Integrand size = 26, antiderivative size = 86 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=-\frac {2 A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b \sqrt {d}}-\frac {2 (b B-A c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} \sqrt {c d-b e}} \]
-2*A*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(1/2)-2*(-A*c+B*b)*arctanh(c^(1/2) *(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b/c^(1/2)/(-b*e+c*d)^(1/2)
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=\frac {2 \left (\frac {(b B-A c) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {c} \sqrt {-c d+b e}}-\frac {A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}\right )}{b} \]
(2*(((b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(Sqrt [c]*Sqrt[-(c*d) + b*e]) - (A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d]))/b
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1197, 25, 1480, 221}
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.
| \(\displaystyle \int \frac {A+B x}{\left (b x+c x^2\right ) \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle 2 \int -\frac {B d-A e-B (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {B d-A e-B (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle 2 \left (\frac {(b B-A c) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b}+\frac {A c \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (-\frac {(b B-A c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} \sqrt {c d-b e}}-\frac {A \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b \sqrt {d}}\right )\) |
2*(-((A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d])) - ((b*B - A*c)*ArcTan h[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*Sqrt[c*d - b*e]))
3.13.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {2 \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{\sqrt {d}}}{b}\) | \(69\) |
| derivativedivides | \(\frac {2 \left (-A c +B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \sqrt {d}}\) | \(70\) |
| default | \(\frac {2 \left (-A c +B b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \sqrt {d}}\) | \(70\) |
2/b*(-(A*c-B*b)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))/((b*e-c*d)*c)^ (1/2)-A*arctanh((e*x+d)^(1/2)/d^(1/2))/d^(1/2))
Time = 0.34 (sec) , antiderivative size = 489, normalized size of antiderivative = 5.69 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=\left [-\frac {\sqrt {c^{2} d - b c e} {\left (B b - A c\right )} d \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {c^{2} d - b c e} \sqrt {e x + d}}{c x + b}\right ) - {\left (A c^{2} d - A b c e\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right )}{b c^{2} d^{2} - b^{2} c d e}, \frac {2 \, \sqrt {-c^{2} d + b c e} {\left (B b - A c\right )} d \arctan \left (\frac {\sqrt {-c^{2} d + b c e} \sqrt {e x + d}}{c e x + c d}\right ) + {\left (A c^{2} d - A b c e\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right )}{b c^{2} d^{2} - b^{2} c d e}, -\frac {\sqrt {c^{2} d - b c e} {\left (B b - A c\right )} d \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {c^{2} d - b c e} \sqrt {e x + d}}{c x + b}\right ) - 2 \, {\left (A c^{2} d - A b c e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right )}{b c^{2} d^{2} - b^{2} c d e}, \frac {2 \, {\left (\sqrt {-c^{2} d + b c e} {\left (B b - A c\right )} d \arctan \left (\frac {\sqrt {-c^{2} d + b c e} \sqrt {e x + d}}{c e x + c d}\right ) + {\left (A c^{2} d - A b c e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right )\right )}}{b c^{2} d^{2} - b^{2} c d e}\right ] \]
[-(sqrt(c^2*d - b*c*e)*(B*b - A*c)*d*log((c*e*x + 2*c*d - b*e + 2*sqrt(c^2 *d - b*c*e)*sqrt(e*x + d))/(c*x + b)) - (A*c^2*d - A*b*c*e)*sqrt(d)*log((e *x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/(b*c^2*d^2 - b^2*c*d*e), (2*sqrt(- c^2*d + b*c*e)*(B*b - A*c)*d*arctan(sqrt(-c^2*d + b*c*e)*sqrt(e*x + d)/(c* e*x + c*d)) + (A*c^2*d - A*b*c*e)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt( d) + 2*d)/x))/(b*c^2*d^2 - b^2*c*d*e), -(sqrt(c^2*d - b*c*e)*(B*b - A*c)*d *log((c*e*x + 2*c*d - b*e + 2*sqrt(c^2*d - b*c*e)*sqrt(e*x + d))/(c*x + b) ) - 2*(A*c^2*d - A*b*c*e)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d))/(b*c^ 2*d^2 - b^2*c*d*e), 2*(sqrt(-c^2*d + b*c*e)*(B*b - A*c)*d*arctan(sqrt(-c^2 *d + b*c*e)*sqrt(e*x + d)/(c*e*x + c*d)) + (A*c^2*d - A*b*c*e)*sqrt(-d)*ar ctan(sqrt(e*x + d)*sqrt(-d)/d))/(b*c^2*d^2 - b^2*c*d*e)]
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (78) = 156\).
