Integrand size = 24, antiderivative size = 132 \[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=\frac {2 b^2 \sqrt {e+f x}}{d^2 f}-\frac {(b c-a d)^2 \sqrt {e+f x}}{d^2 (d e-c f) (c+d x)}+\frac {(b c-a d) (4 b d e-3 b c f-a d f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}} \]
(-a*d+b*c)*(-a*d*f-3*b*c*f+4*b*d*e)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d* e)^(1/2))/d^(5/2)/(-c*f+d*e)^(3/2)+2*b^2*(f*x+e)^(1/2)/d^2/f-(-a*d+b*c)^2* (f*x+e)^(1/2)/d^2/(-c*f+d*e)/(d*x+c)
Time = 0.42 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=\frac {\sqrt {e+f x} \left (2 a b c d f-a^2 d^2 f+b^2 \left (-3 c^2 f+2 d^2 e x+2 c d (e-f x)\right )\right )}{d^2 f (d e-c f) (c+d x)}-\frac {(b c-a d) (-4 b d e+3 b c f+a d f) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2} (-d e+c f)^{3/2}} \]
(Sqrt[e + f*x]*(2*a*b*c*d*f - a^2*d^2*f + b^2*(-3*c^2*f + 2*d^2*e*x + 2*c* d*(e - f*x))))/(d^2*f*(d*e - c*f)*(c + d*x)) - ((b*c - a*d)*(-4*b*d*e + 3* b*c*f + a*d*f)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(5/2 )*(-(d*e) + c*f)^(3/2))
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 27, 90, 73, 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.
| \(\displaystyle \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {\int -\frac {c (2 d e-c f) b^2-2 d (d e-c f) x b^2-2 a d (2 d e-c f) b+a^2 d^2 f}{2 (c+d x) \sqrt {e+f x}}dx}{d^2 (d e-c f)}-\frac {\sqrt {e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {c (2 d e-c f) b^2-2 d (d e-c f) x b^2-2 a d (2 d e-c f) b+a^2 d^2 f}{(c+d x) \sqrt {e+f x}}dx}{2 d^2 (d e-c f)}-\frac {\sqrt {e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {(b c-a d) (-a d f-3 b c f+4 b d e) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx-\frac {4 b^2 \sqrt {e+f x} (d e-c f)}{f}}{2 d^2 (d e-c f)}-\frac {\sqrt {e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {2 (b c-a d) (-a d f-3 b c f+4 b d e) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f}-\frac {4 b^2 \sqrt {e+f x} (d e-c f)}{f}}{2 d^2 (d e-c f)}-\frac {\sqrt {e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {2 (b c-a d) (-a d f-3 b c f+4 b d e) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} \sqrt {d e-c f}}-\frac {4 b^2 \sqrt {e+f x} (d e-c f)}{f}}{2 d^2 (d e-c f)}-\frac {\sqrt {e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}\) |
-(((b*c - a*d)^2*Sqrt[e + f*x])/(d^2*(d*e - c*f)*(c + d*x))) - ((-4*b^2*(d *e - c*f)*Sqrt[e + f*x])/f - (2*(b*c - a*d)*(4*b*d*e - 3*b*c*f - a*d*f)*Ar cTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(Sqrt[d]*Sqrt[d*e - c*f])) /(2*d^2*(d*e - c*f))
3.21.35.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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]
Time = 3.42 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {2 b^{2} \sqrt {f x +e}}{d^{2} f}+\frac {\left (2 a d -2 b c \right ) \left (\frac {f \left (a d -b c \right ) \sqrt {f x +e}}{2 \left (c f -d e \right ) \left (d \left (f x +e \right )+c f -d e \right )}+\frac {\left (a d f +3 b c f -4 b d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{2}}\) | \(140\) |
