Integrand size = 26, antiderivative size = 158 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=-\frac {(A b-(b B-2 A c) x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}-\frac {(2 b B d-4 A c d+A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 \sqrt {d}}+\frac {\left (2 b B c d-4 A c^2 d-b^2 B e+3 A b c e\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c} \sqrt {c d-b e}} \]
-(A*b*e-4*A*c*d+2*B*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(1/2)+(3*A*b *c*e-4*A*c^2*d-B*b^2*e+2*B*b*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d) ^(1/2))/b^3/c^(1/2)/(-b*e+c*d)^(1/2)-(A*b-(-2*A*c+B*b)*x)*(e*x+d)^(1/2)/b^ 2/(c*x^2+b*x)
Time = 0.55 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \sqrt {d+e x} (b B x-A (b+2 c x))}{x (b+c x)}+\frac {\left (4 A c^2 d+b^2 B e-b c (2 B d+3 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {c} \sqrt {-c d+b e}}-\frac {(2 b B d-4 A c d+A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{b^3} \]
((b*Sqrt[d + e*x]*(b*B*x - A*(b + 2*c*x)))/(x*(b + c*x)) + ((4*A*c^2*d + b ^2*B*e - b*c*(2*B*d + 3*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(Sqrt[c]*Sqrt[-(c*d) + b*e]) - ((2*b*B*d - 4*A*c*d + A*b*e)*ArcTan h[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d])/b^3
Time = 0.41 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1234, 27, 25, 1197, 27, 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.
\(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1234 |
\(\displaystyle -\frac {\int \frac {4 A c d-b (2 B d+A e)-(b B-2 A c) e x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -\frac {2 b B d-4 A c d+A b e+(b B-2 A c) e x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 b B d-4 A c d+A b e+(b B-2 A c) e x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle \frac {\int \frac {e (b B d-2 A c d+A b e+(b B-2 A c) (d+e x))}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \int \frac {b B d-2 A c d+A b e+(b B-2 A c) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle \frac {e \left (\frac {c (A b e-4 A c d+2 b B d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {\left (3 A b c e-4 A c^2 d+b^2 (-B) e+2 b B c d\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e \left (\frac {\left (3 A b c e-4 A c^2 d+b^2 (-B) e+2 b B c d\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c} e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (A b e-4 A c d+2 b B d)}{b \sqrt {d} e}\right )}{b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}\) |
-(((A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(b^2*(b*x + c*x^2))) + (e*(-(((2 *b*B*d - 4*A*c*d + A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + ((2*b*B*c*d - 4*A*c^2*d - b^2*B*e + 3*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*Sqrt[c]*e*Sqrt[c*d - b*e])))/b^2
3.13.42.3.1 Defintions of rubi rules used
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)*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_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(-\frac {-4 \left (A \,c^{2} d -\frac {3 b \left (A e +\frac {2 B d}{3}\right ) c}{4}+\frac {b^{2} B e}{4}\right ) x \left (c x +b \right ) \sqrt {d}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\left (x \left (c x +b \right ) \left (-4 A c d +b \left (A e +2 B d \right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, \left (\left (-B x +A \right ) b +2 A c x \right ) \sqrt {d}\, b \right ) \sqrt {\left (b e -c d \right ) c}}{\sqrt {\left (b e -c d \right ) c}\, \sqrt {d}\, \left (c x +b \right ) b^{3} x}\) | \(166\) |
derivativedivides | \(2 e^{2} \left (-\frac {\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A b c e -4 A \,c^{2} d -b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{e^{2} b^{3}}+\frac {-\frac {A b \sqrt {e x +d}}{2 x}-\frac {\left (A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{e^{2} b^{3}}\right )\) | \(172\) |
default | \(2 e^{2} \left (-\frac {\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A b c e -4 A \,c^{2} d -b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{e^{2} b^{3}}+\frac {-\frac {A b \sqrt {e x +d}}{2 x}-\frac {\left (A b e -4 A c d +2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{e^{2} b^{3}}\right )\) | \(172\) |
risch | \(-\frac {A \sqrt {e x +d}}{b^{2} x}-\frac {e \left (\frac {\frac {2 \left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 A b c e -4 A \,c^{2} d -b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}}}{b e}-\frac {\left (-A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}\right )}{b^{2}}\) | \(175\) |
-1/((b*e-c*d)*c)^(1/2)/d^(1/2)*(-4*(A*c^2*d-3/4*b*(A*e+2/3*B*d)*c+1/4*b^2* B*e)*x*(c*x+b)*d^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))+(x*(c*x +b)*(-4*A*c*d+b*(A*e+2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))+(e*x+d)^(1/2)* ((-B*x+A)*b+2*A*c*x)*d^(1/2)*b)*((b*e-c*d)*c)^(1/2))/(c*x+b)/b^3/x
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (140) = 280\).
