Integrand size = 28, antiderivative size = 182 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=-\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{14 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac {\left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{14 c^3 \sqrt {d} \sqrt {a+b x+c x^2}} \]
1/7*(c*x^2+b*x+a)^(3/2)*(2*c*d*x+b*d)^(1/2)/c/d-1/14*(-4*a*c+b^2)*(2*c*d*x +b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/d+1/14*(-4*a*c+b^2)^(9/4)*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)/c^3/d^(1/2)/(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 = 99, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=-\frac {\left (b^2-4 a c\right ) \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
-1/8*((b^2 - 4*a*c)*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometr ic2F1[-3/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^2*d*Sqrt[(c*(a + x* (b + c*x)))/(-b^2 + 4*a*c)])
Time = 0.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1109, 1109, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b d+2 c x d}}dx}{14 c}\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx}{6 c}\right )}{14 c}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{6 c \sqrt {a+b x+c x^2}}\right )}{14 c}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c^2 d \sqrt {a+b x+c x^2}}\right )}{14 c}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}}\right )}{14 c}\) |
(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2))/(7*c*d) - (3*(b^2 - 4*a*c)*( (Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(3*c*d) - ((b^2 - 4*a*c)^(5/4) *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^2*Sqrt[d]*Sqrt[a + b*x + c*x^2])))/(14*c)
3.14.39.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(485\) vs. \(2(154)=308\).
Time = 2.54 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.67
| method | result | size |
| risch | \(\frac {\left (2 c^{2} x^{2}+2 b c x +6 a c -b^{2}\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{14 c^{2} \sqrt {d \left (2 c x +b \right )}}+\frac {\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{14 c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(486\) |
| default | \(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (8 c^{5} x^{5}+16 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{2} c^{2}-8 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a \,b^{2} c +\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, b^{4}+20 b \,x^{4} c^{4}+32 a \,c^{4} x^{3}+12 b^{2} c^{3} x^{3}+48 a b \,c^{3} x^{2}-2 x^{2} b^{3} c^{2}+24 a^{2} c^{3} x +12 a \,c^{2} b^{2} x -2 c x \,b^{4}+12 a^{2} b \,c^{2}-2 a \,b^{3} c \right )}{28 d \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{3}}\) | \(566\) |
| elliptic | \(\text {Expression too large to display}\) | \(1212\) |
1/14*(2*c^2*x^2+2*b*c*x+6*a*c-b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^2/(d*(2 *c*x+b))^(1/2)+1/14*(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*(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*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 = 119, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (2 \, c^{4} x^{2} + 2 \, b c^{3} x - b^{2} c^{2} + 6 \, a c^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{28 \, c^{4} d} \]
1/28*(sqrt(2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c^2*d)*weierstrassPInver se((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(2*c^4*x^2 + 2*b*c^3*x - b ^2*c^2 + 6*a*c^3)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(c^4*d)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\sqrt {d \left (b + 2 c x\right )}}\, dx \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\sqrt {2 \, c d x + b d}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\sqrt {2 \, c d x + b d}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{\sqrt {b\,d+2\,c\,d\,x}} \,d x \]