\(\int \frac {1}{(a-b x^2)^{7/4} (c+d x^2)^2} \, dx\) [489]

Optimal result

Integrand size = 22, antiderivative size = 342 \[ \int \frac {1}{\left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2} \, dx=\frac {b (4 b c-3 a d) x}{6 a c (b c+a d)^2 \left (a-b x^2\right )^{3/4}}+\frac {d x}{2 c (b c+a d) \left (a-b x^2\right )^{3/4} \left (c+d x^2\right )}+\frac {\sqrt {b} (4 b c-3 a d) \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{6 \sqrt {a} c (b c+a d)^2 \left (a-b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} d (9 b c+2 a d) \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c+a d)^3 x}-\frac {\sqrt [4]{a} d (9 b c+2 a d) \sqrt {\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c+a d)^3 x} \] Output:

1/6*b*(-3*a*d+4*b*c)*x/a/c/(a*d+b*c)^2/(-b*x^2+a)^(3/4)+1/2*d*x/c/(a*d+b*c 
)/(-b*x^2+a)^(3/4)/(d*x^2+c)+1/6*b^(1/2)*(-3*a*d+4*b*c)*(1-b*x^2/a)^(3/4)* 
InverseJacobiAM(1/2*arcsin(b^(1/2)*x/a^(1/2)),2^(1/2))/a^(1/2)/c/(a*d+b*c) 
^2/(-b*x^2+a)^(3/4)-1/4*a^(1/4)*d*(2*a*d+9*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)^ 
3/x-1/4*a^(1/4)*d*(2*a*d+9*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)^3/x
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.38 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2} \, dx=\frac {x \left (-b d (-4 b c+3 a d) x^2 \left (1-\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {6 c \left (6 a c \left (6 a^2 d^2-3 a b d \left (-4 c+d x^2\right )+2 b^2 c \left (3 c+2 d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (3 a^2 d^2-3 a b d^2 x^2+4 b^2 c \left (c+d x^2\right )\right ) \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{36 a c^2 (b c+a d)^2 \left (a-b x^2\right )^{3/4}} \] Input:

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

Output:

(x*(-(b*d*(-4*b*c + 3*a*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)]) + (6*c*(6*a*c*(6*a^2*d^2 - 3*a*b*d*(-4*c + 
 d*x^2) + 2*b^2*c*(3*c + 2*d*x^2))*AppellF1[1/2, 3/4, 1, 3/2, (b*x^2)/a, - 
((d*x^2)/c)] + x^2*(3*a^2*d^2 - 3*a*b*d^2*x^2 + 4*b^2*c*(c + d*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)])))/((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)])))))/(36*a*c^2*(b*c + a*d)^2*(a - b*x^2)^(3/4))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {316, 27, 402, 27, 405, 231, 230, 312, 118, 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.

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}-\frac {\int -\frac {-5 b d x^2+4 b c+2 a d}{2 \left (a-b x^2\right )^{7/4} \left (d x^2+c\right )}dx}{2 c (a d+b c)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 (2 b c+a d)-5 b d x^2}{\left (a-b x^2\right )^{7/4} \left (d x^2+c\right )}dx}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {2 \int \frac {4 b^2 c^2+24 a b d c+6 a^2 d^2+b d (4 b c-3 a d) x^2}{2 \left (a-b x^2\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 405

\(\displaystyle \frac {\frac {b (4 b c-3 a d) \int \frac {1}{\left (a-b x^2\right )^{3/4}}dx+3 a d (2 a d+9 b c) \int \frac {1}{\left (a-b x^2\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 231

\(\displaystyle \frac {\frac {\frac {b \left (1-\frac {b x^2}{a}\right )^{3/4} (4 b c-3 a d) \int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}}dx}{\left (a-b x^2\right )^{3/4}}+3 a d (2 a d+9 b c) \int \frac {1}{\left (a-b x^2\right )^{3/4} \left (d x^2+c\right )}dx}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 230

