Integrand size = 28, antiderivative size = 205 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}+\frac {120}{7} c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {72}{7} c d^3 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {60 \left (b^2-4 a c\right )^{9/4} d^{11/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{7 \sqrt {a+b x+c x^2}} \] Output:
-2*d*(2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2)+120/7*c*(-4*a*c+b^2)*d^5*(2*c *d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)+72/7*c*d^3*(2*c*d*x+b*d)^(5/2)*(c*x^2+ b*x+a)^(1/2)+60/7*(-4*a*c+b^2)^(9/4)*d^(11/2)*(-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 = 10.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.84 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 d^5 \sqrt {d (b+2 c x)} \left (-7 b^4+40 b^3 c x+32 b c^2 x \left (-3 a+2 c x^2\right )+24 b^2 c \left (4 a+3 c x^2\right )+16 c^2 \left (-15 a^2-6 a c x^2+2 c^2 x^4\right )+30 \left (b^2-4 a c\right )^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{7 \sqrt {a+x (b+c x)}} \] Input:
Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(3/2),x]
Output:
(2*d^5*Sqrt[d*(b + 2*c*x)]*(-7*b^4 + 40*b^3*c*x + 32*b*c^2*x*(-3*a + 2*c*x ^2) + 24*b^2*c*(4*a + 3*c*x^2) + 16*c^2*(-15*a^2 - 6*a*c*x^2 + 2*c^2*x^4) + 30*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeome tric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(7*Sqrt[a + x*(b + c *x)])
Time = 0.43 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1110, 1116, 1116, 1115, 1113, 762}
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.
| \(\displaystyle \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle 18 c d^2 \int \frac {(b d+2 c x d)^{7/2}}{\sqrt {c x^2+b x+a}}dx-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 18 c d^2 \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{3/2}}{\sqrt {c x^2+b x+a}}dx+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 18 c d^2 \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {1}{3} d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle 18 c d^2 \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {d^2 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle 18 c d^2 \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {2 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 18 c d^2 \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {2 d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )-\frac {2 d (b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}}\) |
Input:
Int[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(3/2),x]
Output:
(-2*d*(b*d + 2*c*d*x)^(9/2))/Sqrt[a + b*x + c*x^2] + 18*c*d^2*((4*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/7 + (5*(b^2 - 4*a*c)*d^2*((4*d*Sqrt [b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/3 + (2*(b^2 - 4*a*c)^(5/4)*d^(3/2)* Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2 *c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(3*c*Sqrt[a + b*x + c*x^2]))) /7)
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[((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[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[1/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*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) 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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Leaf count of result is larger than twice the leaf count of optimal. \(568\) vs. \(2(173)=346\).
Time = 7.90 (sec) , antiderivative size = 569, normalized size of antiderivative = 2.78
| method | result | size |
| default | \(\frac {2 \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{5} \left (64 c^{5} x^{5}+240 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{2} c^{2}-120 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,b^{2} c +15 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{4}+160 b \,x^{4} c^{4}-192 a \,c^{4} x^{3}+208 b^{2} c^{3} x^{3}-288 a b \,c^{3} x^{2}+152 c^{2} x^{2} b^{3}-480 a^{2} c^{3} x +96 a \,b^{2} c^{2} x +26 b^{4} c x -240 a^{2} b \,c^{2}+96 a \,b^{3} c -7 b^{5}\right )}{7 \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(569\) |
| risch | \(-\frac {16 c \left (-4 c^{2} x^{2}-4 c b x +16 a c -5 b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{6}}{7 \sqrt {d \left (2 c x +b \right )}}+\frac {\left (\frac {\left (-448 a^{3} c^{3}+336 a^{2} b^{2} c^{2}-84 a \,b^{4} c +7 b^{6}\right ) \left (\frac {4 c^{2} d x +2 d b c}{\left (4 a c -b^{2}\right ) d c \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +d b c \right )}}+\frac {4 c \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \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 {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \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}}}\, \operatorname {EllipticF}\left (\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 {-\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}+\frac {b}{2 c}}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}\right )}{7}+\frac {88 c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \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 {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \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}}}\, \operatorname {EllipticF}\left (\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 {-\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}+\frac {b}{2 c}}}\right )}{7 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}\right ) d^{6} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(964\) |
