Integrand size = 28, antiderivative size = 244 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {2 \left (5 b^2 c^2-22 a b c d+77 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d^2 e}-\frac {2 b (5 b c-22 a d) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac {2 c^{3/4} \left (5 b^2 c^2-22 a b c d+77 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]
2/11*b^2*(e*x)^(5/2)*(d*x^2+c)^(3/2)/d/e^3-2/77*b*(-22*a*d+5*b*c)*(d*x^2+c )^(3/2)*(e*x)^(1/2)/d^2/e+2/231*(77*a^2*d^2-22*a*b*c*d+5*b^2*c^2)*(e*x)^(1 /2)*(d*x^2+c)^(1/2)/d^2/e+2/231*c^(3/4)*(77*a^2*d^2-22*a*b*c*d+5*b^2*c^2)* (cos(2*arctan(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)/d^(9/4)/e^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {\sqrt {x} \left (\frac {2 \sqrt {x} \left (c+d x^2\right ) \left (77 a^2 d^2+22 a b d \left (2 c+3 d x^2\right )+b^2 \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )\right )}{d^2}+\frac {4 i c \left (5 b^2 c^2-22 a b c d+77 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^2}\right )}{231 \sqrt {e x} \sqrt {c+d x^2}} \]
(Sqrt[x]*((2*Sqrt[x]*(c + d*x^2)*(77*a^2*d^2 + 22*a*b*d*(2*c + 3*d*x^2) + b^2*(-10*c^2 + 6*c*d*x^2 + 21*d^2*x^4)))/d^2 + ((4*I)*c*(5*b^2*c^2 - 22*a* b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt [c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^2)))/(231*Sqrt[e *x]*Sqrt[c + d*x^2])
Time = 0.33 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {367, 27, 363, 248, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx\) |
| \(\Big \downarrow \) 367 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {d x^2+c} \left (11 a^2 d-b (5 b c-22 a d) x^2\right )}{2 \sqrt {e x}}dx}{11 d}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d x^2+c} \left (11 a^2 d-b (5 b c-22 a d) x^2\right )}{\sqrt {e x}}dx}{11 d}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {\left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {e x}}dx}{7 d}-\frac {2 b \sqrt {e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{7 d e}}{11 d}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {\left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \left (\frac {2}{3} c \int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )}{7 d}-\frac {2 b \sqrt {e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{7 d e}}{11 d}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \left (\frac {4 c \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{3 e}+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )}{7 d}-\frac {2 b \sqrt {e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{7 d e}}{11 d}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \left (\frac {2 c^{3/4} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{d} e^{3/2} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )}{7 d}-\frac {2 b \sqrt {e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{7 d e}}{11 d}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}\) |
(2*b^2*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(11*d*e^3) + ((-2*b*(5*b*c - 22*a*d) *Sqrt[e*x]*(c + d*x^2)^(3/2))/(7*d*e) + ((5*b^2*c^2 - 22*a*b*c*d + 77*a^2* d^2)*((2*Sqrt[e*x]*Sqrt[c + d*x^2])/(3*e) + (2*c^(3/4)*(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*A rcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(3*d^(1/4)*e^(3/2)*Sqr t[c + d*x^2])))/(7*d))/(11*d)
3.9.24.3.1 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] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, 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[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[d^2*(e*x)^(m + 3)*((a + b*x^2)^(p + 1)/(b*e^3*(m + 2*p + 5))), x] + Simp[1/(b*(m + 2*p + 5)) Int[(e*x)^m*(a + b*x^2)^p*Simp[b*c^2* (m + 2*p + 5) - d*(a*d*(m + 3) - 2*b*c*(m + 2*p + 5))*x^2, x], x], x] /; Fr eeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + 2*p + 5, 0]
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 = 3.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {2 \left (21 b^{2} d^{2} x^{4}+66 x^{2} a b \,d^{2}+6 x^{2} b^{2} c d +77 a^{2} d^{2}+44 a b c d -10 b^{2} c^{2}\right ) x \sqrt {d \,x^{2}+c}}{231 d^{2} \sqrt {e x}}+\frac {2 c \left (77 a^{2} d^{2}-22 a b c d +5 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(236\) |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {2 b^{2} x^{4} \sqrt {d e \,x^{3}+c e x}}{11 e}+\frac {2 \left (2 a b d +\frac {2}{11} b^{2} c \right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (a^{2} d +2 a b c -\frac {5 c \left (2 a b d +\frac {2}{11} b^{2} c \right )}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (a^{2} c -\frac {c \left (a^{2} d +2 a b c -\frac {5 c \left (2 a b d +\frac {2}{11} b^{2} c \right )}{7 d}\right )}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(289\) |
| default | \(\frac {\frac {2 b^{2} d^{4} x^{7}}{11}+\frac {2 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}}{3}-\frac {4 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d}{21}+\frac {10 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}}{231}+\frac {4 a b \,d^{4} x^{5}}{7}+\frac {18 b^{2} c \,d^{3} x^{5}}{77}+\frac {2 a^{2} d^{4} x^{3}}{3}+\frac {20 x^{3} d^{3} b a c}{21}-\frac {8 b^{2} c^{2} d^{2} x^{3}}{231}+\frac {2 a^{2} c \,d^{3} x}{3}+\frac {8 a b \,c^{2} d^{2} x}{21}-\frac {20 b^{2} d x \,c^{3}}{231}}{\sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{3}}\) | \(401\) |
2/231*(21*b^2*d^2*x^4+66*a*b*d^2*x^2+6*b^2*c*d*x^2+77*a^2*d^2+44*a*b*c*d-1 0*b^2*c^2)*x*(d*x^2+c)^(1/2)/d^2/(e*x)^(1/2)+2/231*c*(77*a^2*d^2-22*a*b*c* d+5*b^2*c^2)/d^3*(-c*d)^(1/2)*((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*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 = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {2 \, {\left (2 \, {\left (5 \, b^{2} c^{3} - 22 \, a b c^{2} d + 77 \, a^{2} c d^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (21 \, b^{2} d^{3} x^{4} - 10 \, b^{2} c^{2} d + 44 \, a b c d^{2} + 77 \, a^{2} d^{3} + 6 \, {\left (b^{2} c d^{2} + 11 \, a b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{231 \, d^{3} e} \]
2/231*(2*(5*b^2*c^3 - 22*a*b*c^2*d + 77*a^2*c*d^2)*sqrt(d*e)*weierstrassPI nverse(-4*c/d, 0, x) + (21*b^2*d^3*x^4 - 10*b^2*c^2*d + 44*a*b*c*d^2 + 77* a^2*d^3 + 6*(b^2*c*d^2 + 11*a*b*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^3* e)
Result contains complex when optimal does not.
Time = 3.77 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {a^{2} \sqrt {c} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {a b \sqrt {c} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} \sqrt {c} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {13}{4}\right )} \]
a**2*sqrt(c)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), d*x**2*exp_pola r(I*pi)/c)/(2*sqrt(e)*gamma(5/4)) + a*b*sqrt(c)*x**(5/2)*gamma(5/4)*hyper( (-1/2, 5/4), (9/4,), d*x**2*exp_polar(I*pi)/c)/(sqrt(e)*gamma(9/4)) + b**2 *sqrt(c)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), d*x**2*exp_polar( I*pi)/c)/(2*sqrt(e)*gamma(13/4))
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\sqrt {e x}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\sqrt {e x}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{\sqrt {e\,x}} \,d x \]