\(\int \frac {(c+d x^2)^2}{(a-b x^2)^{9/4}} \, dx\) [491]

Optimal result

Integrand size = 22, antiderivative size = 155 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{9/4}} \, dx=\frac {2 (b c+a d)^2 x}{5 a b^2 \left (a-b x^2\right )^{5/4}}+\frac {2 (3 b c-7 a d) (b c+a d) x}{5 a^2 b^2 \sqrt [4]{a-b x^2}}-\frac {2 \left (3 b^2 c^2-4 a b c d-12 a^2 d^2\right ) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{3/2} b^{5/2} \sqrt [4]{a-b x^2}} \] Output:

2/5*(a*d+b*c)^2*x/a/b^2/(-b*x^2+a)^(5/4)+2/5*(-7*a*d+3*b*c)*(a*d+b*c)*x/a^ 
2/b^2/(-b*x^2+a)^(1/4)-2/5*(-12*a^2*d^2-4*a*b*c*d+3*b^2*c^2)*(1-b*x^2/a)^( 
1/4)*EllipticE(sin(1/2*arcsin(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(3/2)/b^(5/2) 
/(-b*x^2+a)^(1/4)
 

Mathematica [C] (warning: unable to verify)

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

Time = 14.60 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{9/4}} \, dx=\frac {x \sqrt [4]{1-\frac {b x^2}{a}} \operatorname {Gamma}\left (\frac {1}{4}\right ) \left (7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {7}{2},\frac {b x^2}{a}\right )+18 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {13}{4},\frac {9}{2},\frac {b x^2}{a}\right )+9 b x^2 \left (c+d x^2\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {13}{4};1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{336 a^3 \sqrt [4]{a-b x^2} \operatorname {Gamma}\left (\frac {9}{4}\right )} \] Input:

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

Output:

(x*(1 - (b*x^2)/a)^(1/4)*Gamma[1/4]*(7*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4) 
*Hypergeometric2F1[1/2, 9/4, 7/2, (b*x^2)/a] + 18*b*x^2*(2*c^2 + 3*c*d*x^2 
 + d^2*x^4)*Hypergeometric2F1[3/2, 13/4, 9/2, (b*x^2)/a] + 9*b*x^2*(c + d* 
x^2)^2*HypergeometricPFQ[{3/2, 2, 13/4}, {1, 9/2}, (b*x^2)/a]))/(336*a^3*( 
a - b*x^2)^(1/4)*Gamma[9/4])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {315, 27, 298, 227, 226}

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.

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {6 x \left (\frac {b c^2}{a}-\frac {2 a d^2}{b}-c d\right )}{\sqrt [4]{a-b x^2}}-\left (\frac {3 b c^2}{a}-\frac {12 a d^2}{b}-4 c d\right ) \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{5 a b}+\frac {2 x \left (c+d x^2\right ) (a d+b c)}{5 a b \left (a-b x^2\right )^{5/4}}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {\frac {6 x \left (\frac {b c^2}{a}-\frac {2 a d^2}{b}-c d\right )}{\sqrt [4]{a-b x^2}}-\frac {\sqrt [4]{1-\frac {b x^2}{a}} \left (\frac {3 b c^2}{a}-\frac {12 a d^2}{b}-4 c d\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{\sqrt [4]{a-b x^2}}}{5 a b}+\frac {2 x \left (c+d x^2\right ) (a d+b c)}{5 a b \left (a-b x^2\right )^{5/4}}\)

\(\Big \downarrow \) 226

\(\displaystyle \frac {\frac {6 x \left (\frac {b c^2}{a}-\frac {2 a d^2}{b}-c d\right )}{\sqrt [4]{a-b x^2}}-\frac {2 \sqrt {a} \sqrt [4]{1-\frac {b x^2}{a}} \left (\frac {3 b c^2}{a}-\frac {12 a d^2}{b}-4 c d\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a-b x^2}}}{5 a b}+\frac {2 x \left (c+d x^2\right ) (a d+b c)}{5 a b \left (a-b x^2\right )^{5/4}}\)

Input:

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

Output:

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

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 226

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] 
))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ 
[a, 0] && NegQ[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 315

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(d^2*x^4 + 2*c*d*x^2 + c^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 [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((d*x^2+c)^2/(-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**2 + 2*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*d + 
 int(1/((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**2