Integrand size = 28, antiderivative size = 278 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}-\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}+\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}} \] Output:
-2/3*(2*c*d*x+b*d)^(3/2)/(-4*a*c+b^2)/d/(c*x^2+b*x+a)^(3/2)+4*c*(2*c*d*x+b *d)^(3/2)/(-4*a*c+b^2)^2/d/(c*x^2+b*x+a)^(1/2)-8*c*d^(1/2)*(-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)/(-4*a*c+b^2)^(5/4)/(c*x^2+b*x+a)^(1/2)+8*c*d^(1/2)*(-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)/(-4*a*c+b^2)^(5/4)/(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 = 99, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {32 c (d (b+2 c x))^{3/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+x (b+c x)}} \] Input:
Integrate[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^(5/2),x]
Output:
(32*c*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hyp ergeometric2F1[3/4, 5/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a* c)^2*d*Sqrt[a + x*(b + c*x)])
Time = 0.56 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1111, 1111, 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.
| \(\displaystyle \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {2 c \int \frac {\sqrt {b d+2 c x d}}{\left (c x^2+b x+a\right )^{3/2}}dx}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {2 c \left (\frac {2 c \int \frac {\sqrt {b d+2 c x d}}{\sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle -\frac {2 c \left (\frac {2 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\sqrt {b d+2 c x d}}{\sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1114 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {b d+2 c x d}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{d \sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {\sqrt {\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}+1}}{\sqrt {1-\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 c \left (\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d^{3/2} \left (b^2-4 a c\right )^{3/4} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[Sqrt[b*d + 2*c*d*x]/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(b*d + 2*c*d*x)^(3/2))/(3*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^(3/2)) - ( 2*c*((-2*(b*d + 2*c*d*x)^(3/2))/((b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*((b^2 - 4*a*c)^(3/4)*d^(3/ 2)*EllipticE[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]))/((b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]))) /(b^2 - 4*a*c)
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[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] - Simp[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e , m}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] && !G tQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
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_)^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]
Leaf count of result is larger than twice the leaf count of optimal. \(865\) vs. \(2(240)=480\).
Time = 1.73 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.12
| method | result | size |
| default | \(-\frac {2 \left (24 \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^{3} x^{2}-6 \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} c^{2} x^{2}+24 \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 b \,c^{2} x -6 \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^{3} c x +24 \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^{2} c^{2}-6 \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 \,b^{2} c -24 c^{4} x^{4}-48 b \,c^{3} x^{3}-40 a \,c^{3} x^{2}-26 b^{2} c^{2} x^{2}-40 a b \,c^{2} x -2 b^{3} c x -10 c a \,b^{2}+b^{4}\right ) \sqrt {d \left (2 c x +b \right )}}{3 \left (4 a c -b^{2}\right )^{2} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(866\) |
| elliptic | \(\text {Expression too large to display}\) | \(1144\) |
Input:
int((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(24*(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^3*x^2-6*(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))*b^2*c^2*x^2+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*b*c^2*x-6*(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))*b^3*c*x+24*(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)*Elli pticE(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^2*c^2-6*(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)*(...
Time = 0.10 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (12 \, \sqrt {2} {\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c + {\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} - b^{3} + 10 \, a b c + 4 \, {\left (b^{2} c + 5 \, a c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \] Input:
integrate((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*(12*sqrt(2)*(c^3*x^4 + 2*b*c^2*x^3 + 2*a*b*c*x + a^2*c + (b^2*c + 2*a* c^2)*x^2)*sqrt(c^2*d)*weierstrassZeta((b^2 - 4*a*c)/c^2, 0, weierstrassPIn verse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) + (12*c^3*x^3 + 18*b*c^2*x ^2 - b^3 + 10*a*b*c + 4*(b^2*c + 5*a*c^2)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^ 2 + b*x + a))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6 *a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)
\[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d \left (b + 2 c x\right )}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((2*c*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**(5/2),x)
Output:
Integral(sqrt(d*(b + 2*c*x))/(a + b*x + c*x**2)**(5/2), x)
\[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(2*c*d*x + b*d)/(c*x^2 + b*x + a)^(5/2), x)
\[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(2*c*d*x + b*d)/(c*x^2 + b*x + a)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b\,d+2\,c\,d\,x}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:
int((b*d + 2*c*d*x)^(1/2)/(a + b*x + c*x^2)^(5/2),x)
Output:
int((b*d + 2*c*d*x)^(1/2)/(a + b*x + c*x^2)^(5/2), x)
\[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\sqrt {d}\, \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) \] Input:
int((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(5/2),x)
Output:
sqrt(d)*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2))/(a**3 + 3*a**2*b*x + 3*a**2*c*x**2 + 3*a*b**2*x**2 + 6*a*b*c*x**3 + 3*a*c**2*x**4 + b**3*x**3 + 3*b**2*c*x**4 + 3*b*c**2*x**5 + c**3*x**6),x)