Integrand size = 21, antiderivative size = 254 \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\frac {2 b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {2 \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {a} (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} d \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{(b c-a d)^2 x}-\frac {\sqrt [4]{a} d \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{(b c-a d)^2 x} \]
2/3*b*x/a/(-a*d+b*c)/(b*x^2+a)^(3/4)+2/3*(1+b*x^2/a)^(3/4)*(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)))*EllipticF (sin(1/2*arctan(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2)/(-a*d+b*c)/(b*x^2+a)^ (3/4)/a^(1/2)-a^(1/4)*d*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)^2/x-a^(1/4)*d*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)^2/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.28 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\frac {x \left (-\frac {b d x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (3 a c \left (-3 b c+3 a d-2 b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (c+d x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{9 a (-b c+a d) \left (a+b x^2\right )^{3/4}} \]
(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))
Time = 0.38 (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 |
\(\displaystyle \frac {2 d \sqrt {-\frac {b x^2}{a}} \left (-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 (b c-a d)}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 (b c-a d)}\right )}{x (b c-a d)}+\frac {b \left (\frac {2 \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {a} \sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {2 x}{3 a \left (a+b x^2\right )^{3/4}}\right )}{b c-a d}\) |
(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)
3.4.28.3.1 Defintions of rubi rules used
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]
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])
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]
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]
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] )
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]
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]
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]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (d \,x^{2}+c \right )}d x\]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {7}{4}} \left (c + d x^{2}\right )}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{7/4}\,\left (d\,x^2+c\right )} \,d x \]