Integrand size = 22, antiderivative size = 128 \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=-\frac {2 (f g-e h)}{(d e-c f)^2 \sqrt {e+f x}}-\frac {(d g-c h) \sqrt {e+f x}}{(d e-c f)^2 (c+d x)}+\frac {(3 d f g-2 d e h-c f h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{5/2}} \] Output:
(2*e*h-2*f*g)/(-c*f+d*e)^2/(f*x+e)^(1/2)-(-c*h+d*g)*(f*x+e)^(1/2)/(-c*f+d* e)^2/(d*x+c)+(-c*f*h-2*d*e*h+3*d*f*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+ d*e)^(1/2))/d^(1/2)/(-c*f+d*e)^(5/2)
Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {c (-2 f g+3 e h+f h x)-d (3 f g x+e (g-2 h x))}{(d e-c f)^2 (c+d x) \sqrt {e+f x}}+\frac {(-3 d f g+2 d e h+c f h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d} (-d e+c f)^{5/2}} \] Input:
Integrate[(g + h*x)/((c + d*x)^2*(e + f*x)^(3/2)),x]
Output:
(c*(-2*f*g + 3*e*h + f*h*x) - d*(3*f*g*x + e*(g - 2*h*x)))/((d*e - c*f)^2* (c + d*x)*Sqrt[e + f*x]) + ((-3*d*f*g + 2*d*e*h + c*f*h)*ArcTan[(Sqrt[d]*S qrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(Sqrt[d]*(-(d*e) + c*f)^(5/2))
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 61, 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.
\(\displaystyle \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(-c f h-2 d e h+3 d f g) \int \frac {1}{(c+d x) (e+f x)^{3/2}}dx}{2 d (d e-c f)}-\frac {d g-c h}{d (c+d x) \sqrt {e+f x} (d e-c f)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {(-c f h-2 d e h+3 d f g) \left (\frac {d \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d e-c f}+\frac {2}{\sqrt {e+f x} (d e-c f)}\right )}{2 d (d e-c f)}-\frac {d g-c h}{d (c+d x) \sqrt {e+f x} (d e-c f)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {(-c f h-2 d e h+3 d f g) \left (\frac {2 d \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (d e-c f)}+\frac {2}{\sqrt {e+f x} (d e-c f)}\right )}{2 d (d e-c f)}-\frac {d g-c h}{d (c+d x) \sqrt {e+f x} (d e-c f)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\left (\frac {2}{\sqrt {e+f x} (d e-c f)}-\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{3/2}}\right ) (-c f h-2 d e h+3 d f g)}{2 d (d e-c f)}-\frac {d g-c h}{d (c+d x) \sqrt {e+f x} (d e-c f)}\) |
Input:
Int[(g + h*x)/((c + d*x)^2*(e + f*x)^(3/2)),x]
Output:
-((d*g - c*h)/(d*(d*e - c*f)*(c + d*x)*Sqrt[e + f*x])) - ((3*d*f*g - 2*d*e *h - c*f*h)*(2/((d*e - c*f)*Sqrt[e + f*x]) - (2*Sqrt[d]*ArcTanh[(Sqrt[d]*S qrt[e + f*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(3/2)))/(2*d*(d*e - c*f))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(-\frac {2 \left (-e h +f g \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {\frac {2 \left (\frac {1}{2} c f h -\frac {1}{2} d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (c f h +2 d e h -3 d f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}}}{\left (c f -d e \right )^{2}}\) | \(129\) |
default | \(-\frac {2 \left (-e h +f g \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {\frac {2 \left (\frac {1}{2} c f h -\frac {1}{2} d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (c f h +2 d e h -3 d f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}}}{\left (c f -d e \right )^{2}}\) | \(129\) |
pseudoelliptic | \(\frac {\sqrt {f x +e}\, \left (x d +c \right ) \left (\left (2 e h -3 f g \right ) d +c f h \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+3 \left (\left (-f g x -\frac {e \left (-2 h x +g \right )}{3}\right ) d +c \left (\frac {\left (h x -2 g \right ) f}{3}+e h \right )\right ) \sqrt {\left (c f -d e \right ) d}}{\sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\, \left (c f -d e \right )^{2} \left (x d +c \right )}\) | \(138\) |
Input:
int((h*x+g)/(d*x+c)^2/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*(-e*h+f*g)/(c*f-d*e)^2/(f*x+e)^(1/2)+2/(c*f-d*e)^2*((1/2*c*f*h-1/2*d*f* g)*(f*x+e)^(1/2)/((f*x+e)*d+c*f-d*e)+1/2*(c*f*h+2*d*e*h-3*d*f*g)/((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. 391 vs. \(2 (114) = 228\).
