Integrand size = 28, antiderivative size = 227 \[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 c d}+\frac {\left (b^2-4 a c\right )^{13/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{154 c^3 \sqrt {a+b x+c x^2}} \]
1/11*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(3/2)/c/d-3/154*(-4*a*c+b^2)*(2*c*d *x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c^2/d+1/77*(-4*a*c+b^2)^2*d*(2*c*d*x+b*d )^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2+1/154*(-4*a*c+b^2)^(13/4)*d^(3/2)*Elliptic F((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)/c^3/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.52 \[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2}{11} d \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \left (2 (a+x (b+c x))^2-\frac {\left (b^2-4 a c\right )^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{16 c^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right ) \]
(2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^2 - (( b^2 - 4*a*c)^2*Hypergeometric2F1[-3/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a* c)])/(16*c^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/11
Time = 0.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1109, 1109, 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.
| \(\displaystyle \int \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d}-\frac {3 \left (b^2-4 a c\right ) \int (b d+2 c x d)^{3/2} \sqrt {c x^2+b x+a}dx}{22 c}\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{3/2}}{\sqrt {c x^2+b x+a}}dx}{14 c}\right )}{22 c}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {1}{3} d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\right )}{22 c}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {d^2 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\right )}{22 c}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {2 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\right )}{22 c}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {2 d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\right )}{22 c}\) |
((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(11*c*d) - (3*(b^2 - 4*a*c )*(((b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c*d) - ((b^2 - 4*a*c)* ((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[Sq rt[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])))/(14*c)))/(22*c)
3.14.38.3.1 Defintions of rubi rules used
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[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))) 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] && GtQ[p, 0] && !LtQ[m, -1] && !(IGtQ[(m - 1 )/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && 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*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. \(554\) vs. \(2(193)=386\).
Time = 2.47 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.44
| method | result | size |
| risch | \(\frac {\left (56 c^{4} x^{4}+112 b \,c^{3} x^{3}+104 x^{2} c^{3} a +58 b^{2} c^{2} x^{2}+104 a b \,c^{2} x +2 b^{3} c x +32 a^{2} c^{2}+10 a \,b^{2} c -b^{4}\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d^{2}}{154 c^{2} \sqrt {d \left (2 c x +b \right )}}-\frac {\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{2} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{154 c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(555\) |
| default | \(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d \left (-224 c^{7} x^{7}-784 b \,c^{6} x^{6}-640 a \,c^{6} x^{5}-1016 b^{2} c^{5} x^{5}+64 \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}}}}\, F\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 ) \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-48 \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}}}}\, F\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 ) \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \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}}}}\, F\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 ) \sqrt {-4 a c +b^{2}}\, a \,b^{4} c -\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}}}}\, F\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 ) \sqrt {-4 a c +b^{2}}\, b^{6}-1600 a b \,c^{5} x^{4}-580 b^{3} c^{4} x^{4}-544 a^{2} c^{5} x^{3}-1328 a \,b^{2} c^{4} x^{3}-124 b^{4} c^{3} x^{3}-816 a^{2} b \,c^{4} x^{2}-392 a \,b^{3} c^{3} x^{2}+2 b^{5} c^{2} x^{2}-128 a^{3} c^{4} x -312 a^{2} b^{2} c^{3} x -20 c^{2} a \,b^{4} x +2 b^{6} c x -64 a^{3} c^{3} b -20 a^{2} c^{2} b^{3}+2 a \,b^{5} c \right )}{308 c^{3} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(796\) |
| elliptic | \(\text {Expression too large to display}\) | \(2281\) |
1/154/c^2*(56*c^4*x^4+112*b*c^3*x^3+104*a*c^3*x^2+58*b^2*c^2*x^2+104*a*b*c ^2*x+2*b^3*c*x+32*a^2*c^2+10*a*b^2*c-b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)*d^ 2/(d*(2*c*x+b))^(1/2)-1/154/c^2*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6) *(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/c*b)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(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/c*b))^(1/2))*d^2*(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)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.82 \[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} d {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (56 \, c^{6} d x^{4} + 112 \, b c^{5} d x^{3} + 2 \, {\left (29 \, b^{2} c^{4} + 52 \, a c^{5}\right )} d x^{2} + 2 \, {\left (b^{3} c^{3} + 52 \, a b c^{4}\right )} d x - {\left (b^{4} c^{2} - 10 \, a b^{2} c^{3} - 32 \, a^{2} c^{4}\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{308 \, c^{4}} \]
1/308*(sqrt(2)*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c^2*d )*d*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(56*c ^6*d*x^4 + 112*b*c^5*d*x^3 + 2*(29*b^2*c^4 + 52*a*c^5)*d*x^2 + 2*(b^3*c^3 + 52*a*b*c^4)*d*x - (b^4*c^2 - 10*a*b^2*c^3 - 32*a^2*c^4)*d)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^4
\[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
\[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]