Integrand size = 28, antiderivative size = 228 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac {12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac {40 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\left (b^2-4 a c\right )^{11/4} d^{5/2} \sqrt {a+b x+c x^2}} \] Output:
-2/3/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(3/2)+12*c/(-4*a*c+b ^2)^2/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2)+80*c^2*(c*x^2+b*x+a)^(1/2) /(-4*a*c+b^2)^3/d/(2*c*d*x+b*d)^(3/2)+40*c*(-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)^(11/4)/d^(5/2)/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {32 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{2},\frac {1}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )^2 d (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]
Output:
(-32*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[-3/4, 5/2, 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c)^2*d*(d*(b + 2*c*x ))^(3/2)*Sqrt[a + x*(b + c*x)])
Time = 0.50 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1111, 1111, 1117, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {6 c \int \frac {1}{(b d+2 c x d)^{5/2} \left (c x^2+b x+a\right )^{3/2}}dx}{b^2-4 a c}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {6 c \left (-\frac {10 c \int \frac {1}{(b d+2 c x d)^{5/2} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {6 c \left (-\frac {10 c \left (\frac {\int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx}{3 d^2 \left (b^2-4 a c\right )}+\frac {4 \sqrt {a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle -\frac {6 c \left (-\frac {10 c \left (\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{3 d^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle -\frac {6 c \left (-\frac {10 c \left (\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle -\frac {6 c \left (-\frac {10 c \left (\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}\right )}{b^2-4 a c}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}\) |
Input:
Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x]
Output:
-2/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2)) - (6* c*(-2/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]) - (10* c*((4*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)) + ( 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*(b^2 - 4*a*c)^(3/4)*d^ (5/2)*Sqrt[a + b*x + c*x^2])))/(b^2 - 4*a*c)))/(b^2 - 4*a*c)
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_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[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*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])
Leaf count of result is larger than twice the leaf count of optimal. \(663\) vs. \(2(200)=400\).
Time = 10.46 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.91
| method | result | size |
| elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {\left (-\frac {2 x^{2}}{c^{2} \left (4 a c -b^{2}\right )^{2} d^{3}}-\frac {2 b x}{c^{3} \left (4 a c -b^{2}\right )^{2} d^{3}}-\frac {8 a c +b^{2}}{6 \left (4 a c -b^{2}\right )^{2} d^{3} c^{4}}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}{\left (x^{3}+\frac {3 b \,x^{2}}{2 c}+\frac {\left (2 a c +b^{2}\right ) x}{2 c^{2}}+\frac {a b}{2 c^{2}}\right )^{2}}-\frac {2 c^{2} d \left (\frac {10 x}{d^{3} \left (4 a c -b^{2}\right )^{3}}+\frac {5 b}{c \,d^{3} \left (4 a c -b^{2}\right )^{3}}\right ) \sqrt {2}}{\sqrt {\left (x^{3}+\frac {3 b \,x^{2}}{2 c}+\frac {\left (2 a c +b^{2}\right ) x}{2 c^{2}}+\frac {a b}{2 c^{2}}\right ) c^{2} d}}-\frac {40 c^{2} \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{d^{2} \left (4 a c -b^{2}\right )^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(664\) |
| default | \(-\frac {2 \left (60 \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 ) c^{3} x^{3}+90 \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 \,c^{2} x^{2}+60 \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 ) \sqrt {-4 a c +b^{2}}\, a \,c^{2} x +30 \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^{2} c x +30 \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 ) \sqrt {-4 a c +b^{2}}\, a b c +120 c^{4} x^{4}+240 b \,c^{3} x^{3}+168 a \,c^{3} x^{2}+138 b^{2} c^{2} x^{2}+168 a b \,c^{2} x +18 b^{3} c x +32 a^{2} c^{2}+26 c a \,b^{2}-b^{4}\right ) \sqrt {d \left (2 c x +b \right )}}{3 d^{3} \left (2 c x +b \right )^{2} \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(797\) |
Input:
int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(d*(2*c*x+b)*(c*x^2+b*x+a))^(1/2)/(d*(2*c*x+b))^(1/2)/(c*x^2+b*x+a)^(1/2)* ((-2/c^2/(4*a*c-b^2)^2/d^3*x^2-2*b/c^3/(4*a*c-b^2)^2/d^3*x-1/6*(8*a*c+b^2) /(4*a*c-b^2)^2/d^3/c^4)*(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^ (1/2)/(x^3+3/2*b/c*x^2+1/2/c^2*(2*a*c+b^2)*x+1/2*a/c^2*b)^2-2*c^2*d*(10/d^ 3/(4*a*c-b^2)^3*x+5*b/c/d^3/(4*a*c-b^2)^3)*2^(1/2)/((x^3+3/2*b/c*x^2+1/2/c ^2*(2*a*c+b^2)*x+1/2*a/c^2*b)*c^2*d)^(1/2)-40*c^2/d^2/(4*a*c-b^2)^3*(1/2/c *(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+ b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c ))^(1/2)*((x+1/2*b/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*b/c))^(1/2)*((x-1 /2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4 *a*c+b^2)^(1/2))))^(1/2)/(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d) ^(1/2)*EllipticF(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2) ^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2)) /c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*b/c)) ^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (199) = 398\).
