\(\int \frac {A+B x^2+C x^4+D x^6}{(a+b x^2)^{13/4}} \, dx\) [149]

Optimal result

Integrand size = 29, antiderivative size = 196 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\frac {2 \left (\frac {A}{a}-\frac {b^2 B-a b C+a^2 D}{b^3}\right ) x}{9 \left (a+b x^2\right )^{9/4}}+\frac {2 \left (7 A b^3+a \left (2 b^2 B-11 a b C+20 a^2 D\right )\right ) x}{45 a^2 b^3 \left (a+b x^2\right )^{5/4}}+\frac {2 D x}{b^3 \sqrt [4]{a+b x^2}}+\frac {2 \left (7 A b^3+2 a \left (b^2 B+2 a b C-20 a^2 D\right )\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} b^{7/2} \sqrt [4]{a+b x^2}} \] Output:

2/9*(A/a-(B*b^2-C*a*b+D*a^2)/b^3)*x/(b*x^2+a)^(9/4)+2/45*(7*A*b^3+a*(2*B*b 
^2-11*C*a*b+20*D*a^2))*x/a^2/b^3/(b*x^2+a)^(5/4)+2*D*x/b^3/(b*x^2+a)^(1/4) 
+2/15*(7*A*b^3+2*a*(B*b^2+2*C*a*b-20*D*a^2))*(1+b*x^2/a)^(1/4)*EllipticE(s 
in(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(5/2)/b^(7/2)/(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 = 10.16 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\frac {2 x \left (-60 a^5 D+21 A b^5 x^4+a b^4 x^2 \left (49 A+6 B x^2\right )+2 a^4 b \left (3 C-65 D x^2\right )+a^2 b^3 \left (33 A+14 B x^2+12 C x^4\right )+a^3 b^2 \left (3 B+13 C x^2-75 D x^4\right )\right )-3 \left (7 A b^3+2 a \left (b^2 B+2 a b C-20 a^2 D\right )\right ) x \left (a+b x^2\right )^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{45 a^3 b^3 \left (a+b x^2\right )^{9/4}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(13/4),x]
 

Output:

(2*x*(-60*a^5*D + 21*A*b^5*x^4 + a*b^4*x^2*(49*A + 6*B*x^2) + 2*a^4*b*(3*C 
 - 65*D*x^2) + a^2*b^3*(33*A + 14*B*x^2 + 12*C*x^4) + a^3*b^2*(3*B + 13*C* 
x^2 - 75*D*x^4)) - 3*(7*A*b^3 + 2*a*(b^2*B + 2*a*b*C - 20*a^2*D))*x*(a + b 
*x^2)^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a 
)])/(45*a^3*b^3*(a + b*x^2)^(9/4))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2345, 27, 1471, 27, 298, 227, 225, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}-\frac {2 \int -\frac {\frac {9 a D x^4}{b}+\frac {9 a (b C-a D) x^2}{b^2}+7 A+\frac {2 a \left (D a^2-b C a+b^2 B\right )}{b^3}}{2 \left (b x^2+a\right )^{9/4}}dx}{9 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\frac {9 a D x^4}{b}+\frac {9 a (b C-a D) x^2}{b^2}+7 A+\frac {2 a \left (D a^2-b C a+b^2 B\right )}{b^3}}{\left (b x^2+a\right )^{9/4}}dx}{9 a}+\frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {\frac {2 x \left (\frac {20 a^2 D-11 a b C+2 b^2 B}{b^3}+\frac {7 A}{a}\right )}{5 \left (a+b x^2\right )^{5/4}}-\frac {2 \int -\frac {3 \left (\left (7 A+\frac {2 a \left (-5 D a^2+2 b C a+b^2 B\right )}{b^3}\right ) b^2+15 a^2 D x^2\right )}{2 b^2 \left (b x^2+a\right )^{5/4}}dx}{5 a}}{9 a}+\frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {7 A b^2+15 a^2 D x^2+2 a \left (-\frac {5 D a^2}{b}+2 C a+b B\right )}{\left (b x^2+a\right )^{5/4}}dx}{5 a b^2}+\frac {2 x \left (\frac {20 a^2 D-11 a b C+2 b^2 B}{b^3}+\frac {7 A}{a}\right )}{5 \left (a+b x^2\right )^{5/4}}}{9 a}+\frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {3 \left (\frac {2 x \left (a \left (-25 a^2 D+4 a b C+2 b^2 B\right )+7 A b^3\right )}{a b \sqrt [4]{a+b x^2}}-\frac {\left (-40 a^3 D+4 a^2 b C+2 a b^2 B+7 A b^3\right ) \int \frac {1}{\sqrt [4]{b x^2+a}}dx}{a b}\right )}{5 a b^2}+\frac {2 x \left (\frac {20 a^2 D-11 a b C+2 b^2 B}{b^3}+\frac {7 A}{a}\right )}{5 \left (a+b x^2\right )^{5/4}}}{9 a}+\frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {\frac {3 \left (\frac {2 x \left (a \left (-25 a^2 D+4 a b C+2 b^2 B\right )+7 A b^3\right )}{a b \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (-40 a^3 D+4 a^2 b C+2 a b^2 B+7 A b^3\right ) \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{a b \sqrt [4]{a+b x^2}}\right )}{5 a b^2}+\frac {2 x \left (\frac {20 a^2 D-11 a b C+2 b^2 B}{b^3}+\frac {7 A}{a}\right )}{5 \left (a+b x^2\right )^{5/4}}}{9 a}+\frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}\)

