Integrand size = 25, antiderivative size = 108 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \] Output:
2*(2*a*c*e-b*(a*f+c*d)-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x)/c/(-4*a*c+b^2)/(c *x^2+b*x+a)^(1/2)+f*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^( 3/2)
Time = 1.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (a b f+2 c^2 d x+b^2 f x+b c (d-e x)-2 a c (e+f x)\right )}{c \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {2 f \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{3/2}} \] Input:
Integrate[(d + e*x + f*x^2)/(a + b*x + c*x^2)^(3/2),x]
Output:
(2*(a*b*f + 2*c^2*d*x + b^2*f*x + b*c*(d - e*x) - 2*a*c*(e + f*x)))/(c*(-b ^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + (2*f*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/c^(3/2)
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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.
\(\displaystyle \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {\left (b^2-4 a c\right ) f}{2 c \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {f \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c}+\frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 f \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}+\frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}+\frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[(d + e*x + f*x^2)/(a + b*x + c*x^2)^(3/2),x]
Output:
(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x))/ (c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (f*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]
Leaf count of result is larger than twice the leaf count of optimal. \(200\) vs. \(2(98)=196\).
Time = 1.67 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {2 d \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+f \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\) | \(201\) |
Input:
int((f*x^2+e*x+d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*d*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+e*(-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))+f*(-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. 213 vs. \(2 (98) = 196\).
Time = 0.28 (sec) , antiderivative size = 429, normalized size of antiderivative = 3.97 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} + {\left (b^{3} - 4 \, a b c\right )} f x + {\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (b c^{2} d - 2 \, a c^{2} e + a b c f + {\left (2 \, c^{3} d - b c^{2} e + {\left (b^{2} c - 2 \, a c^{2}\right )} f\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} + {\left (b^{3} - 4 \, a b c\right )} f x + {\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (b c^{2} d - 2 \, a c^{2} e + a b c f + {\left (2 \, c^{3} d - b c^{2} e + {\left (b^{2} c - 2 \, a c^{2}\right )} f\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \] Input:
integrate((f*x^2+e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*(((b^2*c - 4*a*c^2)*f*x^2 + (b^3 - 4*a*b*c)*f*x + (a*b^2 - 4*a^2*c)*f )*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*(b*c^2*d - 2*a*c^2*e + a*b*c*f + (2*c^3*d - b*c^ 2*e + (b^2*c - 2*a*c^2)*f)*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), -(((b^2*c - 4*a*c^ 2)*f*x^2 + (b^3 - 4*a*b*c)*f*x + (a*b^2 - 4*a^2*c)*f)*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*(b *c^2*d - 2*a*c^2*e + a*b*c*f + (2*c^3*d - b*c^2*e + (b^2*c - 2*a*c^2)*f)*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)]
\[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {d + e x + f x^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((f*x**2+e*x+d)/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral((d + e*x + f*x**2)/(a + b*x + c*x**2)**(3/2), x)
Exception generated. \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x^2+e*x+d)/(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 = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.09 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (2 \, c^{2} d - b c e + b^{2} f - 2 \, a c f\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac {b c d - 2 \, a c e + a b f}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {f \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}} \] Input:
integrate((f*x^2+e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
-2*((2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x/(b^2*c - 4*a*c^2) + (b*c*d - 2*a *c*e + a*b*f)/(b^2*c - 4*a*c^2))/sqrt(c*x^2 + b*x + a) - f*log(abs(2*(sqrt (c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(3/2)
Time = 16.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.32 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {f\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{c^{3/2}}-\frac {e\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {d\,\left (\frac {b}{2}+c\,x\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {f\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \] Input:
int((d + e*x + f*x^2)/(a + b*x + c*x^2)^(3/2),x)
Output:
(f*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2)))/c^(3/2) - (e*(4*a + 2*b*x))/((4*a*c - b^2)*(a + b*x + c*x^2)^(1/2)) + (d*(b/2 + c*x))/((a*c - b^2/4)*(a + b*x + c*x^2)^(1/2)) + (f*((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 = 574, normalized size of antiderivative = 5.31 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-4 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} e +2 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} d +4 \sqrt {c \,x^{2}+b x +a}\, c^{3} d x -4 \sqrt {c}\, a^{2} c f +2 \sqrt {c}\, a \,b^{2} f +4 \sqrt {c}\, a \,c^{2} d +2 \sqrt {c}\, b^{3} f x +4 \sqrt {c}\, c^{3} d \,x^{2}+2 \sqrt {c \,x^{2}+b x +a}\, a b c f -4 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} f x +2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c f x -2 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} e x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c f -2 \sqrt {c}\, a b c e -4 \sqrt {c}\, a \,c^{2} f \,x^{2}-2 \sqrt {c}\, b^{2} c e x +2 \sqrt {c}\, b^{2} c f \,x^{2}+4 \sqrt {c}\, b \,c^{2} d x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b c f x -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} f -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} f x -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c f \,x^{2}+4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} f \,x^{2}-4 \sqrt {c}\, a b c f x -2 \sqrt {c}\, b \,c^{2} e \,x^{2}}{c^{2} \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:
int((f*x^2+e*x+d)/(c*x^2+b*x+a)^(3/2),x)
Output:
(2*sqrt(a + b*x + c*x**2)*a*b*c*f - 4*sqrt(a + b*x + c*x**2)*a*c**2*e - 4* sqrt(a + b*x + c*x**2)*a*c**2*f*x + 2*sqrt(a + b*x + c*x**2)*b**2*c*f*x + 2*sqrt(a + b*x + c*x**2)*b*c**2*d - 2*sqrt(a + b*x + c*x**2)*b*c**2*e*x + 4*sqrt(a + b*x + c*x**2)*c**3*d*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**2*c*f - 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**2*f + 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*b*c*f*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*f*x**2 - sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*f*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*f*x**2 - 4*sqrt(c)*a**2*c*f + 2*sqrt(c)*a*b**2*f - 2*sqrt(c)*a*b*c*e - 4*sqrt(c)*a *b*c*f*x + 4*sqrt(c)*a*c**2*d - 4*sqrt(c)*a*c**2*f*x**2 + 2*sqrt(c)*b**3*f *x - 2*sqrt(c)*b**2*c*e*x + 2*sqrt(c)*b**2*c*f*x**2 + 4*sqrt(c)*b*c**2*d*x - 2*sqrt(c)*b*c**2*e*x**2 + 4*sqrt(c)*c**3*d*x**2)/(c**2*(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))