q + (n - q)*p + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p
- 1), x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4
*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q, -(n - q) - 1] && 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*} \text {integral}& = \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}+\frac {\int \frac {\left (-a b-2 \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx}{21 c} \\ & = -\frac {\sqrt {x} \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}+\frac {\int \frac {\sqrt {x} \left (4 a b \left (b^2-6 a c\right )+\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{315 c^2} \\ & = -\frac {\sqrt {x} \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {4 a b \left (b^2-6 a c\right )+\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{315 c^2 \sqrt {a x+b x^3+c x^5}} \\ & = -\frac {\sqrt {x} \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\left (\sqrt {a} \left (8 b^4-57 a b^2 c+84 a^2 c^2\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}{315 c^{5/2} \sqrt {a x+b x^3+c x^5}}+\frac {\left (\sqrt {a} \left (8 b^4-57 a b^2 c+84 a^2 c^2+4 \sqrt {a} b \sqrt {c} \left (b^2-6 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}{315 c^{5/2} \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {x} \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2\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 )}{315 c^{11/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )\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 )}{630 c^{11/4} \sqrt {a x+b x^3+c x^5}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.38 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.25 \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\sqrt {x} \left (4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}+a^2 c \left (24 b+77 c x^2\right )+a \left (-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right )\right )+i \left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-8 b^5+65 a b^3 c-132 a^2 b c^2+8 b^4 \sqrt {b^2-4 a c}-57 a b^2 c \sqrt {b^2-4 a c}+84 a^2 c^2 \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{1260 c^3 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x \left (a+b x^2+c x^4\right )}} \]

[In]

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

[Out]

(Sqrt[x]*(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b^4*x^2 - b^3*c*x^4 + 53*b^2*c^2*x^6 + 85*b*c^3*x^8 + 35*c
^4*x^10 + a^2*c*(24*b + 77*c*x^2) + a*(-4*b^3 + 27*b^2*c*x^2 + 151*b*c^2*x^4 + 112*c^3*x^6)) + I*(8*b^4 - 57*a
*b^2*c + 84*a^2*c^2)*(-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 + Sq
rt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-8*b^5 + 65*a*b^3*c - 132*a^2*b*c^
2 + 8*b^4*Sqrt[b^2 - 4*a*c] - 57*a*b^2*c*Sqrt[b^2 - 4*a*c] + 84*a^2*c^2*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])]))/(1260*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.34

method	result	size
risch	\(\frac {x^{\frac {3}{2}} \left (35 c^{3} x^{6}+50 b \,c^{2} x^{4}+77 a \,c^{2} x^{2}+3 b^{2} c \,x^{2}+24 a b c -4 b^{3}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}{315 c^{2} \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}-\frac {\left (-\frac {a \,b^{3} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {6 c b \,a^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (84 a^{2} c^{2}-57 a \,b^{2} c +8 b^{4}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {x}}{315 c^{2} \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\)	\(654\)
default	\(\text {Expression too large to display}\)	\(1878\)

[In]

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

[Out]

1/315*x^(3/2)*(35*c^3*x^6+50*b*c^2*x^4+77*a*c^2*x^2+3*b^2*c*x^2+24*a*b*c-4*b^3)/c^2*(c*x^4+b*x^2+a)/(x*(c*x^4+
b*x^2+a))^(1/2)-1/315/c^2*(-a*b^3*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2
)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^
2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+6*c*b*a^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a
)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/
2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/
2*(84*a^2*c^2-57*a*b^2*c+8*b^4)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2
)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x
*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1
/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))))*(c*x^4+b*x^2+a)^(1/2)*x
^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00 \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, b^{5} - 57 \, a b^{3} c + 84 \, a^{2} b c^{2}\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (8 \, b^{4} c + 12 \, {\left (7 \, a^{2} + 2 \, a b\right )} c^{3} - {\left (57 \, a b^{2} + 4 \, b^{3}\right )} c^{2}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, b^{5} + 12 \, {\left (7 \, a^{2} b - 2 \, a b^{2}\right )} c^{2} - {\left (57 \, a b^{3} - 4 \, b^{4}\right )} c\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (35 \, c^{5} x^{8} + 50 \, b c^{4} x^{6} + 8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 77 \, a c^{4}\right )} x^{4} - 4 \, {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{630 \, c^{4} x^{2}} \]

[In]

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

[Out]

1/630*(sqrt(1/2)*((8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3)*x^2*sqrt((b^2 - 4*a*c)/c^2) - (8*b^5 - 57*a*b^3*c + 84
*a^2*b*c^2)*x^2)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2
- 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - sqrt(1/2)*((8*b^4*c + 12*(
7*a^2 + 2*a*b)*c^3 - (57*a*b^2 + 4*b^3)*c^2)*x^2*sqrt((b^2 - 4*a*c)/c^2) - (8*b^5 + 12*(7*a^2*b - 2*a*b^2)*c^2
- (57*a*b^3 - 4*b^4)*c)*x^2)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/2)*sqrt
((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 2*(35*c^5*x^8
+ 50*b*c^4*x^6 + 8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3 + (3*b^2*c^3 + 77*a*c^4)*x^4 - 4*(b^3*c^2 - 6*a*b*c^3)*x
^2)*sqrt(c*x^5 + b*x^3 + a*x)*sqrt(x))/(c^4*x^2)

Sympy [F]

\[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\int \sqrt {x} \left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\int { {\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}} \sqrt {x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\int { {\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}} \sqrt {x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\int \sqrt {x}\,{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2} \,d x \]

[In]

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

[Out]

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