Time = 0.10 (sec) , antiderivative size = 796, normalized size of antiderivative = 6.22 \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((h*x+g)/(d*x+c)^2/(f*x+e)^(3/2),x, algorithm="fricas")
Output:
[-1/2*((3*c*d*e*f*g + (3*d^2*f^2*g - (2*d^2*e*f + c*d*f^2)*h)*x^2 - (2*c*d *e^2 + c^2*e*f)*h + (3*(d^2*e*f + c*d*f^2)*g - (2*d^2*e^2 + 3*c*d*e*f + c^ 2*f^2)*h)*x)*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*((d^3*e^2 + c*d^2*e*f - 2*c^2*d*f^2) *g - 3*(c*d^2*e^2 - c^2*d*e*f)*h + (3*(d^3*e*f - c*d^2*f^2)*g - (2*d^3*e^2 - c*d^2*e*f - c^2*d*f^2)*h)*x)*sqrt(f*x + e))/(c*d^4*e^4 - 3*c^2*d^3*e^3* f + 3*c^3*d^2*e^2*f^2 - c^4*d*e*f^3 + (d^5*e^3*f - 3*c*d^4*e^2*f^2 + 3*c^2 *d^3*e*f^3 - c^3*d^2*f^4)*x^2 + (d^5*e^4 - 2*c*d^4*e^3*f + 2*c^3*d^2*e*f^3 - c^4*d*f^4)*x), -((3*c*d*e*f*g + (3*d^2*f^2*g - (2*d^2*e*f + c*d*f^2)*h) *x^2 - (2*c*d*e^2 + c^2*e*f)*h + (3*(d^2*e*f + c*d*f^2)*g - (2*d^2*e^2 + 3 *c*d*e*f + c^2*f^2)*h)*x)*sqrt(-d^2*e + c*d*f)*arctan(sqrt(-d^2*e + c*d*f) *sqrt(f*x + e)/(d*f*x + d*e)) + ((d^3*e^2 + c*d^2*e*f - 2*c^2*d*f^2)*g - 3 *(c*d^2*e^2 - c^2*d*e*f)*h + (3*(d^3*e*f - c*d^2*f^2)*g - (2*d^3*e^2 - c*d ^2*e*f - c^2*d*f^2)*h)*x)*sqrt(f*x + e))/(c*d^4*e^4 - 3*c^2*d^3*e^3*f + 3* c^3*d^2*e^2*f^2 - c^4*d*e*f^3 + (d^5*e^3*f - 3*c*d^4*e^2*f^2 + 3*c^2*d^3*e *f^3 - c^3*d^2*f^4)*x^2 + (d^5*e^4 - 2*c*d^4*e^3*f + 2*c^3*d^2*e*f^3 - c^4 *d*f^4)*x)]
Timed out. \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((h*x+g)/(d*x+c)**2/(f*x+e)**(3/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((h*x+g)/(d*x+c)^2/(f*x+e)^(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(c*f-d*e>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.51 \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=-\frac {{\left (3 \, d f g - 2 \, d e h - c f h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \sqrt {-d^{2} e + c d f}} - \frac {3 \, {\left (f x + e\right )} d f g - 2 \, d e f g + 2 \, c f^{2} g - 2 \, {\left (f x + e\right )} d e h + 2 \, d e^{2} h - {\left (f x + e\right )} c f h - 2 \, c e f h}{{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} {\left ({\left (f x + e\right )}^{\frac {3}{2}} d - \sqrt {f x + e} d e + \sqrt {f x + e} c f\right )}} \] Input:
integrate((h*x+g)/(d*x+c)^2/(f*x+e)^(3/2),x, algorithm="giac")
Output:
-(3*d*f*g - 2*d*e*h - c*f*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/ ((d^2*e^2 - 2*c*d*e*f + c^2*f^2)*sqrt(-d^2*e + c*d*f)) - (3*(f*x + e)*d*f* g - 2*d*e*f*g + 2*c*f^2*g - 2*(f*x + e)*d*e*h + 2*d*e^2*h - (f*x + e)*c*f* h - 2*c*e*f*h)/((d^2*e^2 - 2*c*d*e*f + c^2*f^2)*((f*x + e)^(3/2)*d - sqrt( f*x + e)*d*e + sqrt(f*x + e)*c*f))
