Integrand size = 28, antiderivative size = 193 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}+\frac {\left (21 b^2 c^2-14 a b c d+5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}} \]
-2/7*a^2*(d*x^2+c)^(1/2)/c/e/(e*x)^(7/2)-2/21*a*(-5*a*d+14*b*c)*(d*x^2+c)^ (1/2)/c^2/e^3/(e*x)^(3/2)+1/21*(5*a^2*d^2-14*a*b*c*d+21*b^2*c^2)*(cos(2*ar ctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*( e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^ (1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^( 1/2))^2)^(1/2)/c^(9/4)/d^(1/4)/e^(9/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.15 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {x^{9/2} \left (\frac {2 a \left (c+d x^2\right ) \left (-3 a c-14 b c x^2+5 a d x^2\right )}{c^2 x^{7/2}}+\frac {2 i \left (21 b^2 c^2-14 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{c^2 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 (e x)^{9/2} \sqrt {c+d x^2}} \]
(x^(9/2)*((2*a*(c + d*x^2)*(-3*a*c - 14*b*c*x^2 + 5*a*d*x^2))/(c^2*x^(7/2) ) + ((2*I)*(21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Ell ipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*Sqr t[c])/Sqrt[d]])))/(21*(e*x)^(9/2)*Sqrt[c + d*x^2])
Time = 0.32 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {365, 27, 359, 266, 761}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {7 b^2 c x^2+a (14 b c-5 a d)}{2 (e x)^{5/2} \sqrt {d x^2+c}}dx}{7 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 b^2 c x^2+a (14 b c-5 a d)}{(e x)^{5/2} \sqrt {d x^2+c}}dx}{7 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {\left (21 b^2 c^2-a d (14 b c-5 a d)\right ) \int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx}{3 c e^2}-\frac {2 a \sqrt {c+d x^2} (14 b c-5 a d)}{3 c e (e x)^{3/2}}}{7 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 \left (21 b^2 c^2-a d (14 b c-5 a d)\right ) \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{3 c e^3}-\frac {2 a \sqrt {c+d x^2} (14 b c-5 a d)}{3 c e (e x)^{3/2}}}{7 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (21 b^2 c^2-a d (14 b c-5 a d)\right ) \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 c^{5/4} \sqrt [4]{d} e^{7/2} \sqrt {c+d x^2}}-\frac {2 a \sqrt {c+d x^2} (14 b c-5 a d)}{3 c e (e x)^{3/2}}}{7 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}\) |
(-2*a^2*Sqrt[c + d*x^2])/(7*c*e*(e*x)^(7/2)) + ((-2*a*(14*b*c - 5*a*d)*Sqr t[c + d*x^2])/(3*c*e*(e*x)^(3/2)) + ((21*b^2*c^2 - a*d*(14*b*c - 5*a*d))*( Sqrt[c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sqrt[c]*e + Sqrt[d]*e*x )^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(3*c ^(5/4)*d^(1/4)*e^(7/2)*Sqrt[c + d*x^2]))/(7*c*e^2)
3.9.46.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Time = 3.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, a \left (-5 a d \,x^{2}+14 c b \,x^{2}+3 a c \right )}{21 c^{2} x^{3} e^{4} \sqrt {e x}}+\frac {\left (5 a^{2} d^{2}-14 a b c d +21 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{21 c^{2} d \sqrt {d e \,x^{3}+c e x}\, e^{4} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(212\) |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{7 e^{5} c \,x^{4}}+\frac {2 a \left (5 a d -14 b c \right ) \sqrt {d e \,x^{3}+c e x}}{21 e^{5} c^{2} x^{2}}+\frac {\left (\frac {b^{2}}{e^{4}}+\frac {d a \left (5 a d -14 b c \right )}{21 c^{2} e^{4}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(226\) |
| default | \(\frac {5 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} x^{3}-14 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d \,x^{3}+21 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x^{3}+10 a^{2} d^{3} x^{4}-28 a b c \,d^{2} x^{4}+4 a^{2} c \,d^{2} x^{2}-28 a b \,c^{2} d \,x^{2}-6 a^{2} c^{2} d}{21 \sqrt {d \,x^{2}+c}\, x^{3} d \,c^{2} e^{4} \sqrt {e x}}\) | \(370\) |
-2/21*(d*x^2+c)^(1/2)*a*(-5*a*d*x^2+14*b*c*x^2+3*a*c)/c^2/x^3/e^4/(e*x)^(1 /2)+1/21*(5*a^2*d^2-14*a*b*c*d+21*b^2*c^2)/c^2*(-c*d)^(1/2)/d*((x+(-c*d)^( 1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2) *(-x/(-c*d)^(1/2)*d)^(1/2)/(d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+(-c*d)^(1/2 )/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2))/e^4*(e*x*(d*x^2+c))^(1/2)/(e*x)^(1 /2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left ({\left (21 \, b^{2} c^{2} - 14 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {d e} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (3 \, a^{2} c d + {\left (14 \, a b c d - 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{21 \, c^{2} d e^{5} x^{4}} \]
2/21*((21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*sqrt(d*e)*x^4*weierstrassPInve rse(-4*c/d, 0, x) - (3*a^2*c*d + (14*a*b*c*d - 5*a^2*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(c^2*d*e^5*x^4)
Result contains complex when optimal does not.
Time = 70.80 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {a^{2} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {b^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {9}{2}} \Gamma \left (\frac {5}{4}\right )} \]
a**2*gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), d*x**2*exp_polar(I*pi)/c)/(2* sqrt(c)*e**(9/2)*x**(7/2)*gamma(-3/4)) + a*b*gamma(-3/4)*hyper((-3/4, 1/2) , (1/4,), d*x**2*exp_polar(I*pi)/c)/(sqrt(c)*e**(9/2)*x**(3/2)*gamma(1/4)) + b**2*sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), d*x**2*exp_polar(I*pi )/c)/(2*sqrt(c)*e**(9/2)*gamma(5/4))
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{9/2}\,\sqrt {d\,x^2+c}} \,d x \]