\(\int \frac {(c+d x^2)^2}{(a-b x^2)^{5/4}} \, dx\) [481]

Optimal result

Integrand size = 22, antiderivative size = 137 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\frac {2 (b c+a d)^2 x}{a b^2 \sqrt [4]{a-b x^2}}+\frac {2 d^2 x \left (a-b x^2\right )^{3/4}}{5 b^2}-\frac {2 \left (5 b^2 c^2+20 a b c d+12 a^2 d^2\right ) \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {a} b^{5/2} \sqrt [4]{a-b x^2}} \] Output:

2*(a*d+b*c)^2*x/a/b^2/(-b*x^2+a)^(1/4)+2/5*d^2*x*(-b*x^2+a)^(3/4)/b^2-2/5* 
(12*a^2*d^2+20*a*b*c*d+5*b^2*c^2)*(1-b*x^2/a)^(1/4)*EllipticE(sin(1/2*arcs 
in(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(1/2)/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 = 10.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.23 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\frac {x \sqrt [4]{1-\frac {b x^2}{a}} \operatorname {Gamma}\left (\frac {1}{4}\right ) \left (7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {7}{2},\frac {b x^2}{a}\right )+10 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{4},\frac {9}{2},\frac {b x^2}{a}\right )+5 b x^2 \left (c+d x^2\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {9}{4};1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{420 a^2 \sqrt [4]{a-b x^2} \operatorname {Gamma}\left (\frac {5}{4}\right )} \] Input:

Integrate[(c + d*x^2)^2/(a - b*x^2)^(5/4),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, 5/4, 7/2, (b*x^2)/a] + 10*b*x^2*(2*c^2 + 3*c*d*x^2 
 + d^2*x^4)*Hypergeometric2F1[3/2, 9/4, 9/2, (b*x^2)/a] + 5*b*x^2*(c + d*x 
^2)^2*HypergeometricPFQ[{3/2, 2, 9/4}, {1, 9/2}, (b*x^2)/a]))/(420*a^2*(a 
- b*x^2)^(1/4)*Gamma[5/4])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {315, 27, 299, 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.

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {2 x \left (c+d x^2\right ) (a d+b c)}{a b \sqrt [4]{a-b x^2}}-\frac {\frac {\left (12 a^2 d^2+20 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{5 b}-\frac {2 d x \left (a-b x^2\right )^{3/4} (6 a d+5 b c)}{5 b}}{a b}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {2 x \left (c+d x^2\right ) (a d+b c)}{a b \sqrt [4]{a-b x^2}}-\frac {\frac {\sqrt [4]{1-\frac {b x^2}{a}} \left (12 a^2 d^2+20 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{5 b \sqrt [4]{a-b x^2}}-\frac {2 d x \left (a-b x^2\right )^{3/4} (6 a d+5 b c)}{5 b}}{a b}\)

\(\Big \downarrow \) 226

\(\displaystyle \frac {2 x \left (c+d x^2\right ) (a d+b c)}{a b \sqrt [4]{a-b x^2}}-\frac {\frac {2 \sqrt {a} \sqrt [4]{1-\frac {b x^2}{a}} \left (12 a^2 d^2+20 a b c d+5 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a-b x^2}}-\frac {2 d x \left (a-b x^2\right )^{3/4} (6 a d+5 b c)}{5 b}}{a b}\)

Input:

Int[(c + d*x^2)^2/(a - b*x^2)^(5/4),x]
 

Output:

(2*(b*c + a*d)*x*(c + d*x^2))/(a*b*(a - b*x^2)^(1/4)) - ((-2*d*(5*b*c + 6* 
a*d)*x*(a - b*x^2)^(3/4))/(5*b) + (2*Sqrt[a]*(5*b^2*c^2 + 20*a*b*c*d + 12* 
a^2*d^2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2] 
)/(5*b^(3/2)*(a - b*x^2)^(1/4)))/(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 299

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

\[\int \frac {\left (x^{2} d +c \right )^{2}}{\left (-b \,x^{2}+a \right )^{\frac {5}{4}}}d x\]

Input:

int((d*x^2+c)^2/(-b*x^2+a)^(5/4),x)
 

Output:

int((d*x^2+c)^2/(-b*x^2+a)^(5/4),x)
 

Fricas [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((d*x^2+c)^2/(-b*x^2+a)^(5/4),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\int \frac {\left (c + d x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {5}{4}}}\, dx \] Input:

integrate((d*x**2+c)**2/(-b*x**2+a)**(5/4),x)
 

Output:

Integral((c + d*x**2)**2/(a - b*x**2)**(5/4), x)
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((d*x^2+c)^2/(-b*x^2+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^2/(-b*x^2 + a)^(5/4), x)
 

Giac [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((d*x^2+c)^2/(-b*x^2+a)^(5/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)^2/(-b*x^2 + a)^(5/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (a-b\,x^2\right )}^{5/4}} \,d x \] Input:

int((c + d*x^2)^2/(a - b*x^2)^(5/4),x)
 

Output:

int((c + d*x^2)^2/(a - b*x^2)^(5/4), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^2\right )^2}{\left (a-b x^2\right )^{5/4}} \, dx=\left (\int \frac {x^{4}}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b \,x^{2}}d x \right ) d^{2}+2 \left (\int \frac {x^{2}}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b \,x^{2}}d x \right ) c d +\left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b \,x^{2}}d x \right ) c^{2} \] Input:

int((d*x^2+c)^2/(-b*x^2+a)^(5/4),x)
 

Output:

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