\(\int \sqrt {x} (a x+b x^3+c x^5)^{3/2} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 489 \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {x} \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{315 c^2}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{a} \left (4 \sqrt {a} b \left (b^2-6 a c\right )+\frac {8 b^4-57 a b^2 c+84 a^2 c^2}{\sqrt {c}}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{9/4} \sqrt {a x+b x^3+c x^5}} \] Output:

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)+c^(1/2)*x^2)/(c*x^5+b*x^3+a*x)^(1/2)-1/315*x^(1/2)*(b*(-9*a*c+4*b^2)+6* 
c*(-7*a*c+2*b^2)*x^2)*(c*x^5+b*x^3+a*x)^(1/2)/c^2+1/63*(7*c*x^2+3*b)*(c*x^ 
5+b*x^3+a*x)^(3/2)/c/x^(1/2)-1/315*a^(1/4)*(84*a^2*c^2-57*a*b^2*c+8*b^4)*x 
^(1/2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/ 
2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1 
/2))/c^(11/4)/(c*x^5+b*x^3+a*x)^(1/2)+1/630*a^(1/4)*(4*a^(1/2)*b*(-6*a*c+b 
^2)+(84*a^2*c^2-57*a*b^2*c+8*b^4)/c^(1/2))*x^(1/2)*(a^(1/2)+c^(1/2)*x^2)*( 
(c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^ 
(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(9/4)/(c*x^5+b*x^3+a*x 
)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.02 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.25 \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\sqrt {x} \left (4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}+a^2 c \left (24 b+77 c x^2\right )+a \left (-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right )\right )+i \left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-8 b^5+65 a b^3 c-132 a^2 b c^2+8 b^4 \sqrt {b^2-4 a c}-57 a b^2 c \sqrt {b^2-4 a c}+84 a^2 c^2 \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{1260 c^3 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x \left (a+b x^2+c x^4\right )}} \] Input:

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

Output:

(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 + Sqrt[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*S 
qrt[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 + S 
qrt[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)])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1966, 25, 1992, 25, 2000, 1511, 27, 1416, 1509}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1966

\(\displaystyle \frac {\int -\frac {\left (2 \left (2 b^2-7 a c\right ) x^2+a b\right ) \sqrt {c x^5+b x^3+a x}}{\sqrt {x}}dx}{21 c}+\frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\int \frac {\left (2 \left (2 b^2-7 a c\right ) x^2+a b\right ) \sqrt {c x^5+b x^3+a x}}{\sqrt {x}}dx}{21 c}\)

\(\Big \downarrow \) 1992

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\int -\frac {\sqrt {x} \left (\left (8 b^4-57 a c b^2+84 a^2 c^2\right ) x^2+4 a b \left (b^2-6 a c\right )\right )}{\sqrt {c x^5+b x^3+a x}}dx}{15 c}+\frac {\sqrt {x} \sqrt {a x+b x^3+c x^5} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right )}{15 c}}{21 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{15 c}-\frac {\int \frac {\sqrt {x} \left (\left (8 b^4-57 a c b^2+84 a^2 c^2\right ) x^2+4 a b \left (b^2-6 a c\right )\right )}{\sqrt {c x^5+b x^3+a x}}dx}{15 c}}{21 c}\)

\(\Big \downarrow \) 2000

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{15 c}-\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \int \frac {\left (8 b^4-57 a c b^2+84 a^2 c^2\right ) x^2+4 a b \left (b^2-6 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c \sqrt {a x+b x^3+c x^5}}}{21 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{15 c}-\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt {a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{15 c \sqrt {a x+b x^3+c x^5}}}{21 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{15 c}-\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt {a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{15 c \sqrt {a x+b x^3+c x^5}}}{21 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{15 c}-\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{15 c \sqrt {a x+b x^3+c x^5}}}{21 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt {x}}-\frac {\frac {\sqrt {x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{15 c}-\frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c \sqrt {a x+b x^3+c x^5}}}{21 c}\)

Input:

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

Output:

((3*b + 7*c*x^2)*(a*x + b*x^3 + c*x^5)^(3/2))/(63*c*Sqrt[x]) - ((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])/(1 
5*c) - (Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*(-(((8*b^4 - 57*a*b^2*c + 84*a^2*c 
^2)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sq 
rt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*E 
llipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^( 
1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (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[a] + Sqrt[c]*x^2)*Sq 
rt[(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])/(2*c^(3/4)*Sqrt[a + b*x^2 + 
 c*x^4])))/(15*c*Sqrt[a*x + b*x^3 + c*x^5]))/(21*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

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 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> 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 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[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] && Pos 
Q[c/a]
 

rule 1966

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] + Simp[(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 1992

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ 
.)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> 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] + Simp[(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] &&  !Integ 
erQ[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 2000

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(j_.)))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x 
_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Simp[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] /; Fr 
eeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && Pos 
Q[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q, 1]
 

Maple [A] (verified)

Time = 3.62 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.34

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

Input:

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

Output:

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*(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)))-b^3*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)*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 
*a^2*b*c*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)))*(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)

Time = 0.09 (sec) , antiderivative size = 485, normalized size of antiderivative = 0.99 \[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, b^{5} - 57 \, a b^{3} c + 84 \, a^{2} b c^{2}\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (8 \, b^{4} c + 12 \, {\left (7 \, a^{2} + 2 \, a b\right )} c^{3} - {\left (57 \, a b^{2} + 4 \, b^{3}\right )} c^{2}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, b^{5} + 12 \, {\left (7 \, a^{2} b - 2 \, a b^{2}\right )} c^{2} - {\left (57 \, a b^{3} - 4 \, b^{4}\right )} c\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (35 \, c^{5} x^{8} + 50 \, b c^{4} x^{6} + 8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 77 \, a c^{4}\right )} x^{4} - 4 \, {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{630 \, c^{4} x^{2}} \] Input:

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

Output:

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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2} \, dx=\frac {24 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b c x +77 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,c^{2} x^{3}-4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} x +3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c \,x^{3}+50 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{2} x^{5}+35 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{3} x^{7}-24 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} b c +4 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{3}+84 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} c^{2}-57 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{2} c +8 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{4}}{315 c^{2}} \] Input:

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

Output:

(24*sqrt(a + b*x**2 + c*x**4)*a*b*c*x + 77*sqrt(a + b*x**2 + c*x**4)*a*c** 
2*x**3 - 4*sqrt(a + b*x**2 + c*x**4)*b**3*x + 3*sqrt(a + b*x**2 + c*x**4)* 
b**2*c*x**3 + 50*sqrt(a + b*x**2 + c*x**4)*b*c**2*x**5 + 35*sqrt(a + b*x** 
2 + c*x**4)*c**3*x**7 - 24*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x 
**4),x)*a**2*b*c + 4*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x 
)*a*b**3 + 84*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x 
)*a**2*c**2 - 57*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4 
),x)*a*b**2*c + 8*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x** 
4),x)*b**4)/(315*c**2)