Integrand size = 27, antiderivative size = 131 \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e}+\frac {(e f-d g) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e \sqrt {c d^2-b d e+a e^2}} \] Output:
g*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)/e+(-d*g+e*f)* arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b* x+a)^(1/2))/e/(a*e^2-b*d*e+c*d^2)^(1/2)
Time = 0.73 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08 \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {\frac {2 \sqrt {-c d^2+b d e-a e^2} (-e f+d g) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {g \log \left (e \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{\sqrt {c}}}{e} \] Input:
Integrate[(f + g*x)/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]
Output:
-(((2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(-(e*f) + d*g)*ArcTan[(Sqrt[c]*(d + e *x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/(c*d^2 + e *(-(b*d) + a*e)) + (g*Log[e*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] )/Sqrt[c])/e)
Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1269, 1092, 219, 1154, 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.
\(\displaystyle \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {g \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {2 g \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e}-\frac {2 (e f-d g) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(e f-d g) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}+\frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e}\) |
Input:
Int[(f + g*x)/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]
Output:
(g*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) + ( (e*f - d*g)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e*Sqrt[c*d^2 - b*d*e + a*e^2])
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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 2.52 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.51
method | result | size |
default | \(\frac {g \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}+\frac {\left (d g -e f \right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\) | \(198\) |
Input:
int((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
g/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+(d*g-e*f)/e^2/((a* e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x +d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e) +(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (115) = 230\).
Time = 17.02 (sec) , antiderivative size = 1071, normalized size of antiderivative = 8.18 \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:
integrate((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
[1/2*((c*d^2 - b*d*e + a*e^2)*sqrt(c)*g*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) - sqrt(c*d^2 - b*d*e + a*e^2)*(c*e*f - c*d*g)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - ( 8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a* e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)))/(c^2*d^2 *e - b*c*d*e^2 + a*c*e^3), 1/2*((c*d^2 - b*d*e + a*e^2)*sqrt(c)*g*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) + 2*sqrt(-c*d^2 + b*d*e - a*e^2)*(c*e*f - c*d*g)*arctan(-1/2*sqrt(-c*d ^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/ (a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^ 2 - b^2*d*e + a*b*e^2)*x)))/(c^2*d^2*e - b*c*d*e^2 + a*c*e^3), -1/2*(2*(c* d^2 - b*d*e + a*e^2)*sqrt(-c)*g*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)) + sqrt(c*d^2 - b*d*e + a*e^2)*(c*e*f - c*d*g)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b *c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - ( 3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)))/(c^2*d^2*e - b*c*d*e^2 + a*c*e^3), -((c*d^2 - b*d*e + a*e^2)*sqrt(-c)*g*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)) - sqrt(-c*d^2 + b...
\[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {f + g x}{\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((g*x+f)/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((f + g*x)/((d + e*x)*sqrt(a + b*x + c*x**2)), x)
Exception generated. \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)/(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((b/e-(2*c*d)/e^2)^2>0)', see `as sume?` for
Exception generated. \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {f+g\,x}{\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((f + g*x)/((d + e*x)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int((f + g*x)/((d + e*x)*(a + b*x + c*x^2)^(1/2)), x)
Time = 0.34 (sec) , antiderivative size = 7892, normalized size of antiderivative = 60.24 \[ \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:
int((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)
Output:
(2*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e** 2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2 )*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt( c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*b*c*d*e*g - 2*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) *b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e** 2 + 8*b*c*d*e - 8*c**2*d**2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c) *sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b* d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e* *2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*b*c*e**2*f - 4*sqrt(4*sqrt(c)*s qrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) *c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)*sqrt(a*e**2 - b*d *e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqr t(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b* d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*c** 2*d**2*g + 4*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)* sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8 *c**2*d**2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c *x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*...