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.20 \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {\frac {2\,\left (e\,h-f\,g\right )}{c\,f-d\,e}+\frac {\left (e+f\,x\right )\,\left (c\,f\,h+2\,d\,e\,h-3\,d\,f\,g\right )}{{\left (c\,f-d\,e\right )}^2}}{d\,{\left (e+f\,x\right )}^{3/2}+\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2\right )}{{\left (c\,f-d\,e\right )}^{5/2}}\right )\,\left (c\,f\,h+2\,d\,e\,h-3\,d\,f\,g\right )}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{5/2}} \] Input:
int((g + h*x)/((e + f*x)^(3/2)*(c + d*x)^2),x)
Output:
((2*(e*h - f*g))/(c*f - d*e) + ((e + f*x)*(c*f*h + 2*d*e*h - 3*d*f*g))/(c* f - d*e)^2)/(d*(e + f*x)^(3/2) + (e + f*x)^(1/2)*(c*f - d*e)) + (atan((d^( 1/2)*(e + f*x)^(1/2)*(c^2*f^2 + d^2*e^2 - 2*c*d*e*f))/(c*f - d*e)^(5/2))*( c*f*h + 2*d*e*h - 3*d*f*g))/(d^(1/2)*(c*f - d*e)^(5/2))
Time = 0.16 (sec) , antiderivative size = 476, normalized size of antiderivative = 3.72 \[ \int \frac {g+h x}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {\sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c^{2} f h +2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c d e h -3 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c d f g +\sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c d f h x +2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) d^{2} e h x -3 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) d^{2} f g x +3 c^{2} d e f h -2 c^{2} d \,f^{2} g +c^{2} d \,f^{2} h x -3 c \,d^{2} e^{2} h +c \,d^{2} e f g +c \,d^{2} e f h x -3 c \,d^{2} f^{2} g x +d^{3} e^{2} g -2 d^{3} e^{2} h x +3 d^{3} e f g x}{\sqrt {f x +e}\, d \left (c^{3} d \,f^{3} x -3 c^{2} d^{2} e \,f^{2} x +3 c \,d^{3} e^{2} f x -d^{4} e^{3} x +c^{4} f^{3}-3 c^{3} d e \,f^{2}+3 c^{2} d^{2} e^{2} f -c \,d^{3} e^{3}\right )} \] Input:
int((h*x+g)/(d*x+c)^2/(f*x+e)^(3/2),x)
Output:
(sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqr t(c*f - d*e)))*c**2*f*h + 2*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sq rt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*c*d*e*h - 3*sqrt(d)*sqrt(e + f*x )*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*c*d*f* g + sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)* sqrt(c*f - d*e)))*c*d*f*h*x + 2*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan ((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*d**2*e*h*x - 3*sqrt(d)*sqrt( e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e))) *d**2*f*g*x + 3*c**2*d*e*f*h - 2*c**2*d*f**2*g + c**2*d*f**2*h*x - 3*c*d** 2*e**2*h + c*d**2*e*f*g + c*d**2*e*f*h*x - 3*c*d**2*f**2*g*x + d**3*e**2*g - 2*d**3*e**2*h*x + 3*d**3*e*f*g*x)/(sqrt(e + f*x)*d*(c**4*f**3 - 3*c**3* d*e*f**2 + c**3*d*f**3*x + 3*c**2*d**2*e**2*f - 3*c**2*d**2*e*f**2*x - c*d **3*e**3 + 3*c*d**3*e**2*f*x - d**4*e**3*x))