Integrand size = 24, antiderivative size = 112 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt {f+g x}}+\frac {2 c \sqrt {f+g x}}{e g^2}-\frac {2 \left (c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}} \] Output:
2*(a*g^2+c*f^2)/g^2/(-d*g+e*f)/(g*x+f)^(1/2)+2*c*(g*x+f)^(1/2)/e/g^2-2*(a* e^2+c*d^2)*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))/e^(3/2)/(-d*g+e *f)^(3/2)
Time = 0.49 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=-\frac {2 \left (a e g^2-c d g (f+g x)+c e f (2 f+g x)\right )}{e g^2 (-e f+d g) \sqrt {f+g x}}-\frac {2 \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{3/2} (-e f+d g)^{3/2}} \] Input:
Integrate[(a + c*x^2)/((d + e*x)*(f + g*x)^(3/2)),x]
Output:
(-2*(a*e*g^2 - c*d*g*(f + g*x) + c*e*f*(2*f + g*x)))/(e*g^2*(-(e*f) + d*g) *Sqrt[f + g*x]) - (2*(c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[- (e*f) + d*g]])/(e^(3/2)*(-(e*f) + d*g)^(3/2))
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {649, 25, 1584, 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.
\(\displaystyle \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 649 |
\(\displaystyle \frac {2 \int -\frac {c f^2-2 c (f+g x) f+a g^2+c (f+g x)^2}{(f+g x) (e f-d g-e (f+g x))}d\sqrt {f+g x}}{g^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \int \frac {c f^2-2 c (f+g x) f+a g^2+c (f+g x)^2}{(f+g x) (e f-d g-e (f+g x))}d\sqrt {f+g x}}{g^2}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle -\frac {2 \int \left (\frac {\left (c d^2+a e^2\right ) g^2}{e (e f-d g) (e f-d g-e (f+g x))}-\frac {c}{e}+\frac {c f^2+a g^2}{(e f-d g) (f+g x)}\right )d\sqrt {f+g x}}{g^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (-\frac {g^2 \left (a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}+\frac {a g^2+c f^2}{\sqrt {f+g x} (e f-d g)}+\frac {c \sqrt {f+g x}}{e}\right )}{g^2}\) |
Input:
Int[(a + c*x^2)/((d + e*x)*(f + g*x)^(3/2)),x]
Output:
(2*((c*f^2 + a*g^2)/((e*f - d*g)*Sqrt[f + g*x]) + (c*Sqrt[f + g*x])/e - (( c*d^2 + a*e^2)*g^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(3 /2)*(e*f - d*g)^(3/2))))/g^2
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^(2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x ]], x] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Integ erQ[m + 1/2]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 0.97 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {2 c \sqrt {g x +f}}{e}-\frac {2 g^{2} \left (a \,e^{2}+c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}}{g^{2}}\) | \(112\) |
default | \(\frac {\frac {2 c \sqrt {g x +f}}{e}-\frac {2 g^{2} \left (a \,e^{2}+c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}}{g^{2}}\) | \(112\) |
pseudoelliptic | \(\frac {\frac {2 c \sqrt {g x +f}}{e}-\frac {2 g^{2} \left (a \,e^{2}+c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}}{g^{2}}\) | \(112\) |
risch | \(\frac {2 c \sqrt {g x +f}}{e \,g^{2}}-\frac {2 \left (\frac {g^{2} \left (a \,e^{2}+c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}+\frac {e \left (a \,g^{2}+c \,f^{2}\right )}{\left (d g -e f \right ) \sqrt {g x +f}}\right )}{e \,g^{2}}\) | \(116\) |
Input:
int((c*x^2+a)/(e*x+d)/(g*x+f)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/g^2*(c/e*(g*x+f)^(1/2)-1/e/(d*g-e*f)*g^2*(a*e^2+c*d^2)/((d*g-e*f)*e)^(1/ 2)*arctan(e*(g*x+f)^(1/2)/((d*g-e*f)*e)^(1/2))-(a*g^2+c*f^2)/(d*g-e*f)/(g* x+f)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (98) = 196\).
