Integrand size = 25, antiderivative size = 634 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=-\frac {2 (3 A b-14 a B) \sqrt {a+b x+c x^2}}{105 a x^{5/2}}+\frac {2 \left (4 A b^2-7 a b B-10 a A c\right ) \sqrt {a+b x+c x^2}}{105 a^2 x^{3/2}}+\frac {2 \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) \sqrt {a+b x+c x^2}}{105 a^3 \sqrt {x}}-\frac {2 (3 A+7 B x) \sqrt {a+b x+c x^2}}{21 x^{7/2}}-\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{105 \sqrt {2} a^3 \sqrt {c} \sqrt {a+b x+c x^2}}+\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )+\frac {2 a c \left (4 A b^2-7 a b B-10 a A c\right )}{b+\sqrt {b^2-4 a c}}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{105 \sqrt {2} a^3 \sqrt {c} \sqrt {a+b x+c x^2}} \] Output:
-2/105*(3*A*b-14*B*a)*(c*x^2+b*x+a)^(1/2)/a/x^(5/2)+2/105*(-10*A*a*c+4*A*b ^2-7*B*a*b)*(c*x^2+b*x+a)^(1/2)/a^2/x^(3/2)+2/105*(14*a*B*(-3*a*c+b^2)-A*( -29*a*b*c+8*b^3))*(c*x^2+b*x+a)^(1/2)/a^3/x^(1/2)-2/21*(7*B*x+3*A)*(c*x^2+ b*x+a)^(1/2)/x^(7/2)-1/210*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a*c+b^2)^( 1/2))*(14*a*B*(-3*a*c+b^2)-A*(-29*a*b*c+8*b^3))*(1+2*c*x/(b-(-4*a*c+b^2)^( 1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1 /2)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a *c+b^2)^(1/2)))^(1/2))*2^(1/2)/a^3/c^(1/2)/(c*x^2+b*x+a)^(1/2)+1/210*(-b+( -4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a*c+b^2)^(1/2))*(14*a*B*(-3*a*c+b^2)-A*(-2 9*a*b*c+8*b^3)+2*a*c*(-10*A*a*c+4*A*b^2-7*B*a*b)/(b+(-4*a*c+b^2)^(1/2)))*( 1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/ 2)*EllipticF(2^(1/2)*c^(1/2)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2),((b-(-4 *a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)/a^3/c^(1/2)/(c*x^2 +b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 27.22 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\frac {-4 (a+x (b+c x)) \left (8 A b^3 x^3+3 a^3 (5 A+7 B x)-a b x^2 (4 A b+14 b B x+29 A c x)+a^2 x (7 B x (b+6 c x)+A (3 b+10 c x))\right )+\frac {x^3 \left (4 \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} \left (14 a B \left (-b^2+3 a c\right )+A \left (8 b^3-29 a b c\right )\right ) (a+x (b+c x))+i \left (b-\sqrt {b^2-4 a c}\right ) \left (14 a B \left (-b^2+3 a c\right )+A \left (8 b^3-29 a b c\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (14 a B \left (b^3-4 a b c-b^2 \sqrt {b^2-4 a c}+3 a c \sqrt {b^2-4 a c}\right )+A \left (-8 b^4+37 a b^2 c-20 a^2 c^2+8 b^3 \sqrt {b^2-4 a c}-29 a b c \sqrt {b^2-4 a c}\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}}{210 a^3 x^{7/2} \sqrt {a+x (b+c x)}} \] Input:
Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^(9/2),x]
Output:
(-4*(a + x*(b + c*x))*(8*A*b^3*x^3 + 3*a^3*(5*A + 7*B*x) - a*b*x^2*(4*A*b + 14*b*B*x + 29*A*c*x) + a^2*x*(7*B*x*(b + 6*c*x) + A*(3*b + 10*c*x))) + ( x^3*(4*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(14*a*B*(-b^2 + 3*a*c) + A*(8*b^3 - 29*a*b*c))*(a + x*(b + c*x)) + I*(b - Sqrt[b^2 - 4*a*c])*(14*a*B*(-b^2 + 3*a*c) + A*(8*b^3 - 29*a*b*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)]*EllipticE[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])] + I*(14*a*B*(b^3 - 4 *a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 3*a*c*Sqrt[b^2 - 4*a*c]) + A*(-8*b^4 + 37 *a*b^2*c - 20*a^2*c^2 + 8*b^3*Sqrt[b^2 - 4*a*c] - 29*a*b*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])])/(210*a^3*x^ (7/2)*Sqrt[a + x*(b + c*x)])
Time = 0.85 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1229, 27, 1237, 27, 1237, 27, 1240, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {2 \int \frac {4 A b^2-7 a B b-10 a A c+(3 A b-14 a B) c x}{2 x^{5/2} \sqrt {c x^2+b x+a}}dx}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {4 A b^2-7 a B b-10 a A c+(3 A b-14 a B) c x}{x^{5/2} \sqrt {c x^2+b x+a}}dx}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {2 \int -\frac {14 a B \left (b^2-3 a c\right )-2 A \left (4 b^3-\frac {29 a b c}{2}\right )-c \left (4 A b^2-7 a B b-10 a A c\right ) x}{2 x^{3/2} \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )-c \left (4 A b^2-7 a B b-10 a A c\right ) x}{x^{3/2} \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {\frac {-\frac {2 \int \frac {c \left (a \left (4 A b^2-7 a B b-10 a A c\right )-\left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) x\right )}{2 \sqrt {x} \sqrt {c x^2+b x+a}}dx}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {c \int \frac {a \left (4 A b^2-7 a B b-10 a A c\right )-\left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle -\frac {\frac {-\frac {2 c \int \frac {a \left (4 A b^2-7 a B b-10 a A c\right )-\left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {\frac {-\frac {2 c \left (\frac {\sqrt {a} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {a} \left (-A \left (8 b^3-29 a b c\right )-\sqrt {a} \sqrt {c} \left (-10 a A c-7 a b B+4 A b^2\right )+14 a B \left (b^2-3 a c\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {-\frac {2 c \left (\frac {\left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {a} \left (-A \left (8 b^3-29 a b c\right )-\sqrt {a} \sqrt {c} \left (-10 a A c-7 a b B+4 A b^2\right )+14 a B \left (b^2-3 a c\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\frac {-\frac {2 c \left (\frac {\left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (-A \left (8 b^3-29 a b c\right )-\sqrt {a} \sqrt {c} \left (-10 a A c-7 a b B+4 A b^2\right )+14 a B \left (b^2-3 a c\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x+c x^2}}\right )}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {\frac {-\frac {2 c \left (\frac {\left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (-A \left (8 b^3-29 a b c\right )-\sqrt {a} \sqrt {c} \left (-10 a A c-7 a b B+4 A b^2\right )+14 a B \left (b^2-3 a c\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x+c x^2}}\right )}{a}-\frac {2 \sqrt {a+b x+c x^2} \left (14 a B \left (b^2-3 a c\right )-A \left (8 b^3-29 a b c\right )\right )}{a \sqrt {x}}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} \left (-10 a A c-7 a b B+4 A b^2\right )}{3 a x^{3/2}}}{35 a}-\frac {2 \sqrt {a+b x+c x^2} (x (7 a B+A b)+5 a A)}{35 a x^{7/2}}\) |
Input:
Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^(9/2),x]
Output:
(-2*(5*a*A + (A*b + 7*a*B)*x)*Sqrt[a + b*x + c*x^2])/(35*a*x^(7/2)) - ((-2 *(4*A*b^2 - 7*a*b*B - 10*a*A*c)*Sqrt[a + b*x + c*x^2])/(3*a*x^(3/2)) + ((- 2*(14*a*B*(b^2 - 3*a*c) - A*(8*b^3 - 29*a*b*c))*Sqrt[a + b*x + c*x^2])/(a* Sqrt[x]) - (2*c*(((14*a*B*(b^2 - 3*a*c) - A*(8*b^3 - 29*a*b*c))*(-((Sqrt[x ]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt [c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[ (c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqrt[c] - (a^(1/4)*(14*a*B*(b^2 - 3*a*c) - Sqrt[a]*Sqrt[ c]*(4*A*b^2 - 7*a*b*B - 10*a*A*c) - A*(8*b^3 - 29*a*b*c))*(Sqrt[a] + Sqrt[ c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[( c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x + c*x^2])))/a)/(3*a))/(35*a)
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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Time = 2.68 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.42
| 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}}{7 x^{4}}-\frac {2 \left (A b +7 B a \right ) \sqrt {c \,x^{3}+b \,x^{2}+a x}}{35 a \,x^{3}}-\frac {2 \left (10 A a c -4 b^{2} A +7 a b B \right ) \sqrt {c \,x^{3}+b \,x^{2}+a x}}{105 a^{2} x^{2}}+\frac {2 \left (c \,x^{2}+b x +a \right ) \left (29 A a b c -8 A \,b^{3}-42 B \,a^{2} c +14 B a \,b^{2}\right )}{105 a^{3} \sqrt {x \left (c \,x^{2}+b x +a \right )}}-\frac {\left (10 A a c -4 b^{2} A +7 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}}}}\, \operatorname {EllipticF}\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 )}{105 a^{2} \sqrt {c \,x^{3}+b \,x^{2}+a x}}-\frac {\left (29 A a b c -8 A \,b^{3}-42 B \,a^{2} c +14 B a \,b^{2}\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 ) \operatorname {EllipticE}\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 ) \operatorname {EllipticF}\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 )}{105 a^{3} \sqrt {c \,x^{3}+b \,x^{2}+a x}}\right )}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(903\) |
| risch | \(\text {Expression too large to display}\) | \(1377\) |
| default | \(\text {Expression too large to display}\) | \(3019\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(9/2),x,method=_RETURNVERBOSE)
Output:
(x*(c*x^2+b*x+a))^(1/2)/x^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/7*A/x^4*(c*x^3+b*x ^2+a*x)^(1/2)-2/35*(A*b+7*B*a)/a*(c*x^3+b*x^2+a*x)^(1/2)/x^3-2/105/a^2*(10 *A*a*c-4*A*b^2+7*B*a*b)*(c*x^3+b*x^2+a*x)^(1/2)/x^2+2/105*(c*x^2+b*x+a)/a^ 3*(29*A*a*b*c-8*A*b^3-42*B*a^2*c+14*B*a*b^2)/(x*(c*x^2+b*x+a))^(1/2)-1/105 *(10*A*a*c-4*A*b^2+7*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)*E llipticF(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))-1/105*(29*A*a*b*c-8*A*b^3-42*B*a^2*c +14*B*a*b^2)/a^3*(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+...
