\(\int \frac {x^2}{\sqrt {a x+b x^4}} \, dx\) [108]

Optimal result

Integrand size = 17, antiderivative size = 474 \[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\frac {\left (1+\sqrt {3}\right ) x \left (a+b x^3\right )}{b^{2/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {a x+b x^4}}-\frac {\sqrt [4]{3} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x+b x^4}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x+b x^4}} \] Output:

(1+3^(1/2))*x*(b*x^3+a)/b^(2/3)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)/(b*x^4+a*x 
)^(1/2)-3^(1/4)*a^(1/3)*x*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+ 
b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)*EllipticE((1-(a^(1/3 
)+(1-3^(1/2))*b^(1/3)*x)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2),1/4*6^ 
(1/2)+1/4*2^(1/2))/b^(2/3)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1 
/2))*b^(1/3)*x)^2)^(1/2)/(b*x^4+a*x)^(1/2)-1/6*(1-3^(1/2))*a^(1/3)*x*(a^(1 
/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2 
))*b^(1/3)*x)^2)^(1/2)*InverseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3) 
*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2))*3^(3/4)/b^(2 
/3)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2 
)/(b*x^4+a*x)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\frac {2 x^3 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {b x^3}{a}\right )}{5 \sqrt {x \left (a+b x^3\right )}} \] Input:

Integrate[x^2/Sqrt[a*x + b*x^4],x]
 

Output:

(2*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 5/6, 11/6, -((b*x^3)/a)] 
)/(5*Sqrt[x*(a + b*x^3)])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1938, 851, 837, 25, 766, 2420}

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.

\(\displaystyle \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x^3} \int \frac {x^{3/2}}{\sqrt {b x^3+a}}dx}{\sqrt {a x+b x^4}}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x^3} \int \frac {x^2}{\sqrt {b x^3+a}}d\sqrt {x}}{\sqrt {a x+b x^4}}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x^3} \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}\right )}{\sqrt {a x+b x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x^3} \left (\frac {\int \frac {2 b^{2/3} x^2+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}\right )}{\sqrt {a x+b x^4}}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x^3} \left (\frac {\int \frac {2 b^{2/3} x^2+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt {a x+b x^4}}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x^3} \left (\frac {\frac {\left (1+\sqrt {3}\right ) \sqrt {x} \sqrt {a+b x^3}}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}-\frac {\sqrt [4]{3} \sqrt [3]{a} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt {a x+b x^4}}\)

Input:

Int[x^2/Sqrt[a*x + b*x^4],x]
 

Output:

(2*Sqrt[x]*Sqrt[a + b*x^3]*((((1 + Sqrt[3])*Sqrt[x]*Sqrt[a + b*x^3])/(a^(1 
/3) + (1 + Sqrt[3])*b^(1/3)*x) - (3^(1/4)*a^(1/3)*Sqrt[x]*(a^(1/3) + b^(1/ 
3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqr 
t[3])*b^(1/3)*x)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/( 
a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*x*(a 
^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3 
]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*Sqrt[x]*(a^(1/3) + b^(1/3)*x)*Sqr 
t[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^( 
1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + 
 (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3)*Sqrt[(b^( 
1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[ 
a + b*x^3])))/Sqrt[a*x + b*x^4]
 

Defintions of rubi rules used

rule 25

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

rule 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 837

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 
3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2))   Int[1/Sqrt[ 
a + b*x^6], x], x] - Simp[1/(2*r^2)   Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S 
qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* 
(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 1054, normalized size of antiderivative = 2.22

method	result	size
default	\(\text {Expression too large to display}\)	\(1054\)
elliptic	\(\text {Expression too large to display}\)	\(1054\)

Input:

int(x^2/(b*x^4+a*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{4} + a x}} \,d x } \] Input:

integrate(x^2/(b*x^4+a*x)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^4 + a*x)*x/(b*x^3 + a), x)
 

Sympy [F]

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

integrate(x**2/(b*x**4+a*x)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{4} + a x}} \,d x } \] Input:

integrate(x^2/(b*x^4+a*x)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^2/sqrt(b*x^4 + a*x), x)
 

Giac [F]

\[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{4} + a x}} \,d x } \] Input:

integrate(x^2/(b*x^4+a*x)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^2/sqrt(b*x^4 + a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\int \frac {x^2}{\sqrt {b\,x^4+a\,x}} \,d x \] Input:

int(x^2/(a*x + b*x^4)^(1/2),x)
 

Output:

int(x^2/(a*x + b*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2}{\sqrt {a x+b x^4}} \, dx=\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \] Input:

int(x^2/(b*x^4+a*x)^(1/2),x)
 

Output:

int((sqrt(x)*sqrt(a + b*x**3)*x)/(a + b*x**3),x)