\(\int \frac {1}{(a+b x^2)^{7/4} (c+d x^2)} \, dx\) [443]

Optimal result

Integrand size = 21, antiderivative size = 254 \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\frac {2 b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {2 \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {a} (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} d \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{(b c-a d)^2 x}-\frac {\sqrt [4]{a} d \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{(b c-a d)^2 x} \] Output:

2/3*b*x/a/(-a*d+b*c)/(b*x^2+a)^(3/4)+2/3*b^(1/2)*(1+b*x^2/a)^(3/4)*Inverse 
JacobiAM(1/2*arctan(b^(1/2)*x/a^(1/2)),2^(1/2))/a^(1/2)/(-a*d+b*c)/(b*x^2+ 
a)^(3/4)-a^(1/4)*d*(-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)/(-a*d+b*c)^2/x-a^(1/4)*d*(-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)/(-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 = 9.00 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\frac {x \left (-\frac {b d x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (3 a c \left (-3 b c+3 a d-2 b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (c+d x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{9 a (-b c+a d) \left (a+b x^2\right )^{3/4}} \] Input:

Integrate[1/((a + b*x^2)^(7/4)*(c + d*x^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)])/c) + (6*(3*a*c*(-3*b*c + 3*a*d - 2*b*d*x^2)*AppellF1[1/ 
2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(c + d*x^2)*(4*a*d*App 
ellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 
7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((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*AppellF1[3/2, 7/4, 1, 5/2, 
-((b*x^2)/a), -((d*x^2)/c)])))))/(9*a*(-(b*c) + a*d)*(a + b*x^2)^(3/4))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {302, 215, 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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 302

\(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{7/4}}dx}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/4} \left (d x^2+c\right )}dx}{b c-a d}\)

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 231

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

\(\Big \downarrow \) 229

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

\(\Big \downarrow \) 312

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

\(\Big \downarrow \) 118

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 1542

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

Input:

Int[1/((a + b*x^2)^(7/4)*(c + d*x^2)),x]
 

Output:

(b*((2*x)/(3*a*(a + b*x^2)^(3/4)) + (2*(1 + (b*x^2)/a)^(3/4)*EllipticF[Arc 
Tan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*Sqrt[a]*Sqrt[b]*(a + b*x^2)^(3/4))))/(b 
*c - a*d) + (2*d*Sqrt[-((b*x^2)/a)]*(-1/2*(a^(1/4)*EllipticPi[-((Sqrt[a]*S 
qrt[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))))/((b*c - a*d)*x)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 302

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/( 
b*c - a*d)   Int[(a + b*x^2)^p, x], x] - Simp[d/(b*c - a*d)   Int[(a + b*x^ 
2)^(p + 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 
0] && LtQ[p, -1] && EqQ[Denominator[p], 4] && (EqQ[p, -5/4] || EqQ[p, -7/4] 
)
 

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 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]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (x^{2} d +c \right )}d x\]

Input:

int(1/(b*x^2+a)^(7/4)/(d*x^2+c),x)
 

Output:

int(1/(b*x^2+a)^(7/4)/(d*x^2+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {7}{4}} \left (c + d x^{2}\right )}\, dx \] Input:

integrate(1/(b*x**2+a)**(7/4)/(d*x**2+c),x)
 

Output:

Integral(1/((a + b*x**2)**(7/4)*(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(1/((b*x^2 + a)^(7/4)*(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(7/4)*(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{7/4}\,\left (d\,x^2+c\right )} \,d x \] Input:

int(1/((a + b*x^2)^(7/4)*(c + d*x^2)),x)
 

Output:

int(1/((a + b*x^2)^(7/4)*(c + d*x^2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} a c +\left (b \,x^{2}+a \right )^{\frac {3}{4}} a d \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {3}{4}} b c \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {3}{4}} b d \,x^{4}}d x \] Input:

int(1/(b*x^2+a)^(7/4)/(d*x^2+c),x)
 

Output:

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