\(\int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx\) [274]

Optimal result

Integrand size = 21, antiderivative size = 265 \[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=-\frac {7 a \sqrt {a x^2+b x^5}}{20 b^2 \sqrt {x}}+\frac {x^{5/2} \sqrt {a x^2+b x^5}}{5 b}+\frac {7 a^{5/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{40 \sqrt [4]{3} b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \] Output:

-7/20*a*(b*x^5+a*x^2)^(1/2)/b^2/x^(1/2)+1/5*x^(5/2)*(b*x^5+a*x^2)^(1/2)/b+ 
7/120*a^(5/3)*x^(3/2)*(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/(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^5+a*x^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.32 \[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {x^{3/2} \left (-7 a^2-3 a b x^3+4 b^2 x^6+7 a^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^3}{a}\right )\right )}{20 b^2 \sqrt {x^2 \left (a+b x^3\right )}} \] Input:

Integrate[x^(13/2)/Sqrt[a*x^2 + b*x^5],x]
 

Output:

(x^(3/2)*(-7*a^2 - 3*a*b*x^3 + 4*b^2*x^6 + 7*a^2*Sqrt[1 + (b*x^3)/a]*Hyper 
geometric2F1[1/6, 1/2, 7/6, -((b*x^3)/a)]))/(20*b^2*Sqrt[x^2*(a + b*x^3)])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1930, 1930, 1938, 851, 766}

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.

\(\displaystyle \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx\)

\(\Big \downarrow \) 1930

\(\displaystyle \frac {x^{5/2} \sqrt {a x^2+b x^5}}{5 b}-\frac {7 a \int \frac {x^{7/2}}{\sqrt {b x^5+a x^2}}dx}{10 b}\)

\(\Big \downarrow \) 1930

\(\displaystyle \frac {x^{5/2} \sqrt {a x^2+b x^5}}{5 b}-\frac {7 a \left (\frac {\sqrt {a x^2+b x^5}}{2 b \sqrt {x}}-\frac {a \int \frac {\sqrt {x}}{\sqrt {b x^5+a x^2}}dx}{4 b}\right )}{10 b}\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {x^{5/2} \sqrt {a x^2+b x^5}}{5 b}-\frac {7 a \left (\frac {\sqrt {a x^2+b x^5}}{2 b \sqrt {x}}-\frac {a x \sqrt {a+b x^3} \int \frac {1}{\sqrt {x} \sqrt {b x^3+a}}dx}{4 b \sqrt {a x^2+b x^5}}\right )}{10 b}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {x^{5/2} \sqrt {a x^2+b x^5}}{5 b}-\frac {7 a \left (\frac {\sqrt {a x^2+b x^5}}{2 b \sqrt {x}}-\frac {a x \sqrt {a+b x^3} \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b \sqrt {a x^2+b x^5}}\right )}{10 b}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {x^{5/2} \sqrt {a x^2+b x^5}}{5 b}-\frac {7 a \left (\frac {\sqrt {a x^2+b x^5}}{2 b \sqrt {x}}-\frac {a^{2/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}\right )}{10 b}\)

Input:

Int[x^(13/2)/Sqrt[a*x^2 + b*x^5],x]
 

Output:

(x^(5/2)*Sqrt[a*x^2 + b*x^5])/(5*b) - (7*a*(Sqrt[a*x^2 + b*x^5]/(2*b*Sqrt[ 
x]) - (a^(2/3)*x^(3/2)*(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 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCo 
s[(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*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*x^2 + b*x^5])))/(10*b)
 

Defintions of rubi rules used

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 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 1930

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p 
+ 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1)))   I 
nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, 
x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt 
Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
 

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.57 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.80

method	result	size
risch	\(-\frac {\left (-4 b \,x^{3}+7 a \right ) x^{\frac {3}{2}} \left (b \,x^{3}+a \right )}{20 b^{2} \sqrt {x^{2} \left (b \,x^{3}+a \right )}}+\frac {7 a^{2} \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{\left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right ) \sqrt {x}\, \sqrt {x \left (b \,x^{3}+a \right )}}{20 b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}\, \sqrt {x^{2} \left (b \,x^{3}+a \right )}}\)	\(742\)
default	\(\text {Expression too large to display}\)	\(2017\)

Input:

int(x^(13/2)/(b*x^5+a*x^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/20*(-4*b*x^3+7*a)/b^2*x^(3/2)*(b*x^3+a)/(x^2*(b*x^3+a))^(1/2)+7/20*a^2/ 
b*(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)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2 
)^(1/3))/(-a*b^2)^(1/3)/(b*x*(x-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))*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b* 
(-a*b^2)^(1/3)))^(1/2)*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))/(x^2*(b*x^3+a))^(1/2)*x^(1/2)*(x*(b*x^3+a)) 
^(1/2)
 

Fricas [F]

\[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {x^{\frac {13}{2}}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \] Input:

integrate(x^(13/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^5 + a*x^2)*x^(9/2)/(b*x^3 + a), x)
 

Sympy [F]

\[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{\frac {13}{2}}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \] Input:

integrate(x**(13/2)/(b*x**5+a*x**2)**(1/2),x)
 

Output:

Integral(x**(13/2)/sqrt(x**2*(a + b*x**3)), x)
 

Maxima [F]

\[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {x^{\frac {13}{2}}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \] Input:

integrate(x^(13/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^(13/2)/sqrt(b*x^5 + a*x^2), x)
 

Giac [F]

\[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {x^{\frac {13}{2}}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \] Input:

integrate(x^(13/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^(13/2)/sqrt(b*x^5 + a*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{13/2}}{\sqrt {b\,x^5+a\,x^2}} \,d x \] Input:

int(x^(13/2)/(a*x^2 + b*x^5)^(1/2),x)
 

Output:

int(x^(13/2)/(a*x^2 + b*x^5)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^{13/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {-14 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a +8 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b \,x^{3}+7 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{4}+a x}d x \right ) a^{2}}{40 b^{2}} \] Input:

int(x^(13/2)/(b*x^5+a*x^2)^(1/2),x)
 

Output:

( - 14*sqrt(x)*sqrt(a + b*x**3)*a + 8*sqrt(x)*sqrt(a + b*x**3)*b*x**3 + 7* 
int((sqrt(x)*sqrt(a + b*x**3))/(a*x + b*x**4),x)*a**2)/(40*b**2)