Integrand size = 22, antiderivative size = 96 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (b (A c+a C)+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {C \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \] Output:
(-2*b*(A*c+C*a)-2*(2*A*c^2+(-2*a*c+b^2)*C)*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a) ^(1/2)+C*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)
Time = 1.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.99 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b^2 C x+a C (b-2 c x)+A c (b+2 c x)\right )}{c \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {2 C \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{3/2}} \] Input:
Integrate[(A + C*x^2)/(a + b*x + c*x^2)^(3/2),x]
Output:
(2*(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x)))/(c*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + (2*C*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c* x)])])/c^(3/2)
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2191, 27, 1092, 219}
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.
\(\displaystyle \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle -\frac {2 \int -\frac {\left (b^2-4 a c\right ) C}{2 c \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {C \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 C \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {C \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[(A + C*x^2)/(a + b*x + c*x^2)^(3/2),x]
Output:
(-2*(b*c*(A + (a*C)/c) + (2*A*c^2 + (b^2 - 2*a*c)*C)*x))/(c*(b^2 - 4*a*c)* Sqrt[a + b*x + c*x^2]) + (C*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(3/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 1.53 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.51
method | result | size |
default | \(\frac {2 A \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+C \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\) | \(145\) |
Input:
int((C*x^2+A)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*A*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+C*(-x/c/(c*x^2+b*x+a)^(1/2)- 1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^ (1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (86) = 172\).
Time = 0.15 (sec) , antiderivative size = 403, normalized size of antiderivative = 4.20 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (C a b^{2} - 4 \, C a^{2} c + {\left (C b^{2} c - 4 \, C a c^{2}\right )} x^{2} + {\left (C b^{3} - 4 \, C a b c\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (C a b c + A b c^{2} + {\left (C b^{2} c - 2 \, C a c^{2} + 2 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac {{\left (C a b^{2} - 4 \, C a^{2} c + {\left (C b^{2} c - 4 \, C a c^{2}\right )} x^{2} + {\left (C b^{3} - 4 \, C a b c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (C a b c + A b c^{2} + {\left (C b^{2} c - 2 \, C a c^{2} + 2 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \] Input:
integrate((C*x^2+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((C*a*b^2 - 4*C*a^2*c + (C*b^2*c - 4*C*a*c^2)*x^2 + (C*b^3 - 4*C*a*b* c)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2* c*x + b)*sqrt(c) - 4*a*c) - 4*(C*a*b*c + A*b*c^2 + (C*b^2*c - 2*C*a*c^2 + 2*A*c^3)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a *c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x), -((C*a*b^2 - 4*C*a^2*c + (C*b^2*c - 4*C*a*c^2)*x^2 + (C*b^3 - 4*C*a*b*c)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b *x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(C*a*b*c + A*b*c ^2 + (C*b^2*c - 2*C*a*c^2 + 2*A*c^3)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)]
\[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + C x^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((C*x**2+A)/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral((A + C*x**2)/(a + b*x + c*x**2)**(3/2), x)
Exception generated. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((C*x^2+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.12 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (C b^{2} - 2 \, C a c + 2 \, A c^{2}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac {C a b + A b c}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {C \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}} \] Input:
integrate((C*x^2+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
-2*((C*b^2 - 2*C*a*c + 2*A*c^2)*x/(b^2*c - 4*a*c^2) + (C*a*b + A*b*c)/(b^2 *c - 4*a*c^2))/sqrt(c*x^2 + b*x + a) - C*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(3/2)
Time = 18.53 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.12 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {C\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{c^{3/2}}+\frac {A\,\left (\frac {b}{2}+c\,x\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {C\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \] Input:
int((A + C*x^2)/(a + b*x + c*x^2)^(3/2),x)
Output:
(C*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2)))/c^(3/2) + (A*(b/2 + c*x))/((a*c - b^2/4)*(a + b*x + c*x^2)^(1/2)) + (C*((a*b)/2 - x*(a*c - b^ 2/2)))/(c*(a*c - b^2/4)*(a + b*x + c*x^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.97 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {4 \sqrt {c \,x^{2}+b x +a}\, a b c +2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2}+4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b c x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} x^{2}-\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} x -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c \,x^{2}+2 \sqrt {c}\, a \,b^{2}+2 \sqrt {c}\, b^{3} x +2 \sqrt {c}\, b^{2} c \,x^{2}}{c \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:
int((C*x^2+A)/(c*x^2+b*x+a)^(3/2),x)
Output:
(4*sqrt(a + b*x + c*x**2)*a*b*c + 2*sqrt(a + b*x + c*x**2)*b**2*c*x + 4*sq rt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2 ))*a**2*c - sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqr t(4*a*c - b**2))*a*b**2 + 4*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*x + 4*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*x**2 - sqrt(c)*lo g((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3* x - sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*x**2 + 2*sqrt(c)*a*b**2 + 2*sqrt(c)*b**3*x + 2*sqrt(c)*b** 2*c*x**2)/(c*(4*a**2*c - a*b**2 + 4*a*b*c*x + 4*a*c**2*x**2 - b**3*x - b** 2*c*x**2))