\(\displaystyle \frac {\frac {3 a d (2 a d+9 b c) \int \frac {1}{\left (a-b x^2\right )^{3/4} \left (d x^2+c\right )}dx+\frac {2 \sqrt {a} \sqrt {b} \left (1-\frac {b x^2}{a}\right )^{3/4} (4 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a-b x^2\right )^{3/4}}}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 312

\(\displaystyle \frac {\frac {\frac {3 a d \sqrt {\frac {b x^2}{a}} (2 a d+9 b c) \int \frac {1}{\sqrt {\frac {b x^2}{a}} \left (a-b x^2\right )^{3/4} \left (d x^2+c\right )}dx^2}{2 x}+\frac {2 \sqrt {a} \sqrt {b} \left (1-\frac {b x^2}{a}\right )^{3/4} (4 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a-b x^2\right )^{3/4}}}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 118

\(\displaystyle \frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \left (1-\frac {b x^2}{a}\right )^{3/4} (4 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a-b x^2\right )^{3/4}}-\frac {6 a d \sqrt {\frac {b x^2}{a}} (2 a d+9 b c) \int \frac {1}{\sqrt {1-\frac {x^8}{a}} \left (-d x^8+b c+a d\right )}d\sqrt [4]{a-b x^2}}{x}}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \left (1-\frac {b x^2}{a}\right )^{3/4} (4 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a-b x^2\right )^{3/4}}-\frac {6 a d \sqrt {\frac {b x^2}{a}} (2 a d+9 b c) \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^4}{\sqrt {b c+a d}}\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{a-b x^2}}{2 (a d+b c)}+\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^4}{\sqrt {b c+a d}}+1\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{a-b x^2}}{2 (a d+b c)}\right )}{x}}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\frac {\frac {2 \sqrt {a} \sqrt {b} \left (1-\frac {b x^2}{a}\right )^{3/4} (4 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\left (a-b x^2\right )^{3/4}}-\frac {6 a d \sqrt {\frac {b x^2}{a}} (2 a d+9 b c) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 (a d+b c)}+\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c+a d}},\arcsin \left (\frac {\sqrt [4]{a-b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{2 (a d+b c)}\right )}{x}}{3 a (a d+b c)}+\frac {2 b x (4 b c-3 a d)}{3 a \left (a-b x^2\right )^{3/4} (a d+b c)}}{4 c (a d+b c)}+\frac {d x}{2 c \left (a-b x^2\right )^{3/4} \left (c+d x^2\right ) (a d+b c)}\)

Input:

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

Output:

(d*x)/(2*c*(b*c + a*d)*(a - b*x^2)^(3/4)*(c + d*x^2)) + ((2*b*(4*b*c - 3*a 
*d)*x)/(3*a*(b*c + a*d)*(a - b*x^2)^(3/4)) + ((2*Sqrt[a]*Sqrt[b]*(4*b*c - 
3*a*d)*(1 - (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/ 
(a - b*x^2)^(3/4) - (6*a*d*(9*b*c + 2*a*d)*Sqrt[(b*x^2)/a]*((a^(1/4)*Ellip 
ticPi[-((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)) + (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)/(3*a 
*(b*c + a*d)))/(4*c*(b*c + a*d))
 

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 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 230

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] 
))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ 
[a, 0] && NegQ[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 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a-b x^2\right )^{7/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}} a \,c^{2}+2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a c d \,x^{2}+\left (-b \,x^{2}+a \right )^{\frac {3}{4}} a \,d^{2} x^{4}-\left (-b \,x^{2}+a \right )^{\frac {3}{4}} b \,c^{2} x^{2}-2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b c d \,x^{4}-\left (-b \,x^{2}+a \right )^{\frac {3}{4}} b \,d^{2} x^{6}}d x \] Input:

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

Output:

int(1/((a - b*x**2)**(3/4)*a*c**2 + 2*(a - b*x**2)**(3/4)*a*c*d*x**2 + (a 
- b*x**2)**(3/4)*a*d**2*x**4 - (a - b*x**2)**(3/4)*b*c**2*x**2 - 2*(a - b* 
x**2)**(3/4)*b*c*d*x**4 - (a - b*x**2)**(3/4)*b*d**2*x**6),x)