Integrand size = 28, antiderivative size = 237 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\frac {4 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{c \sqrt [4]{b^2-4 a c} d^{3/2} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{c \sqrt [4]{b^2-4 a c} d^{3/2} \sqrt {a+b x+c x^2}} \] Output:
4*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(1/2)-2*(-c*(c*x^2+b*x+ a)/(-4*a*c+b^2))^(1/2)*EllipticE((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^ (1/2),I)/c/(-4*a*c+b^2)^(1/4)/d^(3/2)/(c*x^2+b*x+a)^(1/2)+2*(-c*(c*x^2+b*x +a)/(-4*a*c+b^2))^(1/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d ^(1/2),I)/c/(-4*a*c+b^2)^(1/4)/d^(3/2)/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.38 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c d \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)}} \] Input:
Integrate[1/((b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
Output:
(-2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-1/4, 1/2 , 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*( b + c*x)])
Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1117, 1115, 1114, 836, 27, 762, 1389, 327}
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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1117 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\int \frac {\sqrt {b d+2 c x d}}{\sqrt {c x^2+b x+a}}dx}{d^2 \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 1115 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\sqrt {b d+2 c x d}}{\sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{d^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1114 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {b d+2 c x d}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{d \sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {\sqrt {\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}+1}}{\sqrt {1-\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d^{3/2} \left (b^2-4 a c\right )^{3/4} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[1/((b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
Output:
(4*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]) - (2*Sqrt[ -((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*((b^2 - 4*a*c)^(3/4)*d^(3/2)*Ellip ticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1] - (b^2 - 4*a*c)^(3/4)*d^(3/2)*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c )^(1/4)*Sqrt[d])], -1]))/(c*(b^2 - 4*a*c)*d^3*Sqrt[a + b*x + c*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symb ol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[x^2/Sqrt[Simp[1 - b^2* (x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym bol] :> Simp[Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/Sqrt[a + b*x + c* x^2] Int[(d + e*x)^m/Sqrt[(-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4*a*c))], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c* d - b*e, 0] && EqQ[m^2, 1/4]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Time = 3.37 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, \left (4 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a c -\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{2}-4 c^{2} x^{2}-4 c b x -4 a c \right )}{c \,d^{2} \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right ) \left (4 a c -b^{2}\right )}\) | \(336\) |
elliptic | \(\text {Expression too large to display}\) | \(1038\) |
Input:
int(1/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(4*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(- 4*a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+ (-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*2^(1/2)*(1/( -4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2),2^(1/2))*a*c-(1/(-4* a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2 )^(1/2))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*El lipticE(1/2*2^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1 /2),2^(1/2))*b^2-4*c^2*x^2-4*c*b*x-4*a*c)/c/d^2/(2*c^2*x^3+3*b*c*x^2+2*a*c *x+b^2*x+a*b)/(4*a*c-b^2)
Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left (\sqrt {2} \sqrt {c^{2} d} {\left (2 \, c x + b\right )} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + 2 \, \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} c\right )}}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} x + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2}} \] Input:
integrate(1/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
2*(sqrt(2)*sqrt(c^2*d)*(2*c*x + b)*weierstrassZeta((b^2 - 4*a*c)/c^2, 0, w eierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) + 2*sqrt(2*c* d*x + b*d)*sqrt(c*x^2 + b*x + a)*c)/(2*(b^2*c^2 - 4*a*c^3)*d^2*x + (b^3*c - 4*a*b*c^2)*d^2)
\[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(1/(2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral(1/((d*(b + 2*c*x))**(3/2)*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} \sqrt {c x^{2} + b x + a}} \,d x } \] Input:
integrate(1/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((2*c*d*x + b*d)^(3/2)*sqrt(c*x^2 + b*x + a)), x)
\[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} \sqrt {c x^{2} + b x + a}} \,d x } \] Input:
integrate(1/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/((2*c*d*x + b*d)^(3/2)*sqrt(c*x^2 + b*x + a)), x)
Timed out. \[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{4 c^{3} x^{4}+8 b \,c^{2} x^{3}+4 a \,c^{2} x^{2}+5 b^{2} c \,x^{2}+4 a b c x +b^{3} x +a \,b^{2}}d x \right )}{d^{2}} \] Input:
int(1/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
(sqrt(d)*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2))/(a*b**2 + 4*a*b*c*x + 4*a*c**2*x**2 + b**3*x + 5*b**2*c*x**2 + 8*b*c**2*x**3 + 4*c**3*x**4),x) )/d**2