Integrand size = 24, antiderivative size = 112 \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {2 (b e-a f)^2}{f^2 (d e-c f) \sqrt {e+f x}}+\frac {2 b^2 \sqrt {e+f x}}{d f^2}-\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}} \]
-2*(-a*d+b*c)^2*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(3/2)/(- c*f+d*e)^(3/2)+2*(-a*f+b*e)^2/f^2/(-c*f+d*e)/(f*x+e)^(1/2)+2*b^2*(f*x+e)^( 1/2)/d/f^2
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {4 a b d e f-2 a^2 d f^2+b^2 (2 c f (e+f x)-2 d e (2 e+f x))}{d f^2 (-d e+c f) \sqrt {e+f x}}-\frac {2 (b c-a d)^2 \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (-d e+c f)^{3/2}} \]
(4*a*b*d*e*f - 2*a^2*d*f^2 + b^2*(2*c*f*(e + f*x) - 2*d*e*(2*e + f*x)))/(d *f^2*(-(d*e) + c*f)*Sqrt[e + f*x]) - (2*(b*c - a*d)^2*ArcTan[(Sqrt[d]*Sqrt [e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(3/2)*(-(d*e) + c*f)^(3/2))
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {98, 2009}
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.
\(\displaystyle \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 98 |
\(\displaystyle \int \left (\frac {(a d-b c)^2}{d (c+d x) \sqrt {e+f x} (d e-c f)}+\frac {(a f-b e)^2}{f (e+f x)^{3/2} (c f-d e)}+\frac {b^2}{d f \sqrt {e+f x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}}+\frac {2 (b e-a f)^2}{f^2 \sqrt {e+f x} (d e-c f)}+\frac {2 b^2 \sqrt {e+f x}}{d f^2}\) |
(2*(b*e - a*f)^2)/(f^2*(d*e - c*f)*Sqrt[e + f*x]) + (2*b^2*Sqrt[e + f*x])/ (d*f^2) - (2*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f] ])/(d^(3/2)*(d*e - c*f)^(3/2))
3.18.78.3.1 Defintions of rubi rules used
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x _)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
Time = 0.99 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} \sqrt {f x +e}}{d}-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) d \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(134\) |
default | \(\frac {\frac {2 b^{2} \sqrt {f x +e}}{d}-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) d \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(134\) |
pseudoelliptic | \(-\frac {2 \left (f^{2} \sqrt {f x +e}\, \left (a d -b c \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (\left (a^{2} f^{2}-2 e b \left (-\frac {b x}{2}+a \right ) f +2 b^{2} e^{2}\right ) d -b^{2} c f \left (f x +e \right )\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, f^{2} d \left (c f -d e \right )}\) | \(137\) |
risch | \(\frac {2 b^{2} \sqrt {f x +e}}{d \,f^{2}}+\frac {-\frac {2 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) d}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{d \,f^{2}}\) | \(140\) |
2/f^2*(b^2/d*(f*x+e)^(1/2)-f^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(c*f-d*e)/d/((c *f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))-(a^2*f^2-2*a* b*e*f+b^2*e^2)/(c*f-d*e)/(f*x+e)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (98) = 196\).