| pseudoelliptic | \(\frac {f \left (d x +c \right ) \left (a d -b c \right ) \left (\left (a f -4 b e \right ) d +3 b c f \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (\left (-2 b^{2} e x +f \,a^{2}\right ) d^{2}-2 \left (\left (-b x +a \right ) f +b e \right ) b c d +3 b^{2} c^{2} f \right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}\, f \,d^{2} \left (c f -d e \right ) \left (d x +c \right )}\) | \(155\) |
| derivativedivides | \(\frac {\frac {2 b^{2} \sqrt {f x +e}}{d^{2}}+\frac {2 f \left (\frac {f \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {f x +e}}{2 \left (c f -d e \right ) \left (d \left (f x +e \right )+c f -d e \right )}+\frac {\left (a^{2} d^{2} f +2 a b c d f -4 a b \,d^{2} e -3 b^{2} c^{2} f +4 b^{2} c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{2}}}{f}\) | \(172\) |
| default | \(\frac {\frac {2 b^{2} \sqrt {f x +e}}{d^{2}}+\frac {2 f \left (\frac {f \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {f x +e}}{2 \left (c f -d e \right ) \left (d \left (f x +e \right )+c f -d e \right )}+\frac {\left (a^{2} d^{2} f +2 a b c d f -4 a b \,d^{2} e -3 b^{2} c^{2} f +4 b^{2} c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\right )}{d^{2}}}{f}\) | \(172\) |
2*b^2*(f*x+e)^(1/2)/d^2/f+1/d^2*(2*a*d-2*b*c)*(1/2*f*(a*d-b*c)/(c*f-d*e)*( f*x+e)^(1/2)/(d*(f*x+e)+c*f-d*e)+1/2*(a*d*f+3*b*c*f-4*b*d*e)/(c*f-d*e)/((c *f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (118) = 236\).
Time = 0.25 (sec) , antiderivative size = 701, normalized size of antiderivative = 5.31 \[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=\left [-\frac {\sqrt {d^{2} e - c d f} {\left (4 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} e f - {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{2} + {\left (4 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} e f - {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} f^{2}\right )} x\right )} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, \sqrt {d^{2} e - c d f} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (2 \, b^{2} c d^{3} e^{2} - {\left (5 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} e f + {\left (3 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{2} + 2 \, {\left (b^{2} d^{4} e^{2} - 2 \, b^{2} c d^{3} e f + b^{2} c^{2} d^{2} f^{2}\right )} x\right )} \sqrt {f x + e}}{2 \, {\left (c d^{5} e^{2} f - 2 \, c^{2} d^{4} e f^{2} + c^{3} d^{3} f^{3} + {\left (d^{6} e^{2} f - 2 \, c d^{5} e f^{2} + c^{2} d^{4} f^{3}\right )} x\right )}}, -\frac {\sqrt {-d^{2} e + c d f} {\left (4 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} e f - {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{2} + {\left (4 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} e f - {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} f^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) - {\left (2 \, b^{2} c d^{3} e^{2} - {\left (5 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} e f + {\left (3 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} f^{2} + 2 \, {\left (b^{2} d^{4} e^{2} - 2 \, b^{2} c d^{3} e f + b^{2} c^{2} d^{2} f^{2}\right )} x\right )} \sqrt {f x + e}}{c d^{5} e^{2} f - 2 \, c^{2} d^{4} e f^{2} + c^{3} d^{3} f^{3} + {\left (d^{6} e^{2} f - 2 \, c d^{5} e f^{2} + c^{2} d^{4} f^{3}\right )} x}\right ] \]