Time = 0.09 (sec) , antiderivative size = 492, normalized size of antiderivative = 4.39 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} + {\left (c d^{2} e + a e^{3}\right )} f g^{2} + {\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt {g x + f}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} + {\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}, \frac {2 \, {\left ({\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} + {\left (c d^{2} e + a e^{3}\right )} f g^{2} + {\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt {g x + f}\right )}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} + {\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}\right ] \] Input:
integrate((c*x^2+a)/(e*x+d)/(g*x+f)^(3/2),x, algorithm="fricas")
Output:
[-(((c*d^2 + a*e^2)*g^3*x + (c*d^2 + a*e^2)*f*g^2)*sqrt(e^2*f - d*e*g)*log ((e*g*x + 2*e*f - d*g + 2*sqrt(e^2*f - d*e*g)*sqrt(g*x + f))/(e*x + d)) - 2*(2*c*e^3*f^3 - 3*c*d*e^2*f^2*g - a*d*e^2*g^3 + (c*d^2*e + a*e^3)*f*g^2 + (c*e^3*f^2*g - 2*c*d*e^2*f*g^2 + c*d^2*e*g^3)*x)*sqrt(g*x + f))/(e^4*f^3* g^2 - 2*d*e^3*f^2*g^3 + d^2*e^2*f*g^4 + (e^4*f^2*g^3 - 2*d*e^3*f*g^4 + d^2 *e^2*g^5)*x), 2*(((c*d^2 + a*e^2)*g^3*x + (c*d^2 + a*e^2)*f*g^2)*sqrt(-e^2 *f + d*e*g)*arctan(sqrt(-e^2*f + d*e*g)*sqrt(g*x + f)/(e*g*x + e*f)) + (2* c*e^3*f^3 - 3*c*d*e^2*f^2*g - a*d*e^2*g^3 + (c*d^2*e + a*e^3)*f*g^2 + (c*e ^3*f^2*g - 2*c*d*e^2*f*g^2 + c*d^2*e*g^3)*x)*sqrt(g*x + f))/(e^4*f^3*g^2 - 2*d*e^3*f^2*g^3 + d^2*e^2*f*g^4 + (e^4*f^2*g^3 - 2*d*e^3*f*g^4 + d^2*e^2* g^5)*x)]
Time = 5.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.35 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c \sqrt {f + g x}}{e g} - \frac {a g^{2} + c f^{2}}{g \sqrt {f + g x} \left (d g - e f\right )} - \frac {g \left (a e^{2} + c d^{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{e^{2} \sqrt {\frac {d g - e f}{e}} \left (d g - e f\right )}\right )}{g} & \text {for}\: g \neq 0 \\\frac {- \frac {c d x}{e^{2}} + \frac {c x^{2}}{2 e} + \frac {\left (a e^{2} + c d^{2}\right ) \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+a)/(e*x+d)/(g*x+f)**(3/2),x)
Output:
Piecewise((2*(c*sqrt(f + g*x)/(e*g) - (a*g**2 + c*f**2)/(g*sqrt(f + g*x)*( d*g - e*f)) - g*(a*e**2 + c*d**2)*atan(sqrt(f + g*x)/sqrt((d*g - e*f)/e))/ (e**2*sqrt((d*g - e*f)/e)*(d*g - e*f)))/g, Ne(g, 0)), ((-c*d*x/e**2 + c*x* *2/(2*e) + (a*e**2 + c*d**2)*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, T rue))/e**2)/f**(3/2), True))
Exception generated. \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+a)/(e*x+d)/(g*x+f)^(3/2),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(e*(d*g-e*f)>0)', see `assume?` f or more de
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {2 \, {\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{{\left (e^{2} f - d e g\right )} \sqrt {-e^{2} f + d e g}} + \frac {2 \, {\left (c f^{2} + a g^{2}\right )}}{{\left (e f g^{2} - d g^{3}\right )} \sqrt {g x + f}} + \frac {2 \, \sqrt {g x + f} c}{e g^{2}} \] Input:
integrate((c*x^2+a)/(e*x+d)/(g*x+f)^(3/2),x, algorithm="giac")
Output:
2*(c*d^2 + a*e^2)*arctan(sqrt(g*x + f)*e/sqrt(-e^2*f + d*e*g))/((e^2*f - d *e*g)*sqrt(-e^2*f + d*e*g)) + 2*(c*f^2 + a*g^2)/((e*f*g^2 - d*g^3)*sqrt(g* x + f)) + 2*sqrt(g*x + f)*c/(e*g^2)
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.26 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )\,\left (e^2\,f-d\,e\,g\right )}{\sqrt {e}\,\left (2\,c\,d^2+2\,a\,e^2\right )\,{\left (d\,g-e\,f\right )}^{3/2}}\right )\,\left (c\,d^2+a\,e^2\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {2\,c\,\sqrt {f+g\,x}}{e\,g^2}-\frac {2\,\left (c\,e\,f^2+a\,e\,g^2\right )}{e\,g^2\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )} \] Input:
int((a + c*x^2)/((f + g*x)^(3/2)*(d + e*x)),x)
Output:
(2*atan((2*(f + g*x)^(1/2)*(a*e^2 + c*d^2)*(e^2*f - d*e*g))/(e^(1/2)*(2*a* e^2 + 2*c*d^2)*(d*g - e*f)^(3/2)))*(a*e^2 + c*d^2))/(e^(3/2)*(d*g - e*f)^( 3/2)) + (2*c*(f + g*x)^(1/2))/(e*g^2) - (2*(a*e*g^2 + c*e*f^2))/(e*g^2*(f + g*x)^(1/2)*(d*g - e*f))
Time = 0.26 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.97 \[ \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {-2 \sqrt {e}\, \sqrt {g x +f}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) a \,e^{2} g^{2}-2 \sqrt {e}\, \sqrt {g x +f}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) c \,d^{2} g^{2}-2 a d \,e^{2} g^{3}+2 a \,e^{3} f \,g^{2}+2 c \,d^{2} e f \,g^{2}+2 c \,d^{2} e \,g^{3} x -6 c d \,e^{2} f^{2} g -4 c d \,e^{2} f \,g^{2} x +4 c \,e^{3} f^{3}+2 c \,e^{3} f^{2} g x}{\sqrt {g x +f}\, e^{2} g^{2} \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )} \] Input:
int((c*x^2+a)/(e*x+d)/(g*x+f)^(3/2),x)
Output:
(2*( - sqrt(e)*sqrt(f + g*x)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt( e)*sqrt(d*g - e*f)))*a*e**2*g**2 - sqrt(e)*sqrt(f + g*x)*sqrt(d*g - e*f)*a tan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*c*d**2*g**2 - a*d*e**2*g* *3 + a*e**3*f*g**2 + c*d**2*e*f*g**2 + c*d**2*e*g**3*x - 3*c*d*e**2*f**2*g - 2*c*d*e**2*f*g**2*x + 2*c*e**3*f**3 + c*e**3*f**2*g*x))/(sqrt(f + g*x)* e**2*g**2*(d**2*g**2 - 2*d*e*f*g + e**2*f**2))