Time = 0.52 (sec) , antiderivative size = 1574, normalized size of antiderivative = 9.96 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[1/2*(sqrt(c^2*d - b*c*e)*((2*(B*b*c^2 - 2*A*c^3)*d^2 - (B*b^2*c - 3*A*b*c ^2)*d*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^2 - (B*b^3 - 3*A*b^2*c)*d*e)*x)* 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*b^2*c^2*e^2 - 2*(B*b*c^3 - 2*A*c^4)*d^2 + (2*B*b^2*c^2 - 5*A*b*c^3) *d*e)*x^2 + (A*b^3*c*e^2 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (2*B*b^3*c - 5* A*b^2*c^2)*d*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(A*b^2*c^2*d^2 - A*b^3*c*d*e - ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - (B*b^3*c - 2*A*b^2*c^2)*d*e)*x)*sqrt(e*x + d))/((b^3*c^3*d^2 - b^4*c^2*d*e)*x^2 + (b ^4*c^2*d^2 - b^5*c*d*e)*x), -1/2*(2*sqrt(-c^2*d + b*c*e)*((2*(B*b*c^2 - 2* A*c^3)*d^2 - (B*b^2*c - 3*A*b*c^2)*d*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^2 - (B*b^3 - 3*A*b^2*c)*d*e)*x)*arctan(sqrt(-c^2*d + b*c*e)*sqrt(e*x + d)/( c*e*x + c*d)) + ((A*b^2*c^2*e^2 - 2*(B*b*c^3 - 2*A*c^4)*d^2 + (2*B*b^2*c^2 - 5*A*b*c^3)*d*e)*x^2 + (A*b^3*c*e^2 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (2 *B*b^3*c - 5*A*b^2*c^2)*d*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(A*b^2*c^2*d^2 - A*b^3*c*d*e - ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - (B*b^3*c - 2*A*b^2*c^2)*d*e)*x)*sqrt(e*x + d))/((b^3*c^3*d^2 - b^4*c^2* d*e)*x^2 + (b^4*c^2*d^2 - b^5*c*d*e)*x), -1/2*(2*((A*b^2*c^2*e^2 - 2*(B*b* c^3 - 2*A*c^4)*d^2 + (2*B*b^2*c^2 - 5*A*b*c^3)*d*e)*x^2 + (A*b^3*c*e^2 - 2 *(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (2*B*b^3*c - 5*A*b^2*c^2)*d*e)*x)*sqrt(-d)* arctan(sqrt(e*x + d)*sqrt(-d)/d) - sqrt(c^2*d - b*c*e)*((2*(B*b*c^2 - 2...
