Integrand size = 26, antiderivative size = 86 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} d} \] Output:
-2/(-4*a*c+b^2)/d/(c*x^2+b*x+a)^(1/2)-4*c^(1/2)*arctan(2*c^(1/2)*(c*x^2+b* x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/d
Time = 0.76 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-\frac {2}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}-\frac {8 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{\left (-b^2+4 a c\right )^{3/2}}}{d} \] Input:
Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]
Output:
(-2/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]) - (8*Sqrt[c]*ArcTanh[(Sqrt[c]*Sq rt[-b^2 + 4*a*c]*x)/(Sqrt[a]*(b + 2*c*x) - b*Sqrt[a + x*(b + c*x)])])/(-b^ 2 + 4*a*c)^(3/2))/d
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1111, 27, 1112, 218}
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.
\(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)} \, dx\) |
\(\Big \downarrow \) 1111 |
\(\displaystyle -\frac {4 c \int \frac {1}{d (b+2 c x) \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 c \int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{d \left (b^2-4 a c\right )}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1112 |
\(\displaystyle -\frac {16 c^2 \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{d \left (b^2-4 a c\right )}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {4 \sqrt {c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}}-\frac {2}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]
Output:
-2/((b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (4*Sqrt[c]*ArcTan[(2*Sqrt[c]* Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*d)
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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] - Simp[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*c))) 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] && LtQ[p, -1] && !G tQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb ol] :> Simp[4*c Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Time = 1.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {\frac {2}{\sqrt {c \,x^{2}+b x +a}}-\frac {4 c \,\operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {\left (4 a c -b^{2}\right ) c}}\right )}{\sqrt {\left (4 a c -b^{2}\right ) c}}}{\left (4 a c -b^{2}\right ) d}\) | \(78\) |
default | \(\frac {\frac {4 c}{\left (4 a c -b^{2}\right ) \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}-\frac {8 c \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}}{2 d c}\) | \(162\) |
Input:
int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/(4*a*c-b^2)*(1/(c*x^2+b*x+a)^(1/2)-2*c/((4*a*c-b^2)*c)^(1/2)*arctanh(2*( c*x^2+b*x+a)^(1/2)*c/((4*a*c-b^2)*c)^(1/2)))/d
Time = 0.16 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.62 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {2 \, {\left ({\left (c x^{2} + b x + a\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + \sqrt {c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 4 \, a b c\right )} d x + {\left (a b^{2} - 4 \, a^{2} c\right )} d}, -\frac {2 \, {\left (2 \, {\left (c x^{2} + b x + a\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + \sqrt {c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 4 \, a b c\right )} d x + {\left (a b^{2} - 4 \, a^{2} c\right )} d}\right ] \] Input:
integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
[-2*((c*x^2 + b*x + a)*sqrt(-c/(b^2 - 4*a*c))*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c + 4*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*sqrt(-c/(b^2 - 4*a*c)) )/(4*c^2*x^2 + 4*b*c*x + b^2)) + sqrt(c*x^2 + b*x + a))/((b^2*c - 4*a*c^2) *d*x^2 + (b^3 - 4*a*b*c)*d*x + (a*b^2 - 4*a^2*c)*d), -2*(2*(c*x^2 + b*x + a)*sqrt(c/(b^2 - 4*a*c))*arctan(-1/2*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*s qrt(c/(b^2 - 4*a*c))/(c^2*x^2 + b*c*x + a*c)) + sqrt(c*x^2 + b*x + a))/((b ^2*c - 4*a*c^2)*d*x^2 + (b^3 - 4*a*b*c)*d*x + (a*b^2 - 4*a^2*c)*d)]
\[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\int \frac {1}{a b \sqrt {a + b x + c x^{2}} + 2 a c x \sqrt {a + b x + c x^{2}} + b^{2} x \sqrt {a + b x + c x^{2}} + 3 b c x^{2} \sqrt {a + b x + c x^{2}} + 2 c^{2} x^{3} \sqrt {a + b x + c x^{2}}}\, dx}{d} \] Input:
integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral(1/(a*b*sqrt(a + b*x + c*x**2) + 2*a*c*x*sqrt(a + b*x + c*x**2) + b**2*x*sqrt(a + b*x + c*x**2) + 3*b*c*x**2*sqrt(a + b*x + c*x**2) + 2*c**2 *x**3*sqrt(a + b*x + c*x**2)), x)/d
Exception generated. \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(2*c*d*x+b*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
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (74) = 148\).
Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {8 \, c \arctan \left (\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{\sqrt {b^{2} c - 4 \, a c^{2}} {\left (b^{2} d - 4 \, a c d\right )}} - \frac {2 \, {\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )}}{{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )} \sqrt {c x^{2} + b x + a}} \] Input:
integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
8*c*arctan((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2* c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c^2)*(b^2*d - 4*a*c*d)) - 2*(b^4*d - 8*a*b ^2*c*d + 16*a^2*c^2*d)/((b^6*d^2 - 12*a*b^4*c*d^2 + 48*a^2*b^2*c^2*d^2 - 6 4*a^3*c^3*d^2)*sqrt(c*x^2 + b*x + a))
Timed out. \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (b\,d+2\,c\,d\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int(1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x)
Output:
int(1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)), x)
Time = 0.20 (sec) , antiderivative size = 508, normalized size of antiderivative = 5.91 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a +4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b x +4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c \,x^{2}-4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a -4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b x -4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c \,x^{2}+8 \sqrt {c \,x^{2}+b x +a}\, a c -2 \sqrt {c \,x^{2}+b x +a}\, b^{2}}{d \left (16 a^{2} c^{3} x^{2}-8 a \,b^{2} c^{2} x^{2}+b^{4} c \,x^{2}+16 a^{2} b \,c^{2} x -8 a \,b^{3} c x +b^{5} x +16 a^{3} c^{2}-8 a^{2} b^{2} c +a \,b^{4}\right )} \] Input:
int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x)
Output:
(2*(2*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sq rt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a + 2*sqrt(c)*sqrt(4 *a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*x + 2*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt (4*a*c - b**2))*c*x**2 - 2*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b* *2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a - 2*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*x - 2*sqrt(c)*sqrt(4*a* c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*c*x**2 + 4*sqrt(a + b*x + c*x**2)*a*c - sqrt(a + b*x + c*x**2)*b**2))/(d*(16*a**3*c**2 - 8*a**2*b**2*c + 16*a**2*b*c**2* x + 16*a**2*c**3*x**2 + a*b**4 - 8*a*b**3*c*x - 8*a*b**2*c**2*x**2 + b**5* x + b**4*c*x**2))