[-1/2*(sqrt(d^2*e - c*d*f)*(4*(b^2*c^2*d - a*b*c*d^2)*e*f - (3*b^2*c^3 - 2 *a*b*c^2*d - a^2*c*d^2)*f^2 + (4*(b^2*c*d^2 - a*b*d^3)*e*f - (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*f^2)*x)*log((d*f*x + 2*d*e - c*f - 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) - 2*(2*b^2*c*d^3*e^2 - (5*b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f + (3*b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2 + 2*(b^2*d^4*e^2 - 2*b^2*c*d^3*e*f + b^2*c^2*d^2*f^2)*x)*sqrt(f*x + e))/( c*d^5*e^2*f - 2*c^2*d^4*e*f^2 + c^3*d^3*f^3 + (d^6*e^2*f - 2*c*d^5*e*f^2 + c^2*d^4*f^3)*x), -(sqrt(-d^2*e + c*d*f)*(4*(b^2*c^2*d - a*b*c*d^2)*e*f - (3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2)*f^2 + (4*(b^2*c*d^2 - a*b*d^3)*e*f - (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*f^2)*x)*arctan(sqrt(-d^2*e + c*d*f) *sqrt(f*x + e)/(d*f*x + d*e)) - (2*b^2*c*d^3*e^2 - (5*b^2*c^2*d^2 - 2*a*b* c*d^3 + a^2*d^4)*e*f + (3*b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2 + 2*( b^2*d^4*e^2 - 2*b^2*c*d^3*e*f + b^2*c^2*d^2*f^2)*x)*sqrt(f*x + e))/(c*d^5* e^2*f - 2*c^2*d^4*e*f^2 + c^3*d^3*f^3 + (d^6*e^2*f - 2*c*d^5*e*f^2 + c^2*d ^4*f^3)*x)]
\[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=\int \frac {\left (a + b x\right )^{2}}{\left (c + d x\right )^{2} \sqrt {e + f x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=-\frac {{\left (4 \, b^{2} c d e - 4 \, a b d^{2} e - 3 \, b^{2} c^{2} f + 2 \, a b c d f + a^{2} d^{2} f\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{3} e - c d^{2} f\right )} \sqrt {-d^{2} e + c d f}} + \frac {2 \, \sqrt {f x + e} b^{2}}{d^{2} f} - \frac {\sqrt {f x + e} b^{2} c^{2} f - 2 \, \sqrt {f x + e} a b c d f + \sqrt {f x + e} a^{2} d^{2} f}{{\left (d^{3} e - c d^{2} f\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}} \]
-(4*b^2*c*d*e - 4*a*b*d^2*e - 3*b^2*c^2*f + 2*a*b*c*d*f + a^2*d^2*f)*arcta n(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^3*e - c*d^2*f)*sqrt(-d^2*e + c *d*f)) + 2*sqrt(f*x + e)*b^2/(d^2*f) - (sqrt(f*x + e)*b^2*c^2*f - 2*sqrt(f *x + e)*a*b*c*d*f + sqrt(f*x + e)*a^2*d^2*f)/((d^3*e - c*d^2*f)*((f*x + e) *d - d*e + c*f))
Time = 1.93 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^2}{(c+d x)^2 \sqrt {e+f x}} \, dx=\frac {2\,b^2\,\sqrt {e+f\,x}}{d^2\,f}+\frac {\sqrt {e+f\,x}\,\left (f\,a^2\,d^2-2\,f\,a\,b\,c\,d+f\,b^2\,c^2\right )}{\left (c\,f-d\,e\right )\,\left (d^3\,\left (e+f\,x\right )-d^3\,e+c\,d^2\,f\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,\left (a\,d-b\,c\right )\,\left (a\,d\,f+3\,b\,c\,f-4\,b\,d\,e\right )}{\sqrt {c\,f-d\,e}\,\left (f\,a^2\,d^2+2\,f\,a\,b\,c\,d-4\,e\,a\,b\,d^2-3\,f\,b^2\,c^2+4\,e\,b^2\,c\,d\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d\,f+3\,b\,c\,f-4\,b\,d\,e\right )}{d^{5/2}\,{\left (c\,f-d\,e\right )}^{3/2}} \]
(2*b^2*(e + f*x)^(1/2))/(d^2*f) + ((e + f*x)^(1/2)*(a^2*d^2*f + b^2*c^2*f - 2*a*b*c*d*f))/((c*f - d*e)*(d^3*(e + f*x) - d^3*e + c*d^2*f)) + (atan((d ^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)*(a*d*f + 3*b*c*f - 4*b*d*e))/((c*f - d* e)^(1/2)*(a^2*d^2*f - 3*b^2*c^2*f - 4*a*b*d^2*e + 4*b^2*c*d*e + 2*a*b*c*d* f)))*(a*d - b*c)*(a*d*f + 3*b*c*f - 4*b*d*e))/(d^(5/2)*(c*f - d*e)^(3/2))