Integrand size = 26, antiderivative size = 95 \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right )^{3/2} d^3} \] Output:
(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/d^3/(2*c*x+b)^2+1/2*arctan(2*c^(1/2)*(c*x ^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(1/2)/(-4*a*c+b^2)^(3/2)/d^3
Time = 10.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {a+x (b+c x)} \left (\frac {2 \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac {\text {arctanh}\left (2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right )}{\sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )}{2 \left (b^2-4 a c\right )^2 d^3} \] Input:
Integrate[1/((b*d + 2*c*d*x)^3*Sqrt[a + b*x + c*x^2]),x]
Output:
(Sqrt[a + x*(b + c*x)]*((2*(b^2 - 4*a*c))/(b + 2*c*x)^2 + ArcTanh[2*Sqrt[( c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]]/Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4 *a*c)]))/(2*(b^2 - 4*a*c)^2*d^3)
Time = 0.24 (sec) , antiderivative size = 95, 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 = {1117, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} (b d+2 c d x)^3} \, dx\) |
\(\Big \downarrow \) 1117 |
\(\displaystyle \frac {\int \frac {1}{d (b+2 c x) \sqrt {c x^2+b x+a}}dx}{2 d^2 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{2 d^3 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}\) |
\(\Big \downarrow \) 1112 |
\(\displaystyle \frac {2 c \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{d^3 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} d^3 \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}\) |
Input:
Int[1/((b*d + 2*c*d*x)^3*Sqrt[a + b*x + c*x^2]),x]
Output:
Sqrt[a + b*x + c*x^2]/((b^2 - 4*a*c)*d^3*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c ]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[c]*(b^2 - 4*a*c)^(3/2) *d^3)
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[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]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Time = 1.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {c \,x^{2}+b x +a}}{4 c^{2} x^{2}+4 c b x +b^{2}}+\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {\left (4 a c -b^{2}\right ) c}}\right )}{2 \sqrt {\left (4 a c -b^{2}\right ) c}}}{\left (4 a c -b^{2}\right ) d^{3}}\) | \(97\) |
default | \(\frac {-\frac {2 c \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {4 c^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}}{8 d^{3} c^{3}}\) | \(174\) |
Input:
int(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/(4*a*c-b^2)*(-(c*x^2+b*x+a)^(1/2)/(4*c^2*x^2+4*b*c*x+b^2)+1/2/((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^3
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (81) = 162\).
Time = 0.16 (sec) , antiderivative size = 433, normalized size of antiderivative = 4.56 \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\left [\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x + {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{3}\right )}}, -\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (-\frac {2 \, \sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x + {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{3}\right )}}\right ] \] Input:
integrate(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
[1/4*((4*c^2*x^2 + 4*b*c*x + b^2)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c + 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/( 4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a))/( 4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^2 + 4*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x + (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^3), -1/2* ((4*c^2*x^2 + 4*b*c*x + b^2)*sqrt(b^2*c - 4*a*c^2)*arctan(-2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(b^2 - 4*a*c)) - 2*(b^2*c - 4*a*c^2)*sqrt(c *x^2 + b*x + a))/(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^2 + 4*(b^5* c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x + (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^ 2*c^3)*d^3)]
\[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {\int \frac {1}{b^{3} \sqrt {a + b x + c x^{2}} + 6 b^{2} c x \sqrt {a + b x + c x^{2}} + 12 b c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 8 c^{3} x^{3} \sqrt {a + b x + c x^{2}}}\, dx}{d^{3}} \] Input:
integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral(1/(b**3*sqrt(a + b*x + c*x**2) + 6*b**2*c*x*sqrt(a + b*x + c*x**2 ) + 12*b*c**2*x**2*sqrt(a + b*x + c*x**2) + 8*c**3*x**3*sqrt(a + b*x + c*x **2)), x)/d**3
Exception generated. \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(1/2),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. 267 vs. \(2 (81) = 162\).
Time = 0.39 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.81 \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {\arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )} \sqrt {b^{2} c - 4 \, a c^{2}}} - \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b \sqrt {c} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c + a b \sqrt {c}}{{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )} {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{2}} \] Input:
integrate(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/((b^2*d^3 - 4*a*c*d^3)*sqrt(b^2*c - 4*a*c^2)) - (2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*sq rt(c) + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2 + 2*(sqrt(c)*x - sqrt(c*x^ 2 + b*x + a))*a*c + a*b*sqrt(c))/((b^2*d^3 - 4*a*c*d^3)*(2*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^2)
Timed out. \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^(1/2)), x)
Time = 0.27 (sec) , antiderivative size = 529, normalized size of antiderivative = 5.57 \[ \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {-\sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}-4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b c x -4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{2} x^{2}+\sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}+4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b c x +4 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{2} x^{2}-8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2}+2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c}{2 c \,d^{3} \left (64 a^{2} c^{4} x^{2}-32 a \,b^{2} c^{3} x^{2}+4 b^{4} c^{2} x^{2}+64 a^{2} b \,c^{3} x -32 a \,b^{3} c^{2} x +4 b^{5} c x +16 a^{2} b^{2} c^{2}-8 a \,b^{4} c +b^{6}\right )} \] Input:
int(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(1/2),x)
Output:
( - 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**2 - 4*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*c*x - 4*sqrt(c)*sqrt(4*a*c - b**2)*lo g(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s qrt(4*a*c - b**2))*c**2*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) )*b**2 + 4*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*c*x + 4*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**2*x**2 - 8*sqrt(a + b*x + c*x**2)* a*c**2 + 2*sqrt(a + b*x + c*x**2)*b**2*c)/(2*c*d**3*(16*a**2*b**2*c**2 + 6 4*a**2*b*c**3*x + 64*a**2*c**4*x**2 - 8*a*b**4*c - 32*a*b**3*c**2*x - 32*a *b**2*c**3*x**2 + b**6 + 4*b**5*c*x + 4*b**4*c**2*x**2))