Integrand size = 25, antiderivative size = 421 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=\frac {2 \left (2 A b^2-5 a b B-6 a A c\right ) \sqrt {a+b x+c x^2}}{15 a^2 \sqrt {x}}-\frac {2 (3 a A+(A b+5 a B) x) \sqrt {a+b x+c x^2}}{15 a x^{5/2}}+\frac {2 \sqrt {c} \left (5 a b B-2 A \left (b^2-3 a c\right )\right ) \sqrt {x} \sqrt {a+b x+c x^2}}{15 a^2 \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt [4]{c} \left (5 a b B-2 A \left (b^2-3 a c\right )\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 a^{7/4} \sqrt {a+b x+c x^2}}-\frac {\left (b+2 \sqrt {a} \sqrt {c}\right ) \left (2 A b-5 a B-3 \sqrt {a} A \sqrt {c}\right ) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 a^{7/4} \sqrt {a+b x+c x^2}} \]
-2/15*(3*a*A+(A*b+5*B*a)*x)*(c*x^2+b*x+a)^(1/2)/a/x^(5/2)+2/15*(-6*A*a*c+2 *A*b^2-5*B*a*b)*(c*x^2+b*x+a)^(1/2)/a^2/x^(1/2)+2/15*(5*a*b*B-2*A*(-3*a*c+ b^2))*c^(1/2)*x^(1/2)*(c*x^2+b*x+a)^(1/2)/a^2/(a^(1/2)+x*c^(1/2))-2/15*c^( 1/4)*(5*a*b*B-2*A*(-3*a*c+b^2))*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2) ^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/ 4)*x^(1/2)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x*c^(1/2))* ((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/a^(7/4)/(c*x^2+b*x+a)^(1/2)-1/ 15*c^(1/4)*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c ^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))), 1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x*c^(1/2))*(b+2*a^(1/2)*c^(1/2)) *(2*A*b-5*B*a-3*A*a^(1/2)*c^(1/2))*((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2)^( 1/2)/a^(7/4)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.48 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=\frac {-4 (a+x (b+c x)) \left (-2 A b^2 x^2+a^2 (3 A+5 B x)+a x (5 b B x+A (b+6 c x))\right )+\frac {x^2 \left (4 \left (-2 A b^2+5 a b B+6 a A c\right ) \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))+i \left (-b+\sqrt {b^2-4 a c}\right ) \left (-5 a b B+2 A \left (b^2-3 a c\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (5 a B \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )+2 A \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}}{30 a^2 x^{5/2} \sqrt {a+x (b+c x)}} \]
(-4*(a + x*(b + c*x))*(-2*A*b^2*x^2 + a^2*(3*A + 5*B*x) + a*x*(5*b*B*x + A *(b + 6*c*x))) + (x^2*(4*(-2*A*b^2 + 5*a*b*B + 6*a*A*c)*Sqrt[a/(b + Sqrt[b ^2 - 4*a*c])]*(a + x*(b + c*x)) + I*(-b + Sqrt[b^2 - 4*a*c])*(-5*a*b*B + 2 *A*(b^2 - 3*a*c))*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt [(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*Ellipt icE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqr t[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(5*a*B*(b^2 - 4*a*c - b*Sqrt[ b^2 - 4*a*c]) + 2*A*(-b^3 + 4*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 3*a*c*Sqrt[b ^2 - 4*a*c]))*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(4* a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[ I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^ 2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/(3 0*a^2*x^(5/2)*Sqrt[a + x*(b + c*x)])
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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {2 \int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{2 x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {5 a b B-2 A \left (b^2-3 a c\right )-(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {2 A b^2-5 a B b-6 a A c+(A b-10 a B) c x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{15 a}-\frac {2 \sqrt {a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}}\) |
3.11.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(403)=806\).
Time = 1.86 (sec) , antiderivative size = 842, normalized size of antiderivative = 2.00
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 A \sqrt {c \,x^{3}+b \,x^{2}+a x}}{5 x^{3}}-\frac {2 \left (A b +5 B a \right ) \sqrt {c \,x^{3}+b \,x^{2}+a x}}{15 a \,x^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right ) \left (6 A a c -2 A \,b^{2}+5 a b B \right )}{15 a^{2} \sqrt {x \left (c \,x^{2}+b x +a \right )}}+\frac {\left (B c -\frac {c \left (A b +5 B a \right )}{15 a}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \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 {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, F\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {\left (6 A a c -2 A \,b^{2}+5 a b B \right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \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 {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{15 a^{2} \sqrt {c \,x^{3}+b \,x^{2}+a x}}\right )}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(842\) |
| risch | \(\text {Expression too large to display}\) | \(1059\) |
| default | \(\text {Expression too large to display}\) | \(2108\) |
(x*(c*x^2+b*x+a))^(1/2)/x^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/5*A*(c*x^3+b*x^2+a *x)^(1/2)/x^3-2/15*(A*b+5*B*a)/a*(c*x^3+b*x^2+a*x)^(1/2)/x^2-2/15*(c*x^2+b *x+a)/a^2*(6*A*a*c-2*A*b^2+5*B*a*b)/(x*(c*x^2+b*x+a))^(1/2)+(B*c-1/15*c*(A *b+5*B*a)/a)*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2 ))/c)/(b+(-4*a*c+b^2)^(1/2))*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* x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*x^3+b*x^2+a*x)^(1/2)*EllipticF(2^(1/2)* ((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2* (b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+ b^2)^(1/2))))^(1/2))+1/15*(6*A*a*c-2*A*b^2+5*B*a*b)/a^2*(b+(-4*a*c+b^2)^(1 /2))*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*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*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/( c*x^3+b*x^2+a*x)^(1/2)*((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b ^2)^(1/2)))*EllipticE(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c +b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^ 2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^ (1/2))*EllipticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2) ^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1 /2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.61 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=-\frac {2 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3} - 3 \, {\left (10 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {c} x^{3} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (6 \, A a c^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} c\right )} \sqrt {c} x^{3} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (3 \, A a^{2} c + {\left (5 \, B a^{2} + A a b\right )} c x + {\left (6 \, A a c^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} c\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{45 \, a^{2} c x^{3}} \]
-2/45*((5*B*a*b^2 - 2*A*b^3 - 3*(10*B*a^2 - 3*A*a*b)*c)*sqrt(c)*x^3*weiers trassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c *x + b)/c) + 3*(6*A*a*c^2 + (5*B*a*b - 2*A*b^2)*c)*sqrt(c)*x^3*weierstrass Zeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, weierstrassPInver se(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c)) + 3*(3*A*a^2*c + (5*B*a^2 + A*a*b)*c*x + (6*A*a*c^2 + (5*B*a*b - 2*A*b^2) *c)*x^2)*sqrt(c*x^2 + b*x + a)*sqrt(x))/(a^2*c*x^3)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{\frac {7}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{7/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^{7/2}} \,d x \]