Integrand size = 21, antiderivative size = 274 \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\frac {2 b x}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}+\frac {2 \sqrt {b} (3 b c-8 a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{3/2} (b c-a d)^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} d^{3/2} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{(-b c+a d)^{5/2} x}-\frac {\sqrt [4]{a} d^{3/2} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{(-b c+a d)^{5/2} x} \]
2/5*b*x/a/(-a*d+b*c)/(b*x^2+a)^(5/4)+2/5*(-8*a*d+3*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)/a^(3/2 )/(-a*d+b*c)^2/(b*x^2+a)^(1/4)+a^(1/4)*d^(3/2)*EllipticPi((b*x^2+a)^(1/4)/ a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*(-b*x^2/a)^(1/2)/(a*d-b*c)^(5/ 2)/x-a^(1/4)*d^(3/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a *d-b*c)^(1/2),I)*(-b*x^2/a)^(1/2)/(a*d-b*c)^(5/2)/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.70 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\frac {x \left (\frac {b d (-3 b c+8 a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}-\frac {6 \left (3 a c \left (5 a^3 d^2+3 b^3 c x^2 \left (c+2 d x^2\right )-a^2 b d \left (10 c+13 d x^2\right )+a b^2 \left (5 c^2-16 d^2 x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (c+d x^2\right ) \left (9 a^2 d-3 b^2 c x^2-4 a b \left (c-2 d x^2\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{15 a^2 (b c-a d)^2 \sqrt [4]{a+b x^2}} \]
(x*((b*d*(-3*b*c + 8*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 - (6*(3*a*c*(5*a^3*d^2 + 3*b^3*c*x^2*( c + 2*d*x^2) - a^2*b*d*(10*c + 13*d*x^2) + a*b^2*(5*c^2 - 16*d^2*x^4))*App ellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + b*x^2*(c + d*x^2)*(9 *a^2*d - 3*b^2*c*x^2 - 4*a*b*(c - 2*d*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)])))/((a + b*x^2)*(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)])))))/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(1/4))
Time = 0.55 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.28, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}-\frac {2 \int -\frac {3 b d x^2+3 b c-5 a d}{2 \left (b x^2+a\right )^{5/4} \left (d x^2+c\right )}dx}{5 a (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 b d x^2+3 b c-5 a d}{\left (b x^2+a\right )^{5/4} \left (d x^2+c\right )}dx}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {2 \int \frac {3 b^2 c^2-8 a b d c-5 a^2 d^2+b d (3 b c-8 a d) x^2}{2 \sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\int \frac {3 b^2 c^2-8 a b d c-5 a^2 d^2+b d (3 b c-8 a d) x^2}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {b (3 b c-8 a d) \int \frac {1}{\sqrt [4]{b x^2+a}}dx-5 a^2 d^2 \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 227 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (3 b c-8 a d) \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{\sqrt [4]{a+b x^2}}-5 a^2 d^2 \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 225 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (3 b c-8 a d) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{\sqrt [4]{a+b x^2}}-5 a^2 d^2 \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (3 b c-8 a d) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}-5 a^2 d^2 \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 310 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (3 b c-8 a d) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}-\frac {10 a^2 d^2 \sqrt {-\frac {b x^2}{a}} \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c-a d+d \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (3 b c-8 a d) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}-\frac {10 a^2 d^2 \sqrt {-\frac {b x^2}{a}} \left (\frac {\int \frac {1}{\left (\sqrt {a d-b c}+\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {a d-b c}-\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}\right )}{x}}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {2 b x (3 b c-8 a d)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (3 b c-8 a d) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}-\frac {10 a^2 d^2 \sqrt {-\frac {b x^2}{a}} \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}\right )}{x}}{a (b c-a d)}}{5 a (b c-a d)}+\frac {2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\) |
(2*b*x)/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)) + ((2*b*(3*b*c - 8*a*d)*x)/(a* (b*c - a*d)*(a + b*x^2)^(1/4)) - ((b*(3*b*c - 8*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)/Sq rt[a]]/2, 2])/Sqrt[b]))/(a + b*x^2)^(1/4) - (10*a^2*d^2*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)*Ell ipticPi[(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)))/(5*a*(b*c - a*d))
3.4.29.3.1 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]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {9}{4}} \left (d \,x^{2}+c \right )}d x\]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {9}{4}} \left (c + d x^{2}\right )}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{9/4}\,\left (d\,x^2+c\right )} \,d x \]