Integrand size = 22, antiderivative size = 448 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=-\frac {2 (a+b x) \sqrt {a+b x+c x^2}}{3 a d (d x)^{3/2}}+\frac {b \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt {2} a \sqrt {c} d^{5/2} \sqrt {a+x (b+c x)}}-\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt {2} a \sqrt {c} d^{5/2} \sqrt {a+x (b+c x)}} \] Output:
-2/3*(b*x+a)*(c*x^2+b*x+a)^(1/2)/a/d/(d*x)^(3/2)+1/6*b*(-b+(-4*a*c+b^2)^(1 /2))^(1/2)*(b+(-4*a*c+b^2)^(1/2))*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*( 1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1/2)*(d*x)^(1/2 )/(-b+(-4*a*c+b^2)^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c +b^2)^(1/2)))^(1/2))*2^(1/2)/a/c^(1/2)/d^(5/2)/(a+x*(c*x+b))^(1/2)-1/6*(-b +(-4*a*c+b^2)^(1/2))^(1/2)*(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))*(1+2*c*x/(b-(- 4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF( 2^(1/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2)/d^(1/2),((b-(-4* a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)/a/c^(1/2)/d^(5/2)/( a+x*(c*x+b))^(1/2)
Result contains complex when optimal does not.
Time = 12.08 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=\frac {i x \left (4 i a \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))-b \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{5/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{5/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{6 a \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (d x)^{5/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[Sqrt[a + b*x + c*x^2]/(d*x)^(5/2),x]
Output:
((I/6)*x*((4*I)*a*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(a + x*(b + c*x)) - b*(- b + Sqrt[b^2 - 4*a*c])*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(5/2) *Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*E llipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(5/2)*Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*A rcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(a*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(d* x)^(5/2)*Sqrt[a + x*(b + c*x)])
Time = 0.88 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1161, 1237, 27, 1241, 1240, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {\int \frac {b+2 c x}{(d x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {c d (2 a+b x)}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{a d^2}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \int \frac {2 a+b x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{a d}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 1241 |
| \(\displaystyle \frac {\frac {c \sqrt {x} \int \frac {2 a+b x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{a d \sqrt {d x}}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {\frac {2 c \sqrt {x} \int \frac {2 a+b x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{a d \sqrt {d x}}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {2 c \sqrt {x} \left (\sqrt {a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {\sqrt {a} b \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 c \sqrt {x} \left (\sqrt {a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {b \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {2 c \sqrt {x} \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {b \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {2 c \sqrt {x} \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {b \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{a d \sqrt {d x}}-\frac {2 b \sqrt {a+b x+c x^2}}{a d \sqrt {d x}}}{3 d}-\frac {2 \sqrt {a+b x+c x^2}}{3 d (d x)^{3/2}}\) |
Input:
Int[Sqrt[a + b*x + c*x^2]/(d*x)^(5/2),x]
Output:
(-2*Sqrt[a + b*x + c*x^2])/(3*d*(d*x)^(3/2)) + ((-2*b*Sqrt[a + b*x + c*x^2 ])/(a*d*Sqrt[d*x]) + (2*c*Sqrt[x]*(-((b*(-((Sqrt[x]*Sqrt[a + b*x + c*x^2]) /(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c *x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4 )], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqrt[c ]) + (a^(1/4)*(2*Sqrt[a] + b/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^( 1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x + c*x^2])))/( a*d*Sqrt[d*x]))/(3*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ )^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Time = 1.66 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.58
| method | result | size |
| default | \(\frac {2 \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, a c x +4 \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, a b c x -\sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, b^{3} x -\sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, b^{2} x -2 b \,c^{2} x^{3}-2 a \,c^{2} x^{2}-2 b^{2} c \,x^{2}-4 a b c x -2 a^{2} c}{3 x \sqrt {c \,x^{2}+b x +a}\, d^{2} \sqrt {d x}\, a c}\) | \(707\) |
| risch | \(-\frac {2 \sqrt {c \,x^{2}+b x +a}\, \left (b x +a \right )}{3 x a \,d^{2} \sqrt {d x}}+\frac {c \left (\frac {b \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {2 a \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right ) \sqrt {d x \left (c \,x^{2}+b x +a \right )}}{3 a \,d^{2} \sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(760\) |
| elliptic | \(\frac {\sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{3 d^{3} x^{2}}-\frac {2 \left (c d \,x^{2}+b d x +a d \right ) b}{3 d^{3} a \sqrt {x \left (c d \,x^{2}+b d x +a d \right )}}+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{3 d^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {b \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{3 a \,d^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right )}{\sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(789\) |
Input:
int((c*x^2+b*x+a)^(1/2)/(d*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*(2*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+( -4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a *c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/ 2))^(1/2))*(-4*a*c+b^2)^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2 )^(1/2)))^(1/2)*a*c*x+4*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2)) ^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^ (1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2)) /(-4*a*c+b^2)^(1/2))^(1/2))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^ (1/2)))^(1/2)*a*b*c*x-((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^( 1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1 /2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/( -4*a*c+b^2)^(1/2))^(1/2))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1 /2)))^(1/2)*b^3*x-((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2) *(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+ b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a *c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+ (-4*a*c+b^2)^(1/2)))^(1/2)*b^2*x-2*b*c^2*x^3-2*a*c^2*x^2-2*b^2*c*x^2-4*a*b *c*x-2*a^2*c)/x/(c*x^2+b*x+a)^(1/2)/d^2/(d*x)^(1/2)/a/c
Time = 0.08 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c d} b c x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + {\left (b^{2} - 6 \, a c\right )} \sqrt {c d} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (b c x + a c\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{9 \, a c d^{3} x^{2}} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(d*x)^(5/2),x, algorithm="fricas")
Output:
-2/9*(3*sqrt(c*d)*b*c*x^2*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2* b^3 - 9*a*b*c)/c^3, weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^ 3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c)) + (b^2 - 6*a*c)*sqrt(c*d)*x^2*weiers trassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c *x + b)/c) + 3*(b*c*x + a*c)*sqrt(c*x^2 + b*x + a)*sqrt(d*x))/(a*c*d^3*x^2 )
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(1/2)/(d*x)**(5/2),x)
Output:
Integral(sqrt(a + b*x + c*x**2)/(d*x)**(5/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(d*x)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + b*x + a)/(d*x)^(5/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(d*x)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + b*x + a)/(d*x)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*x + c*x^2)^(1/2)/(d*x)^(5/2),x)
Output:
int((a + b*x + c*x^2)^(1/2)/(d*x)^(5/2), x)
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d x)^{5/2}} \, dx=\frac {\sqrt {d}\, \left (-2 \sqrt {c \,x^{2}+b x +a}-\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{5}+b \,x^{4}+a \,x^{3}}d x \right ) a x +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) c x \right )}{2 \sqrt {x}\, d^{3} x} \] Input:
int((c*x^2+b*x+a)^(1/2)/(d*x)^(5/2),x)
Output:
(sqrt(d)*( - 2*sqrt(a + b*x + c*x**2) - sqrt(x)*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x**3 + b*x**4 + c*x**5),x)*a*x + sqrt(x)*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)*c*x))/(2*sqrt(x)*d**3*x)