\(\int \frac {1}{(a+b x^2)^{3/4} (c+d x^2)^2} \, dx\) [1546]

Optimal result

Integrand size = 21, antiderivative size = 292 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (5 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d)^2 x}+\frac {\sqrt [4]{a} (5 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d)^2 x} \] Output:

-1/2*d*x*(b*x^2+a)^(1/4)/c/(-a*d+b*c)/(d*x^2+c)-1/2*a^(1/2)*b^(1/2)*(1+b*x 
^2/a)^(3/4)*InverseJacobiAM(1/2*arctan(b^(1/2)*x/a^(1/2)),2^(1/2))/c/(-a*d 
+b*c)/(b*x^2+a)^(3/4)+1/4*a^(1/4)*(-2*a*d+5*b*c)*(-b*x^2/a)^(1/2)*Elliptic 
Pi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c/(-a*d+b*c 
)^2/x+1/4*a^(1/4)*(-2*a*d+5*b*c)*(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/ 
4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c/(-a*d+b*c)^2/x
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 0.30 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\frac {x \left (\frac {b d x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-b c+a d}+\frac {c \left (36 a c \left (-2 b c+2 a d+b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 d x^2 \left (a+b x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{(b c-a d) \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 c^2 \left (a+b x^2\right )^{3/4}} \] Input:

Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]
 

Output:

(x*((b*d*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a) 
, -((d*x^2)/c)])/(-(b*c) + a*d) + (c*(36*a*c*(-2*b*c + 2*a*d + b*d*x^2)*Ap 
pellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 6*d*x^2*(a + b*x^2) 
*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*App 
ellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((b*c - a*d)*(c + d 
*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2 
*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*App 
ellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(12*c^2*(a + b*x^ 
2)^(3/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {316, 27, 405, 231, 229, 312, 118, 25, 925, 1542}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 405

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

\(\Big \downarrow \) 231

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

\(\Big \downarrow \) 229

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

\(\Big \downarrow \) 312

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

\(\Big \downarrow \) 118

\(\displaystyle \frac {-\frac {2 \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \int -\frac {1}{\sqrt {1-\frac {x^8}{a}} \left (d x^8+b c-a d\right )}d\sqrt [4]{b x^2+a}}{x}-\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{4 c (b c-a d)}-\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \int \frac {1}{\sqrt {1-\frac {x^8}{a}} \left (d x^8+b c-a d\right )}d\sqrt [4]{b x^2+a}}{x}-\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{4 c (b c-a d)}-\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {-\frac {2 \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \left (-\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^4}{\sqrt {a d-b c}}\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{b x^2+a}}{2 (b c-a d)}-\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^4}{\sqrt {a d-b c}}+1\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{b x^2+a}}{2 (b c-a d)}\right )}{x}-\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{4 c (b c-a d)}-\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {-\frac {2 \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \left (-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 (b c-a d)}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 (b c-a d)}\right )}{x}-\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a+b x^2\right )^{3/4}}}{4 c (b c-a d)}-\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}\)

Input:

Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x]
 

Output:

-1/2*(d*x*(a + b*x^2)^(1/4))/(c*(b*c - a*d)*(c + d*x^2)) + ((-2*Sqrt[a]*Sq 
rt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/( 
a + b*x^2)^(3/4) - (2*(5*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*(-1/2*(a^(1/4)*El 
lipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4) 
/a^(1/4)], -1])/(b*c - a*d) - (a^(1/4)*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[- 
(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(2*(b*c - a*d))))/x) 
/(4*c*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 118

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
3/4)), x_] :> Simp[-4   Subst[Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + 
 d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] & 
& GtQ[-f/(d*e - c*f), 0]
 

rule 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 231

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

rule 312

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[Sqrt[(-b)*(x^2/a)]/(2*x)   Subst[Int[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)* 
(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 316

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

rule 405

Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 
), x_Symbol] :> Simp[f/d   Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d 
 Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{2}}d x\]

Input:

int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
 

Output:

int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c)**2,x)
 

Output:

Integral(1/((a + b*x**2)**(3/4)*(c + d*x**2)**2), x)
 

Maxima [F]

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

integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^2), x)
 

Giac [F]

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

integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/4}\,{\left (d\,x^2+c\right )}^2} \,d x \] Input:

int(1/((a + b*x^2)^(3/4)*(c + d*x^2)^2),x)
                                                                                    
                                                                                    
 

Output:

int(1/((a + b*x^2)^(3/4)*(c + d*x^2)^2), x)
 

Reduce [F]

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

int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^2,x)
 

Output:

int(1/((a + b*x**2)**(3/4)*c**2 + 2*(a + b*x**2)**(3/4)*c*d*x**2 + (a + b* 
x**2)**(3/4)*d**2*x**4),x)