\(\Big \downarrow \) 225

\(\displaystyle \frac {\frac {3 \left (\frac {2 x \left (a \left (-25 a^2 D+4 a b C+2 b^2 B\right )+7 A b^3\right )}{a b \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (-40 a^3 D+4 a^2 b C+2 a b^2 B+7 A b^3\right ) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{a b \sqrt [4]{a+b x^2}}\right )}{5 a b^2}+\frac {2 x \left (\frac {20 a^2 D-11 a b C+2 b^2 B}{b^3}+\frac {7 A}{a}\right )}{5 \left (a+b x^2\right )^{5/4}}}{9 a}+\frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {2 x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{9 a \left (a+b x^2\right )^{9/4}}+\frac {\frac {2 x \left (\frac {20 a^2 D-11 a b C+2 b^2 B}{b^3}+\frac {7 A}{a}\right )}{5 \left (a+b x^2\right )^{5/4}}+\frac {3 \left (\frac {2 x \left (a \left (-25 a^2 D+4 a b C+2 b^2 B\right )+7 A b^3\right )}{a b \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right ) \left (-40 a^3 D+4 a^2 b C+2 a b^2 B+7 A b^3\right )}{a b \sqrt [4]{a+b x^2}}\right )}{5 a b^2}}{9 a}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^(13/4),x]
 

Output:

(2*(A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x)/(9*a*(a + b*x^2)^(9/4)) + ((2* 
((7*A)/a + (2*b^2*B - 11*a*b*C + 20*a^2*D)/b^3)*x)/(5*(a + b*x^2)^(5/4)) + 
 (3*((2*(7*A*b^3 + a*(2*b^2*B + 4*a*b*C - 25*a^2*D))*x)/(a*b*(a + b*x^2)^( 
1/4)) - ((7*A*b^3 + 2*a*b^2*B + 4*a^2*b*C - 40*a^3*D)*(1 + (b*x^2)/a)^(1/4 
)*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt[b]*x)/S 
qrt[a]]/2, 2])/Sqrt[b]))/(a*b*(a + b*x^2)^(1/4))))/(5*a*b^2))/(9*a)
 

Defintions of rubi rules used

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 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 225

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

rule 227

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

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [F]

\[\int \frac {D x^{6}+C \,x^{4}+x^{2} B +A}{\left (b \,x^{2}+a \right )^{\frac {13}{4}}}d x\]

Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(13/4),x)
 

Output:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(13/4),x)
 

Fricas [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {13}{4}}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(13/4),x, algorithm="fricas")
 

Output:

integral((D*x^6 + C*x^4 + B*x^2 + A)*(b*x^2 + a)^(3/4)/(b^4*x^8 + 4*a*b^3* 
x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 19.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\frac {A x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {13}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {13}{4}}} + \frac {B x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {13}{4} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {13}{4}}} + \frac {C x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {13}{4} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {13}{4}}} + \frac {D x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {13}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac {13}{4}}} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(13/4),x)
 

Output:

A*x*hyper((1/2, 13/4), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(13/4) + B*x** 
3*hyper((3/2, 13/4), (5/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(13/4)) + C*x 
**5*hyper((5/2, 13/4), (7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(13/4)) + D 
*x**7*hyper((13/4, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/(7*a**(13/4))
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {13}{4}}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(13/4),x, algorithm="maxima")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/(b*x^2 + a)^(13/4), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {13}{4}}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(13/4),x, algorithm="giac")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/(b*x^2 + a)^(13/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^{13/4}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^(13/4),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^(13/4), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{13/4}} \, dx=\left (\int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{3} x^{6}}d x \right ) d +\left (\int \frac {x^{4}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{3} x^{6}}d x \right ) c +\left (\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{3} x^{6}}d x \right ) b +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{3} x^{6}}d x \right ) a \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(13/4),x)
 

Output:

int(x**6/((a + b*x**2)**(1/4)*a**3 + 3*(a + b*x**2)**(1/4)*a**2*b*x**2 + 3 
*(a + b*x**2)**(1/4)*a*b**2*x**4 + (a + b*x**2)**(1/4)*b**3*x**6),x)*d + i 
nt(x**4/((a + b*x**2)**(1/4)*a**3 + 3*(a + b*x**2)**(1/4)*a**2*b*x**2 + 3* 
(a + b*x**2)**(1/4)*a*b**2*x**4 + (a + b*x**2)**(1/4)*b**3*x**6),x)*c + in 
t(x**2/((a + b*x**2)**(1/4)*a**3 + 3*(a + b*x**2)**(1/4)*a**2*b*x**2 + 3*( 
a + b*x**2)**(1/4)*a*b**2*x**4 + (a + b*x**2)**(1/4)*b**3*x**6),x)*b + int 
(1/((a + b*x**2)**(1/4)*a**3 + 3*(a + b*x**2)**(1/4)*a**2*b*x**2 + 3*(a + 
b*x**2)**(1/4)*a*b**2*x**4 + (a + b*x**2)**(1/4)*b**3*x**6),x)*a