Integrand size = 28, antiderivative size = 325 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}-\frac {56 c \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {168 c \sqrt {a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}+\frac {84 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \left (b^2-4 a c\right )^{9/4} d^{7/2} \sqrt {a+b x+c x^2}}-\frac {84 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{5 \left (b^2-4 a c\right )^{9/4} d^{7/2} \sqrt {a+b x+c x^2}} \]
-2/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(1/2)-56/5*c*(c*x^2+b* x+a)^(1/2)/(-4*a*c+b^2)^2/d/(2*c*d*x+b*d)^(5/2)-168/5*c*(c*x^2+b*x+a)^(1/2 )/(-4*a*c+b^2)^3/d^3/(2*c*d*x+b*d)^(1/2)+84/5*EllipticE((2*c*d*x+b*d)^(1/2 )/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/(-4* a*c+b^2)^(9/4)/d^(7/2)/(c*x^2+b*x+a)^(1/2)-84/5*EllipticF((2*c*d*x+b*d)^(1 /2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/(- 4*a*c+b^2)^(9/4)/d^(7/2)/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.30 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {8 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{5 \left (b^2-4 a c\right ) d (d (b+2 c x))^{5/2} \sqrt {a+x (b+c x)}} \]
(8*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-5/4, 3/2, -1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c)*d*(d*(b + 2*c*x))^(5 /2)*Sqrt[a + x*(b + c*x)])
Time = 0.65 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1111, 1117, 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.
\(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 1111 |
\(\displaystyle -\frac {14 c \int \frac {1}{(b d+2 c x d)^{7/2} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 1117 |
\(\displaystyle -\frac {14 c \left (\frac {3 \int \frac {1}{(b d+2 c x d)^{3/2} \sqrt {c x^2+b x+a}}dx}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 1117 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\int \frac {\sqrt {b d+2 c x d}}{\sqrt {c x^2+b x+a}}dx}{d^2 \left (b^2-4 a c\right )}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 1115 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\sqrt {b d+2 c x d}}{\sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{d^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 1114 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {b d+2 c x d}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{d \sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {\sqrt {\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}+1}}{\sqrt {1-\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {14 c \left (\frac {3 \left (\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d^{3/2} \left (b^2-4 a c\right )^{3/4} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )}{5 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}\) |
-2/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2]) - (14*c*( (4*Sqrt[a + b*x + c*x^2])/(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + (3*( (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])))/(5* (b^2 - 4*a*c)*d^2)))/(b^2 - 4*a*c)
3.14.88.3.1 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[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_))^(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]
Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(277)=554\).
Time = 4.69 (sec) , antiderivative size = 876, normalized size of antiderivative = 2.70
method | result | size |
default | \(-\frac {2 \left (336 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,c^{3} x^{2}-84 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2} c^{2} x^{2}+336 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a b \,c^{2} x -84 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{3} c x +84 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{2} c -21 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{4}-336 c^{4} x^{4}-672 b \,c^{3} x^{3}-224 x^{2} c^{3} a -448 b^{2} c^{2} x^{2}-224 a b \,c^{2} x -112 b^{3} c x +32 a^{2} c^{2}-72 a \,b^{2} c -5 b^{4}\right ) \sqrt {d \left (2 c x +b \right )}}{5 d^{4} \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right )^{3} \left (4 a c -b^{2}\right )^{3}}\) | \(876\) |
elliptic | \(\text {Expression too large to display}\) | \(1188\) |
-2/5*(336*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x +b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^ (1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2 ))^(1/2)*2^(1/2),2^(1/2))*a*c^3*x^2-84*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a *c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4* a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+ b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^2*c^2*x^2+336*((b +2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b ^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)* EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1 /2),2^(1/2))*a*b*c^2*x-84*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2) )^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2 ))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/( -4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^3*c*x+84*((b+2*c*x+(-4*a*c+b^2 )^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*( (-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+ 2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a*b^2 *c-21*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/ (-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2 ))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (42 \, \sqrt {2} {\left (8 \, c^{4} x^{5} + 20 \, b c^{3} x^{4} + a b^{3} + 2 \, {\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{3} + {\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} x^{2} + {\left (b^{4} + 6 \, a b^{2} c\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (336 \, c^{4} x^{4} + 672 \, b c^{3} x^{3} + 5 \, b^{4} + 72 \, a b^{2} c - 32 \, a^{2} c^{2} + 224 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 112 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{5 \, {\left (8 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \, {\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} + {\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} + {\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x + {\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\right )} d^{4}\right )}} \]
-2/5*(42*sqrt(2)*(8*c^4*x^5 + 20*b*c^3*x^4 + a*b^3 + 2*(9*b^2*c^2 + 4*a*c^ 3)*x^3 + (7*b^3*c + 12*a*b*c^2)*x^2 + (b^4 + 6*a*b^2*c)*x)*sqrt(c^2*d)*wei erstrassZeta((b^2 - 4*a*c)/c^2, 0, weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) + (336*c^4*x^4 + 672*b*c^3*x^3 + 5*b^4 + 72*a*b^2*c - 32*a^2*c^2 + 224*(2*b^2*c^2 + a*c^3)*x^2 + 112*(b^3*c + 2*a*b*c^2)*x)*s qrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(8*(b^6*c^4 - 12*a*b^4*c^5 + 48* a^2*b^2*c^6 - 64*a^3*c^7)*d^4*x^5 + 20*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^ 3*c^5 - 64*a^3*b*c^6)*d^4*x^4 + 2*(9*b^8*c^2 - 104*a*b^6*c^3 + 384*a^2*b^4 *c^4 - 384*a^3*b^2*c^5 - 256*a^4*c^6)*d^4*x^3 + (7*b^9*c - 72*a*b^7*c^2 + 192*a^2*b^5*c^3 + 128*a^3*b^3*c^4 - 768*a^4*b*c^5)*d^4*x^2 + (b^10 - 6*a*b ^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*d^4*x + (a*b^9 - 12*a^2*b^7*c + 48*a^3*b^5*c^2 - 64*a^4*b^3*c^3)*d^4)
\[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]