Time = 0.12 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.62 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (30 \, \sqrt {2} {\left (4 \, c^{4} x^{6} + 12 \, b c^{3} x^{5} + {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{4} + a^{2} b^{2} + 2 \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} x^{3} + {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (120 \, c^{4} x^{4} + 240 \, b c^{3} x^{3} - b^{4} + 26 \, a b^{2} c + 32 \, a^{2} c^{2} + 6 \, {\left (23 \, b^{2} c^{2} + 28 \, a c^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c + 28 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} \] Input:
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*(30*sqrt(2)*(4*c^4*x^6 + 12*b*c^3*x^5 + (13*b^2*c^2 + 8*a*c^3)*x^4 + a ^2*b^2 + 2*(3*b^3*c + 8*a*b*c^2)*x^3 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^2 + 2*(a*b^3 + 2*a^2*b*c)*x)*sqrt(c^2*d)*weierstrassPInverse((b^2 - 4*a*c)/c ^2, 0, 1/2*(2*c*x + b)/c) + (120*c^4*x^4 + 240*b*c^3*x^3 - b^4 + 26*a*b^2* c + 32*a^2*c^2 + 6*(23*b^2*c^2 + 28*a*c^3)*x^2 + 6*(3*b^3*c + 28*a*b*c^2)* x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(4*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^ 2*b^3*c^5 - 64*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2* b^4*c^4 - 448*a^3*b^2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a*b^7*c ^2 + 48*a^2*b^5*c^3 + 192*a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10 - 2 *a*b^8*c - 68*a^2*b^6*c^2 + 368*a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^ 5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c + 24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 1 28*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^ 2*c^3)*d^3)
\[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)
Output:
Integral(1/((d*(b + 2*c*x))**(5/2)*(a + b*x + c*x**2)**(5/2)), x)
\[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)), x)
\[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(5/2)), x)
Timed out. \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:
int(1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)),x)
Output:
int(1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2)), x)
\[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{8 c^{6} x^{9}+36 b \,c^{5} x^{8}+24 a \,c^{5} x^{7}+66 b^{2} c^{4} x^{7}+84 a b \,c^{4} x^{6}+63 b^{3} c^{3} x^{6}+24 a^{2} c^{4} x^{5}+114 a \,b^{2} c^{3} x^{5}+33 b^{4} c^{2} x^{5}+60 a^{2} b \,c^{3} x^{4}+75 a \,b^{3} c^{2} x^{4}+9 b^{5} c \,x^{4}+8 a^{3} c^{3} x^{3}+54 a^{2} b^{2} c^{2} x^{3}+24 a \,b^{4} c \,x^{3}+b^{6} x^{3}+12 a^{3} b \,c^{2} x^{2}+21 a^{2} b^{3} c \,x^{2}+3 a \,b^{5} x^{2}+6 a^{3} b^{2} c x +3 a^{2} b^{4} x +a^{3} b^{3}}d x \right )}{d^{3}} \] Input:
int(1/(2*c*d*x+b*d)^(5/2)/(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*b**3 + 6*a**3* b**2*c*x + 12*a**3*b*c**2*x**2 + 8*a**3*c**3*x**3 + 3*a**2*b**4*x + 21*a** 2*b**3*c*x**2 + 54*a**2*b**2*c**2*x**3 + 60*a**2*b*c**3*x**4 + 24*a**2*c** 4*x**5 + 3*a*b**5*x**2 + 24*a*b**4*c*x**3 + 75*a*b**3*c**2*x**4 + 114*a*b* *2*c**3*x**5 + 84*a*b*c**4*x**6 + 24*a*c**5*x**7 + b**6*x**3 + 9*b**5*c*x* *4 + 33*b**4*c**2*x**5 + 63*b**3*c**3*x**6 + 66*b**2*c**4*x**7 + 36*b*c**5 *x**8 + 8*c**6*x**9),x))/d**3