Integrand size = 28, antiderivative size = 258 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {28 c d^3 (b d+2 c d x)^{3/2}}{3 \sqrt {a+b x+c x^2}}+\frac {56 c \left (b^2-4 a c\right )^{3/4} d^{9/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\sqrt {a+b x+c x^2}}-\frac {56 c \left (b^2-4 a c\right )^{3/4} d^{9/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\sqrt {a+b x+c x^2}} \] Output:
-2/3*d*(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(3/2)-28/3*c*d^3*(2*c*d*x+b*d)^(3 /2)/(c*x^2+b*x+a)^(1/2)+56*c*(-4*a*c+b^2)^(3/4)*d^(9/2)*(-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*x^2+b*x+a)^(1/2)-56*c*(-4*a*c+b^2)^(3/4)*d^(9/2)*(-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*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.47 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {16 d^3 (d (b+2 c x))^{3/2} \left (b^2-3 b c x-c \left (7 a+3 c x^2\right )+14 c (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 (a+x (b+c x))^{3/2}} \] Input:
Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-16*d^3*(d*(b + 2*c*x))^(3/2)*(b^2 - 3*b*c*x - c*(7*a + 3*c*x^2) + 14*c*( a + x*(b + c*x))*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric 2F1[3/4, 5/2, 7/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(a + x*(b + c*x))^(3/ 2))
Time = 0.51 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1110, 1110, 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.
\(\displaystyle \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1110 |
\(\displaystyle \frac {14}{3} c d^2 \int \frac {(b d+2 c x d)^{5/2}}{\left (c x^2+b x+a\right )^{3/2}}dx-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1110 |
\(\displaystyle \frac {14}{3} c d^2 \left (6 c d^2 \int \frac {\sqrt {b d+2 c x d}}{\sqrt {c x^2+b x+a}}dx-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1115 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {6 c d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\sqrt {b d+2 c x d}}{\sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1114 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {12 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {b d+2 c x d}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {12 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{d \sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {12 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d \sqrt {b^2-4 a c} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}\right )}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {12 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {12 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {\sqrt {\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}+1}}{\sqrt {1-\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {14}{3} c d^2 \left (\frac {12 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d^{3/2} \left (b^2-4 a c\right )^{3/4} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d (b d+2 c d x)^{7/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*d*(b*d + 2*c*d*x)^(7/2))/(3*(a + b*x + c*x^2)^(3/2)) + (14*c*d^2*((-2* d*(b*d + 2*c*d*x)^(3/2))/Sqrt[a + b*x + c*x^2] + (12*d*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[Sq rt[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]))/Sqrt[a + b*x + c*x^2]))/3
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_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && 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. \(858\) vs. \(2(218)=436\).
Time = 8.24 (sec) , antiderivative size = 859, normalized size of antiderivative = 3.33
method | result | size |
default | \(\frac {2 \sqrt {d \left (2 c x +b \right )}\, \left (168 \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}-42 \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}+168 \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 -42 \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 +168 \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}-42 \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 -72 c^{4} x^{4}-144 b \,c^{3} x^{3}-56 a \,c^{3} x^{2}-94 b^{2} c^{2} x^{2}-56 a b \,c^{2} x -22 b^{3} c x -14 c a \,b^{2}-b^{4}\right ) d^{4}}{3 \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(859\) |
elliptic | \(\text {Expression too large to display}\) | \(1108\) |
Input:
int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3*(d*(2*c*x+b))^(1/2)*(168*(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^2-42*(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)*Ellipti cE(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+168*(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-42*(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+168*(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^2*c^2-42*(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...
Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.89 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (84 \, \sqrt {2} {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (36 \, c^{3} d^{4} x^{3} + 54 \, b c^{2} d^{4} x^{2} + 4 \, {\left (5 \, b^{2} c + 7 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 14 \, a b c\right )} d^{4}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \] Input:
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
-2/3*(84*sqrt(2)*(c^3*d^4*x^4 + 2*b*c^2*d^4*x^3 + 2*a*b*c*d^4*x + a^2*c*d^ 4 + (b^2*c + 2*a*c^2)*d^4*x^2)*sqrt(c^2*d)*weierstrassZeta((b^2 - 4*a*c)/c ^2, 0, weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) + (36 *c^3*d^4*x^3 + 54*b*c^2*d^4*x^2 + 4*(5*b^2*c + 7*a*c^2)*d^4*x + (b^3 + 14* a*b*c)*d^4)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^ 3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)
Timed out. \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(5/2), x)
\[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(5/2), x)
Timed out. \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:
int((b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2),x)
Output:
int((b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(5/2), x)
\[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*sqrt(d)*d**4*(336*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*c**2 + 11 2*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2*c + 560*sqrt(b + 2*c*x)*sq rt(a + b*x + c*x**2)*a*b*c**2*x - 19*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2 )*b**4 + 40*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**3*c*x + 360*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**2*c**2*x**2 + 240*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b*c**3*x**3 + 1680*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c* x**2)*x**2)/(a**3*b + 2*a**3*c*x + 3*a**2*b**2*x + 9*a**2*b*c*x**2 + 6*a** 2*c**2*x**3 + 3*a*b**3*x**2 + 12*a*b**2*c*x**3 + 15*a*b*c**2*x**4 + 6*a*c* *3*x**5 + b**4*x**3 + 5*b**3*c*x**4 + 9*b**2*c**2*x**5 + 7*b*c**3*x**6 + 2 *c**4*x**7),x)*a**4*c**4 - 840*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2) *x**2)/(a**3*b + 2*a**3*c*x + 3*a**2*b**2*x + 9*a**2*b*c*x**2 + 6*a**2*c** 2*x**3 + 3*a*b**3*x**2 + 12*a*b**2*c*x**3 + 15*a*b*c**2*x**4 + 6*a*c**3*x* *5 + b**4*x**3 + 5*b**3*c*x**4 + 9*b**2*c**2*x**5 + 7*b*c**3*x**6 + 2*c**4 *x**7),x)*a**3*b**2*c**3 + 3360*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2 )*x**2)/(a**3*b + 2*a**3*c*x + 3*a**2*b**2*x + 9*a**2*b*c*x**2 + 6*a**2*c* *2*x**3 + 3*a*b**3*x**2 + 12*a*b**2*c*x**3 + 15*a*b*c**2*x**4 + 6*a*c**3*x **5 + b**4*x**3 + 5*b**3*c*x**4 + 9*b**2*c**2*x**5 + 7*b*c**3*x**6 + 2*c** 4*x**7),x)*a**3*b*c**4*x + 3360*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2 )*x**2)/(a**3*b + 2*a**3*c*x + 3*a**2*b**2*x + 9*a**2*b*c*x**2 + 6*a**2*c* *2*x**3 + 3*a*b**3*x**2 + 12*a*b**2*c*x**3 + 15*a*b*c**2*x**4 + 6*a*c**...