Time = 0.24 (sec) , antiderivative size = 600, normalized size of antiderivative = 5.36 \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{2}\right )} \sqrt {d^{2} e - c d f} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {d^{2} e - c d f} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (2 \, b^{2} d^{3} e^{3} - a^{2} c d^{2} f^{3} - {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{2} f + {\left (b^{2} c^{2} d + 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} + {\left (b^{2} d^{3} e^{2} f - 2 \, b^{2} c d^{2} e f^{2} + b^{2} c^{2} d f^{3}\right )} x\right )} \sqrt {f x + e}}{d^{4} e^{3} f^{2} - 2 \, c d^{3} e^{2} f^{3} + c^{2} d^{2} e f^{4} + {\left (d^{4} e^{2} f^{3} - 2 \, c d^{3} e f^{4} + c^{2} d^{2} f^{5}\right )} x}, \frac {2 \, {\left ({\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{2}\right )} \sqrt {-d^{2} e + c d f} \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (2 \, b^{2} d^{3} e^{3} - a^{2} c d^{2} f^{3} - {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{2} f + {\left (b^{2} c^{2} d + 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} + {\left (b^{2} d^{3} e^{2} f - 2 \, b^{2} c d^{2} e f^{2} + b^{2} c^{2} d f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{d^{4} e^{3} f^{2} - 2 \, c d^{3} e^{2} f^{3} + c^{2} d^{2} e f^{4} + {\left (d^{4} e^{2} f^{3} - 2 \, c d^{3} e f^{4} + c^{2} d^{2} f^{5}\right )} x}\right ] \]
[-(((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^3*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2 )*e*f^2)*sqrt(d^2*e - c*d*f)*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*d^3*e^3 - a^2*c*d^2*f^3 - (3*b^2* c*d^2 + 2*a*b*d^3)*e^2*f + (b^2*c^2*d + 2*a*b*c*d^2 + a^2*d^3)*e*f^2 + (b^ 2*d^3*e^2*f - 2*b^2*c*d^2*e*f^2 + b^2*c^2*d*f^3)*x)*sqrt(f*x + e))/(d^4*e^ 3*f^2 - 2*c*d^3*e^2*f^3 + c^2*d^2*e*f^4 + (d^4*e^2*f^3 - 2*c*d^3*e*f^4 + c ^2*d^2*f^5)*x), 2*(((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^3*x + (b^2*c^2 - 2*a *b*c*d + a^2*d^2)*e*f^2)*sqrt(-d^2*e + c*d*f)*arctan(sqrt(-d^2*e + c*d*f)* sqrt(f*x + e)/(d*f*x + d*e)) + (2*b^2*d^3*e^3 - a^2*c*d^2*f^3 - (3*b^2*c*d ^2 + 2*a*b*d^3)*e^2*f + (b^2*c^2*d + 2*a*b*c*d^2 + a^2*d^3)*e*f^2 + (b^2*d ^3*e^2*f - 2*b^2*c*d^2*e*f^2 + b^2*c^2*d*f^3)*x)*sqrt(f*x + e))/(d^4*e^3*f ^2 - 2*c*d^3*e^2*f^3 + c^2*d^2*e*f^4 + (d^4*e^2*f^3 - 2*c*d^3*e*f^4 + c^2* d^2*f^5)*x)]
Time = 5.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} \sqrt {e + f x}}{d f} - \frac {\left (a f - b e\right )^{2}}{f \sqrt {e + f x} \left (c f - d e\right )} - \frac {f \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{2} \sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {b^{2} x^{2}}{2 d} + \frac {x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac {\left (a d - b c\right )^{2} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**2*sqrt(e + f*x)/(d*f) - (a*f - b*e)**2/(f*sqrt(e + f*x)*( c*f - d*e)) - f*(a*d - b*c)**2*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d* *2*sqrt((c*f - d*e)/d)*(c*f - d*e)))/f, Ne(f, 0)), ((b**2*x**2/(2*d) + x*( 2*a*b*d - b**2*c)/d**2 + (a*d - b*c)**2*Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d**2)/e**(3/2), True))
Exception generated. \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{2} e - c d f\right )} \sqrt {-d^{2} e + c d f}} + \frac {2 \, {\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )}}{{\left (d e f^{2} - c f^{3}\right )} \sqrt {f x + e}} + \frac {2 \, \sqrt {f x + e} b^{2}}{d f^{2}} \]
2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d *f))/((d^2*e - c*d*f)*sqrt(-d^2*e + c*d*f)) + 2*(b^2*e^2 - 2*a*b*e*f + a^2 *f^2)/((d*e*f^2 - c*f^3)*sqrt(f*x + e)) + 2*sqrt(f*x + e)*b^2/(d*f^2)
Time = 1.43 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {2\,b^2\,\sqrt {e+f\,x}}{d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {e+f\,x}\,\left (d^2\,e-c\,d\,f\right )\,{\left (a\,d-b\,c\right )}^2}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{3/2}\,\left (2\,a^2\,d^2-4\,a\,b\,c\,d+2\,b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{3/2}\,{\left (c\,f-d\,e\right )}^{3/2}}-\frac {2\,\left (d\,a^2\,f^2-2\,d\,a\,b\,e\,f+d\,b^2\,e^2\right )}{d\,f^2\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )} \]