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (2 \, B b c d - 4 \, A c^{2} d - B b^{2} e + 3 \, A b c e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3}} + \frac {{\left (2 \, B b d - 4 \, A c d + A b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} B b e - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} A c e - \sqrt {e x + d} B b d e + 2 \, \sqrt {e x + d} A c d e - \sqrt {e x + d} A b e^{2}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2}} \]
-(2*B*b*c*d - 4*A*c^2*d - B*b^2*e + 3*A*b*c*e)*arctan(sqrt(e*x + d)*c/sqrt (-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3) + (2*B*b*d - 4*A*c*d + A*b*e) *arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt(-d)) + ((e*x + d)^(3/2)*B*b*e - 2*(e*x + d)^(3/2)*A*c*e - sqrt(e*x + d)*B*b*d*e + 2*sqrt(e*x + d)*A*c*d*e - sqrt(e*x + d)*A*b*e^2)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2)
Time = 11.03 (sec) , antiderivative size = 2558, normalized size of antiderivative = 16.19 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
(atan((((-c*(b*e - c*d))^(1/2)*((2*(d + e*x)^(1/2)*(10*A^2*b^2*c^3*e^4 + 3 2*A^2*c^5*d^2*e^2 + B^2*b^4*c*e^4 + 8*B^2*b^2*c^3*d^2*e^2 - 6*A*B*b^3*c^2* e^4 - 32*A^2*b*c^4*d*e^3 - 4*B^2*b^3*c^2*d*e^3 - 32*A*B*b*c^4*d^2*e^2 + 24 *A*B*b^2*c^3*d*e^3))/b^4 + (((4*A*b^7*c^2*e^4 - 8*A*b^6*c^3*d*e^3 + 4*B*b^ 7*c^2*d*e^3)/b^6 + ((4*b^7*c^2*e^3 - 8*b^6*c^3*d*e^2)*(-c*(b*e - c*d))^(1/ 2)*(d + e*x)^(1/2)*(4*A*c^2*d + B*b^2*e - 3*A*b*c*e - 2*B*b*c*d))/(b^4*(b^ 3*c^2*d - b^4*c*e)))*(-c*(b*e - c*d))^(1/2)*(4*A*c^2*d + B*b^2*e - 3*A*b*c *e - 2*B*b*c*d))/(2*(b^3*c^2*d - b^4*c*e)))*(4*A*c^2*d + B*b^2*e - 3*A*b*c *e - 2*B*b*c*d)*1i)/(2*(b^3*c^2*d - b^4*c*e)) + ((-c*(b*e - c*d))^(1/2)*(( 2*(d + e*x)^(1/2)*(10*A^2*b^2*c^3*e^4 + 32*A^2*c^5*d^2*e^2 + B^2*b^4*c*e^4 + 8*B^2*b^2*c^3*d^2*e^2 - 6*A*B*b^3*c^2*e^4 - 32*A^2*b*c^4*d*e^3 - 4*B^2* b^3*c^2*d*e^3 - 32*A*B*b*c^4*d^2*e^2 + 24*A*B*b^2*c^3*d*e^3))/b^4 - (((4*A *b^7*c^2*e^4 - 8*A*b^6*c^3*d*e^3 + 4*B*b^7*c^2*d*e^3)/b^6 - ((4*b^7*c^2*e^ 3 - 8*b^6*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d + B *b^2*e - 3*A*b*c*e - 2*B*b*c*d))/(b^4*(b^3*c^2*d - b^4*c*e)))*(-c*(b*e - c *d))^(1/2)*(4*A*c^2*d + B*b^2*e - 3*A*b*c*e - 2*B*b*c*d))/(2*(b^3*c^2*d - b^4*c*e)))*(4*A*c^2*d + B*b^2*e - 3*A*b*c*e - 2*B*b*c*d)*1i)/(2*(b^3*c^2*d - b^4*c*e)))/((2*(6*A^3*b^2*c^3*e^5 + 32*A^3*c^5*d^2*e^3 - 4*B^3*b^3*c^2* d^2*e^3 + A*B^2*b^4*c*e^5 - 32*A^3*b*c^4*d*e^4 + 2*B^3*b^4*c*d*e^4 - 5*A^2 *B*b^3*c^2*e^5 + 24*A*B^2*b^2*c^3*d^2*e^3 - 16*A*B^2*b^3*c^2*d*e^4 - 48...