\(\int \frac {x^2 (c+d x^2)}{(a+b x^2)^{9/4}} \, dx\) [420]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=-\frac {2 (b c-a d) x}{5 b^2 \left (a+b x^2\right )^{5/4}}+\frac {2 d x}{b^2 \sqrt [4]{a+b x^2}}+\frac {4 (b c-6 a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {a} b^{5/2} \sqrt [4]{a+b x^2}} \] Output:

-2/5*(-a*d+b*c)*x/b^2/(b*x^2+a)^(5/4)+2*d*x/b^2/(b*x^2+a)^(1/4)+4/5*(-6*a* 
d+b*c)*(1+b*x^2/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1 
/2))/a^(1/2)/b^(5/2)/(b*x^2+a)^(1/4)
 

Mathematica [C] (verified)

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

Time = 10.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {2 x \left (-6 a^2 d+2 b^2 c x^2+a b \left (c-7 d x^2\right )+(-b c+6 a d) \left (a+b x^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{5 a b^2 \left (a+b x^2\right )^{5/4}} \] Input:

Integrate[(x^2*(c + d*x^2))/(a + b*x^2)^(9/4),x]
 

Output:

(2*x*(-6*a^2*d + 2*b^2*c*x^2 + a*b*(c - 7*d*x^2) + (-(b*c) + 6*a*d)*(a + b 
*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)] 
))/(5*a*b^2*(a + b*x^2)^(5/4))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {362, 250, 213, 212}

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.

\(\displaystyle \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {2 x^3 (b c-a d)}{5 a b \left (a+b x^2\right )^{5/4}}-\frac {(b c-6 a d) \int \frac {x^2}{\left (b x^2+a\right )^{5/4}}dx}{5 a b}\)

\(\Big \downarrow \) 250

\(\displaystyle \frac {2 x^3 (b c-a d)}{5 a b \left (a+b x^2\right )^{5/4}}-\frac {(b c-6 a d) \left (\frac {2 x}{b \sqrt [4]{a+b x^2}}-\frac {2 a \int \frac {1}{\left (b x^2+a\right )^{5/4}}dx}{b}\right )}{5 a b}\)

\(\Big \downarrow \) 213

\(\displaystyle \frac {2 x^3 (b c-a d)}{5 a b \left (a+b x^2\right )^{5/4}}-\frac {(b c-6 a d) \left (\frac {2 x}{b \sqrt [4]{a+b x^2}}-\frac {2 \sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx}{b \sqrt [4]{a+b x^2}}\right )}{5 a b}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {2 x^3 (b c-a d)}{5 a b \left (a+b x^2\right )^{5/4}}-\frac {(b c-6 a d) \left (\frac {2 x}{b \sqrt [4]{a+b x^2}}-\frac {4 \sqrt {a} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^2}}\right )}{5 a b}\)

Input:

Int[(x^2*(c + d*x^2))/(a + b*x^2)^(9/4),x]
 

Output:

(2*(b*c - a*d)*x^3)/(5*a*b*(a + b*x^2)^(5/4)) - ((b*c - 6*a*d)*((2*x)/(b*( 
a + b*x^2)^(1/4)) - (4*Sqrt[a]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqr 
t[b]*x)/Sqrt[a]]/2, 2])/(b^(3/2)*(a + b*x^2)^(1/4))))/(5*a*b)
 

Defintions of rubi rules used

rule 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 213

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a*(a + b*x^2)^(1/4))   Int[1/(1 + b*(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, 
 x] && PosQ[a] && PosQ[b/a]
 

rule 250

Int[((c_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2)^(5/4), x_Symbol] :> Simp[2*c*(( 
c*x)^(m - 1)/(b*(2*m - 3)*(a + b*x^2)^(1/4))), x] - Simp[2*a*c^2*((m - 1)/( 
b*(2*m - 3)))   Int[(c*x)^(m - 2)/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b, 
 c}, x] && PosQ[b/a] && IntegerQ[2*m] && GtQ[m, 3/2]
 

rule 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
 

Maple [F]

\[\int \frac {x^{2} \left (x^{2} d +c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}d x\]

Input:

int(x^2*(d*x^2+c)/(b*x^2+a)^(9/4),x)
 

Output:

int(x^2*(d*x^2+c)/(b*x^2+a)^(9/4),x)
 

Fricas [F]

\[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \] Input:

integrate(x^2*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="fricas")
 

Output:

integral((d*x^4 + c*x^2)*(b*x^2 + a)^(3/4)/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2* 
b*x^2 + a^3), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.54 \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {c x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {9}{4}}} + \frac {d x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {9}{4}}} \] Input:

integrate(x**2*(d*x**2+c)/(b*x**2+a)**(9/4),x)
 

Output:

c*x**3*hyper((3/2, 9/4), (5/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(9/4)) + 
d*x**5*hyper((9/4, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(9/4))
 

Maxima [F]

\[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \] Input:

integrate(x^2*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)*x^2/(b*x^2 + a)^(9/4), x)
 

Giac [F]

\[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \] Input:

integrate(x^2*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)*x^2/(b*x^2 + a)^(9/4), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int \frac {x^2\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \] Input:

int((x^2*(c + d*x^2))/(a + b*x^2)^(9/4),x)
 

Output:

int((x^2*(c + d*x^2))/(a + b*x^2)^(9/4), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\left (\int \frac {x^{4}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2}+2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a b \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} x^{4}}d x \right ) d +\left (\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2}+2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a b \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} x^{4}}d x \right ) c \] Input:

int(x^2*(d*x^2+c)/(b*x^2+a)^(9/4),x)
 

Output:

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