Time = 10.56 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.00 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {A e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b \sqrt {- d}} + \frac {e \left (- A c + B b\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b c \sqrt {\frac {b e - c d}{c}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {B \log {\left (b x + c x^{2} \right )}}{2 c} + \left (A - \frac {B b}{2 c}\right ) \left (- \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\- \frac {\log {\left (b - 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b} - \frac {2 c \left (\begin {cases} \frac {\frac {b}{2 c} + x}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + 2 c \left (\frac {b}{2 c} + x\right ) \right )}}{2 c} & \text {otherwise} \end {cases}\right )}{b}\right )}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Piecewise((2*(A*e*atan(sqrt(d + e*x)/sqrt(-d))/(b*sqrt(-d)) + e*(-A*c + B* b)*atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*c*sqrt((b*e - c*d)/c)))/e, N e(e, 0)), ((B*log(b*x + c*x**2)/(2*c) + (A - B*b/(2*c))*(-2*c*Piecewise((( b/(2*c) + x)/b, Eq(c, 0)), (-log(b - 2*c*(b/(2*c) + x))/(2*c), True))/b - 2*c*Piecewise(((b/(2*c) + x)/b, Eq(c, 0)), (log(b + 2*c*(b/(2*c) + x))/(2* c), True))/b))/sqrt(d), True))
Exception generated. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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-c*d>0)', see `assume?` for m ore detail
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=\frac {2 \, {\left (B b - A c\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b} + \frac {2 \, A \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b \sqrt {-d}} \]
2*(B*b - A*c)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b) + 2*A*arctan(sqrt(e*x + d)/sqrt(-d))/(b*sqrt(-d))
Time = 10.61 (sec) , antiderivative size = 1130, normalized size of antiderivative = 13.14 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx=-\frac {2\,A\,\mathrm {atanh}\left (\frac {16\,A^3\,c^2\,e^3\,\sqrt {d+e\,x}}{d^{3/2}\,\left (\frac {16\,A^3\,c^2\,e^3}{d}-32\,A^2\,B\,c^2\,e^2+16\,A\,B^2\,b\,c\,e^2\right )}-\frac {32\,A^2\,B\,c^2\,e^2\,\sqrt {d+e\,x}}{\sqrt {d}\,\left (\frac {16\,A^3\,c^2\,e^3}{d}-32\,A^2\,B\,c^2\,e^2+16\,A\,B^2\,b\,c\,e^2\right )}+\frac {16\,A\,B^2\,b\,c\,e^2\,\sqrt {d+e\,x}}{\sqrt {d}\,\left (\frac {16\,A^3\,c^2\,e^3}{d}-32\,A^2\,B\,c^2\,e^2+16\,A\,B^2\,b\,c\,e^2\right )}\right )}{b\,\sqrt {d}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {d+e\,x}\,\left (16\,A^2\,c^3\,e^2-16\,A\,B\,b\,c^2\,e^2+8\,B^2\,b^2\,c\,e^2\right )+\frac {\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (8\,B\,b^2\,c^2\,d\,e^2-8\,A\,b^2\,c^2\,e^3+\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}}{b\,c^2\,d-b^2\,c\,e}\right )}{b\,c^2\,d-b^2\,c\,e}\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,1{}\mathrm {i}}{b\,c^2\,d-b^2\,c\,e}+\frac {\left (\sqrt {d+e\,x}\,\left (16\,A^2\,c^3\,e^2-16\,A\,B\,b\,c^2\,e^2+8\,B^2\,b^2\,c\,e^2\right )+\frac {\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (8\,A\,b^2\,c^2\,e^3-8\,B\,b^2\,c^2\,d\,e^2+\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}}{b\,c^2\,d-b^2\,c\,e}\right )}{b\,c^2\,d-b^2\,c\,e}\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,1{}\mathrm {i}}{b\,c^2\,d-b^2\,c\,e}}{\frac {\left (\sqrt {d+e\,x}\,\left (16\,A^2\,c^3\,e^2-16\,A\,B\,b\,c^2\,e^2+8\,B^2\,b^2\,c\,e^2\right )+\frac {\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (8\,B\,b^2\,c^2\,d\,e^2-8\,A\,b^2\,c^2\,e^3+\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}}{b\,c^2\,d-b^2\,c\,e}\right )}{b\,c^2\,d-b^2\,c\,e}\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}}{b\,c^2\,d-b^2\,c\,e}-\frac {\left (\sqrt {d+e\,x}\,\left (16\,A^2\,c^3\,e^2-16\,A\,B\,b\,c^2\,e^2+8\,B^2\,b^2\,c\,e^2\right )+\frac {\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (8\,A\,b^2\,c^2\,e^3-8\,B\,b^2\,c^2\,d\,e^2+\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\sqrt {d+e\,x}}{b\,c^2\,d-b^2\,c\,e}\right )}{b\,c^2\,d-b^2\,c\,e}\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}}{b\,c^2\,d-b^2\,c\,e}+16\,A^2\,B\,c^2\,e^2-16\,A\,B^2\,b\,c\,e^2}\right )\,\left (A\,c-B\,b\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,2{}\mathrm {i}}{b\,c^2\,d-b^2\,c\,e} \]
- (2*A*atanh((16*A^3*c^2*e^3*(d + e*x)^(1/2))/(d^(3/2)*((16*A^3*c^2*e^3)/d - 32*A^2*B*c^2*e^2 + 16*A*B^2*b*c*e^2)) - (32*A^2*B*c^2*e^2*(d + e*x)^(1/ 2))/(d^(1/2)*((16*A^3*c^2*e^3)/d - 32*A^2*B*c^2*e^2 + 16*A*B^2*b*c*e^2)) + (16*A*B^2*b*c*e^2*(d + e*x)^(1/2))/(d^(1/2)*((16*A^3*c^2*e^3)/d - 32*A^2* B*c^2*e^2 + 16*A*B^2*b*c*e^2))))/(b*d^(1/2)) - (atan(((((d + e*x)^(1/2)*(1 6*A^2*c^3*e^2 + 8*B^2*b^2*c*e^2 - 16*A*B*b*c^2*e^2) + ((A*c - B*b)*(-c*(b* e - c*d))^(1/2)*(8*B*b^2*c^2*d*e^2 - 8*A*b^2*c^2*e^3 + ((8*b^3*c^2*e^3 - 1 6*b^2*c^3*d*e^2)*(A*c - B*b)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2))/(b*c^ 2*d - b^2*c*e)))/(b*c^2*d - b^2*c*e))*(A*c - B*b)*(-c*(b*e - c*d))^(1/2)*1 i)/(b*c^2*d - b^2*c*e) + (((d + e*x)^(1/2)*(16*A^2*c^3*e^2 + 8*B^2*b^2*c*e ^2 - 16*A*B*b*c^2*e^2) + ((A*c - B*b)*(-c*(b*e - c*d))^(1/2)*(8*A*b^2*c^2* e^3 - 8*B*b^2*c^2*d*e^2 + ((8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2)*(A*c - B*b)* (-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2))/(b*c^2*d - b^2*c*e)))/(b*c^2*d - b ^2*c*e))*(A*c - B*b)*(-c*(b*e - c*d))^(1/2)*1i)/(b*c^2*d - b^2*c*e))/((((d + e*x)^(1/2)*(16*A^2*c^3*e^2 + 8*B^2*b^2*c*e^2 - 16*A*B*b*c^2*e^2) + ((A* c - B*b)*(-c*(b*e - c*d))^(1/2)*(8*B*b^2*c^2*d*e^2 - 8*A*b^2*c^2*e^3 + ((8 *b^3*c^2*e^3 - 16*b^2*c^3*d*e^2)*(A*c - B*b)*(-c*(b*e - c*d))^(1/2)*(d + e *x)^(1/2))/(b*c^2*d - b^2*c*e)))/(b*c^2*d - b^2*c*e))*(A*c - B*b)*(-c*(b*e - c*d))^(1/2))/(b*c^2*d - b^2*c*e) - (((d + e*x)^(1/2)*(16*A^2*c^3*e^2 + 8*B^2*b^2*c*e^2 - 16*A*B*b*c^2*e^2) + ((A*c - B*b)*(-c*(b*e - c*d))^(1/...