Integrand size = 21, antiderivative size = 314 \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=-\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac {\sqrt {b} (4 b c+a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} \sqrt {d} (7 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (-b c+a d)^{5/2} x}+\frac {\sqrt [4]{a} \sqrt {d} (7 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (-b c+a d)^{5/2} x} \]
-1/2*d*x/c/(-a*d+b*c)/(b*x^2+a)^(1/4)/(d*x^2+c)+1/2*(a*d+4*b*c)*(1+b*x^2/a )^(1/4)*(cos(1/2*arctan(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x*b^(1 /2)/a^(1/2)))*EllipticE(sin(1/2*arctan(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2 )/c/(-a*d+b*c)^2/(b*x^2+a)^(1/4)/a^(1/2)-1/4*a^(1/4)*(-2*a*d+7*b*c)*Ellipt icPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*d^(1/2)*( -b*x^2/a)^(1/2)/c/(a*d-b*c)^(5/2)/x+1/4*a^(1/4)*(-2*a*d+7*b*c)*EllipticPi( (b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*d^(1/2)*(-b*x^2 /a)^(1/2)/c/(a*d-b*c)^(5/2)/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.33 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\frac {x \left (-b d (4 b c+a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {c \left (36 a c \left (2 a^2 d^2+a b d \left (-4 c+d x^2\right )+2 b^2 c \left (c+2 d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 x^2 \left (a^2 d^2+a b d^2 x^2+4 b^2 c \left (c+d x^2\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 a c^2 (b c-a d)^2 \sqrt [4]{a+b x^2}} \]
(x*(-(b*d*(4*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/ 2, -((b*x^2)/a), -((d*x^2)/c)]) + (c*(36*a*c*(2*a^2*d^2 + a*b*d*(-4*c + d* x^2) + 2*b^2*c*(c + 2*d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -(( d*x^2)/c)] - 6*x^2*(a^2*d^2 + a*b*d^2*x^2 + 4*b^2*c*(c + d*x^2))*(4*a*d*Ap pellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5 /4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(6*a*c*AppellF1[1/ 2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4 , 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -(( b*x^2)/a), -((d*x^2)/c)])))))/(12*a*c^2*(b*c - a*d)^2*(a + b*x^2)^(1/4))
Time = 0.54 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {316, 27, 402, 27, 405, 227, 225, 212, 310, 993, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {-3 b d x^2+4 b c-2 a d}{2 \left (b x^2+a\right )^{5/4} \left (d x^2+c\right )}dx}{2 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 (2 b c-a d)-3 b d x^2}{\left (b x^2+a\right )^{5/4} \left (d x^2+c\right )}dx}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {2 \int \frac {4 b^2 c^2+8 a b d c-2 a^2 d^2+b d (4 b c+a d) x^2}{2 \sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\int \frac {b d (4 b c+a d) x^2+2 \left (2 b^2 c^2+4 a b d c-a^2 d^2\right )}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {b (a d+4 b c) \int \frac {1}{\sqrt [4]{b x^2+a}}dx+a d (7 b c-2 a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 227 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{\sqrt [4]{a+b x^2}}+a d (7 b c-2 a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 225 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{\sqrt [4]{a+b x^2}}+a d (7 b c-2 a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {a d (7 b c-2 a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 310 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {2 a d \sqrt {-\frac {b x^2}{a}} (7 b c-2 a d) \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c-a d+d \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {2 a d \sqrt {-\frac {b x^2}{a}} (7 b c-2 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a d-b c}+\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {a d-b c}-\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}\right )}{x}+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {2 b x (a d+4 b c)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {2 a d \sqrt {-\frac {b x^2}{a}} (7 b c-2 a d) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}\right )}{x}+\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}\) |
-1/2*(d*x)/(c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)) + ((2*b*(4*b*c + a*d)*x)/(a*(b*c - a*d)*(a + b*x^2)^(1/4)) - ((b*(4*b*c + a*d)*(1 + (b*x^2) /a)^(1/4)*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt [b]*x)/Sqrt[a]]/2, 2])/Sqrt[b]))/(a + b*x^2)^(1/4) + (2*a*d*(7*b*c - 2*a*d )*Sqrt[-((b*x^2)/a)]*((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*Sqrt[d]*Sqrt[-(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*Sqrt[d]*Sqrt[-(b*c) + a*d])))/x)/(a*(b* c - a*d)))/(4*c*(b*c - a*d))
3.4.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
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]
Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim p[2*(Sqrt[(-b)*(x^2/a)]/x) Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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]
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]
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]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (d \,x^{2}+c \right )^{2}}d x\]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{4}} \left (c + d x^{2}\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]