| elliptic | \(\text {Expression too large to display}\) | \(1441\) |
Input:
int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/7*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^5*(64*c^5*x^5+240*(-4*a*c+b^ 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*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)*EllipticF(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^2-120*(-4*a*c+b^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*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)*EllipticF (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^2*c+15*(-4*a*c+b^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*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)*EllipticF(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^4+160*b*x^4*c^4- 192*a*c^4*x^3+208*b^2*c^3*x^3-288*a*b*c^3*x^2+152*c^2*x^2*b^3-480*a^2*c^3* x+96*a*b^2*c^2*x+26*b^4*c*x-240*a^2*b*c^2+96*a*b^3*c-7*b^5)/(2*c^2*x^3+3*b *c*x^2+2*a*c*x+b^2*x+a*b)
Time = 0.09 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.28 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (15 \, \sqrt {2} {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{5} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{5} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d^{5}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (32 \, c^{5} d^{5} x^{4} + 64 \, b c^{4} d^{5} x^{3} + 24 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{5} x^{2} + 8 \, {\left (5 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} d^{5} x - {\left (7 \, b^{4} c - 96 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{7 \, {\left (c^{2} x^{2} + b c x + a c\right )}} \] Input:
integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
2/7*(15*sqrt(2)*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*x^2 + (b^5 - 8*a*b ^3*c + 16*a^2*b*c^2)*d^5*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d^5)*sqrt( c^2*d)*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + (32* c^5*d^5*x^4 + 64*b*c^4*d^5*x^3 + 24*(3*b^2*c^3 - 4*a*c^4)*d^5*x^2 + 8*(5*b ^3*c^2 - 12*a*b*c^3)*d^5*x - (7*b^4*c - 96*a*b^2*c^2 + 240*a^2*c^3)*d^5)*s qrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(c^2*x^2 + b*c*x + a*c)
Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((2*c*d*x+b*d)**(11/2)/(c*x**2+b*x+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^(3/2), x)
\[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^(3/2), x)
Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{11/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int((b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(3/2),x)
Output:
int((b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(3/2), x)
\[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(3/2),x)
Output:
(2*sqrt(d)*d**5*(192*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b**2*c**2 - 192*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b*c**3*x - 192*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*c**4*x**2 - 80*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**4*c + 176*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b* *3*c**2*x + 240*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2*c**3*x**2 + 128*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b*c**4*x**3 + 64*sqrt(b + 2*c *x)*sqrt(a + b*x + c*x**2)*a*c**5*x**4 + 7*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**6 - 40*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**5*c*x - 72*sqr t(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**4*c**2*x**2 - 64*sqrt(b + 2*c*x)*sq rt(a + b*x + c*x**2)*b**3*c**3*x**3 - 32*sqrt(b + 2*c*x)*sqrt(a + b*x + c* x**2)*b**2*c**4*x**4 + 1920*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x* *2)/(2*a**3*b*c + 4*a**3*c**2*x - a**2*b**3 + 2*a**2*b**2*c*x + 12*a**2*b* c**2*x**2 + 8*a**2*c**3*x**3 - 2*a*b**4*x - 4*a*b**3*c*x**2 + 4*a*b**2*c** 2*x**3 + 10*a*b*c**3*x**4 + 4*a*c**4*x**5 - b**5*x**2 - 4*b**4*c*x**3 - 5* b**3*c**2*x**4 - 2*b**2*c**3*x**5),x)*a**5*c**6 - 2400*int((sqrt(b + 2*c*x )*sqrt(a + b*x + c*x**2)*x**2)/(2*a**3*b*c + 4*a**3*c**2*x - a**2*b**3 + 2 *a**2*b**2*c*x + 12*a**2*b*c**2*x**2 + 8*a**2*c**3*x**3 - 2*a*b**4*x - 4*a *b**3*c*x**2 + 4*a*b**2*c**2*x**3 + 10*a*b*c**3*x**4 + 4*a*c**4*x**5 - b** 5*x**2 - 4*b**4*c*x**3 - 5*b**3*c**2*x**4 - 2*b**2*c**3*x**5),x)*a**4*b**2 *c**5 + 1920*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**3*...