Time = 0.10 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.51 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\frac {2 \, {\left ({\left (14 \, B a b^{3} - 8 \, A b^{4} - 30 \, A a^{2} c^{2} - {\left (63 \, B a^{2} b - 41 \, A a b^{2}\right )} c\right )} \sqrt {c} x^{4} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) - 3 \, {\left ({\left (42 \, B a^{2} - 29 \, A a b\right )} c^{2} - 2 \, {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} c\right )} \sqrt {c} x^{4} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (15 \, A a^{3} c + {\left ({\left (42 \, B a^{2} - 29 \, A a b\right )} c^{2} - 2 \, {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} c\right )} x^{3} + 3 \, {\left (7 \, B a^{3} + A a^{2} b\right )} c x + {\left (10 \, A a^{2} c^{2} + {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} c\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{315 \, a^{3} c x^{4}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(9/2),x, algorithm="fricas")
Output:
2/315*((14*B*a*b^3 - 8*A*b^4 - 30*A*a^2*c^2 - (63*B*a^2*b - 41*A*a*b^2)*c) *sqrt(c)*x^4*weierstrassPInverse(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*((42*B*a^2 - 29*A*a*b)*c^2 - 2*(7*B*a*b^ 2 - 4*A*b^3)*c)*sqrt(c)*x^4*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*( 2*b^3 - 9*a*b*c)/c^3, weierstrassPInverse(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*(15*A*a^3*c + ((42*B*a^2 - 29* A*a*b)*c^2 - 2*(7*B*a*b^2 - 4*A*b^3)*c)*x^3 + 3*(7*B*a^3 + A*a^2*b)*c*x + (10*A*a^2*c^2 + (7*B*a^2*b - 4*A*a*b^2)*c)*x^2)*sqrt(c*x^2 + b*x + a)*sqrt (x))/(a^3*c*x^4)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{\frac {9}{2}}}\, dx \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**(9/2),x)
Output:
Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**(9/2), x)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(9/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^(9/2), x)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {9}{2}}} \,d x } \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(9/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^(9/2), x)
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^{9/2}} \,d x \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^(9/2),x)
Output:
int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^(9/2), x)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{9/2}} \, dx=\frac {-36 \sqrt {c \,x^{2}+b x +a}\, a -84 \sqrt {c \,x^{2}+b x +a}\, b x -66 \sqrt {x}\, \left (\int \frac {\sqrt {c \,x^{2}+b x +a}}{\sqrt {x}\, a \,x^{3}+\sqrt {x}\, b \,x^{4}+\sqrt {x}\, c \,x^{5}}d x \right ) a b \,x^{3}-55 \sqrt {x}\, \left (\int \frac {\sqrt {c \,x^{2}+b x +a}}{\sqrt {x}\, a \,x^{2}+\sqrt {x}\, b \,x^{3}+\sqrt {x}\, c \,x^{4}}d x \right ) a c \,x^{3}+91 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{5}+b \,x^{4}+a \,x^{3}}d x \right ) a c \,x^{3}-42 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{5}+b \,x^{4}+a \,x^{3}}d x \right ) b^{2} x^{3}}{126 \sqrt {x}\, x^{3}} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(9/2),x)
Output:
( - 36*sqrt(a + b*x + c*x**2)*a - 84*sqrt(a + b*x + c*x**2)*b*x - 66*sqrt( x)*int(sqrt(a + b*x + c*x**2)/(sqrt(x)*a*x**3 + sqrt(x)*b*x**4 + sqrt(x)*c *x**5),x)*a*b*x**3 - 55*sqrt(x)*int(sqrt(a + b*x + c*x**2)/(sqrt(x)*a*x**2 + sqrt(x)*b*x**3 + sqrt(x)*c*x**4),x)*a*c*x**3 + 91*sqrt(x)*int((sqrt(x)* sqrt(a + b*x + c*x**2))/(a*x**3 + b*x**4 + c*x**5),x)*a*c*x**3 - 42*sqrt(x )*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x**3 + b*x**4 + c*x**5),x)*b**2* x**3)/(126*sqrt(x)*x**3)