Integrand size = 22, antiderivative size = 119 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=-\frac {2 (b e-a f)}{3 f (d e-c f) (e+f x)^{3/2}}-\frac {2 (b c-a d)}{(d e-c f)^2 \sqrt {e+f x}}+\frac {2 \sqrt {d} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{5/2}} \] Output:
1/3*(2*a*f-2*b*e)/f/(-c*f+d*e)/(f*x+e)^(3/2)-2*(-a*d+b*c)/(-c*f+d*e)^2/(f* x+e)^(1/2)+2*d^(1/2)*(-a*d+b*c)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^( 1/2))/(-c*f+d*e)^(5/2)
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=-\frac {2 \left (b d e^2+b c f (2 e+3 f x)+a f (-4 d e+c f-3 d f x)\right )}{3 f (d e-c f)^2 (e+f x)^{3/2}}+\frac {2 \sqrt {d} (-b c+a d) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{5/2}} \] Input:
Integrate[(a + b*x)/((c + d*x)*(e + f*x)^(5/2)),x]
Output:
(-2*(b*d*e^2 + b*c*f*(2*e + 3*f*x) + a*f*(-4*d*e + c*f - 3*d*f*x)))/(3*f*( d*e - c*f)^2*(e + f*x)^(3/2)) + (2*Sqrt[d]*(-(b*c) + a*d)*ArcTan[(Sqrt[d]* Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(-(d*e) + c*f)^(5/2)
Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(b c-a d) \int \frac {1}{(c+d x) (e+f x)^{3/2}}dx}{d e-c f}-\frac {2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {(b c-a d) \left (\frac {d \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d e-c f}+\frac {2}{\sqrt {e+f x} (d e-c f)}\right )}{d e-c f}-\frac {2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {(b c-a d) \left (\frac {2 d \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (d e-c f)}+\frac {2}{\sqrt {e+f x} (d e-c f)}\right )}{d e-c f}-\frac {2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {(b c-a d) \left (\frac {2}{\sqrt {e+f x} (d e-c f)}-\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{3/2}}\right )}{d e-c f}-\frac {2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}\) |
Input:
Int[(a + b*x)/((c + d*x)*(e + f*x)^(5/2)),x]
Output:
(-2*(b*e - a*f))/(3*f*(d*e - c*f)*(e + f*x)^(3/2)) - ((b*c - a*d)*(2/((d*e - c*f)*Sqrt[e + f*x]) - (2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d *e - c*f]])/(d*e - c*f)^(3/2)))/(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 = 0.51 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {2 d f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a f -b e \right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f \left (a d -b c \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f}\) | \(116\) |
default | \(\frac {\frac {2 d f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a f -b e \right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f \left (a d -b c \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f}\) | \(116\) |
pseudoelliptic | \(\frac {2 d \left (a d -b c \right ) f \left (f x +e \right )^{\frac {3}{2}} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )-\frac {2 \sqrt {\left (c f -d e \right ) d}\, \left (\left (-3 a d x +c \left (3 b x +a \right )\right ) f^{2}+2 e \left (-2 a d +b c \right ) f +b d \,e^{2}\right )}{3}}{f \left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}\, \left (f x +e \right )^{\frac {3}{2}}}\) | \(127\) |
Input:
int((b*x+a)/(d*x+c)/(f*x+e)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/f*(-1/3*(a*f-b*e)/(c*f-d*e)/(f*x+e)^(3/2)+f*(a*d-b*c)/(c*f-d*e)^2/(f*x+e )^(1/2)+d*f*(a*d-b*c)/(c*f-d*e)^2/((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. 228 vs. \(2 (103) = 206\).
Time = 0.09 (sec) , antiderivative size = 486, normalized size of antiderivative = 4.08 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b c - a d\right )} f^{3} x^{2} + 2 \, {\left (b c - a d\right )} e f^{2} x + {\left (b c - a d\right )} e^{2} f\right )} \sqrt {\frac {d}{d e - c f}} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, {\left (d e - c f\right )} \sqrt {f x + e} \sqrt {\frac {d}{d e - c f}}}{d x + c}\right ) + 2 \, {\left (b d e^{2} + a c f^{2} + 3 \, {\left (b c - a d\right )} f^{2} x + 2 \, {\left (b c - 2 \, a d\right )} e f\right )} \sqrt {f x + e}}{3 \, {\left (d^{2} e^{4} f - 2 \, c d e^{3} f^{2} + c^{2} e^{2} f^{3} + {\left (d^{2} e^{2} f^{3} - 2 \, c d e f^{4} + c^{2} f^{5}\right )} x^{2} + 2 \, {\left (d^{2} e^{3} f^{2} - 2 \, c d e^{2} f^{3} + c^{2} e f^{4}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left ({\left (b c - a d\right )} f^{3} x^{2} + 2 \, {\left (b c - a d\right )} e f^{2} x + {\left (b c - a d\right )} e^{2} f\right )} \sqrt {-\frac {d}{d e - c f}} \arctan \left (\sqrt {f x + e} \sqrt {-\frac {d}{d e - c f}}\right ) + {\left (b d e^{2} + a c f^{2} + 3 \, {\left (b c - a d\right )} f^{2} x + 2 \, {\left (b c - 2 \, a d\right )} e f\right )} \sqrt {f x + e}\right )}}{3 \, {\left (d^{2} e^{4} f - 2 \, c d e^{3} f^{2} + c^{2} e^{2} f^{3} + {\left (d^{2} e^{2} f^{3} - 2 \, c d e f^{4} + c^{2} f^{5}\right )} x^{2} + 2 \, {\left (d^{2} e^{3} f^{2} - 2 \, c d e^{2} f^{3} + c^{2} e f^{4}\right )} x\right )}}\right ] \] Input:
integrate((b*x+a)/(d*x+c)/(f*x+e)^(5/2),x, algorithm="fricas")
Output:
[-1/3*(3*((b*c - a*d)*f^3*x^2 + 2*(b*c - a*d)*e*f^2*x + (b*c - a*d)*e^2*f) *sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e - c*f - 2*(d*e - c*f)*sqrt(f*x + e )*sqrt(d/(d*e - c*f)))/(d*x + c)) + 2*(b*d*e^2 + a*c*f^2 + 3*(b*c - a*d)*f ^2*x + 2*(b*c - 2*a*d)*e*f)*sqrt(f*x + e))/(d^2*e^4*f - 2*c*d*e^3*f^2 + c^ 2*e^2*f^3 + (d^2*e^2*f^3 - 2*c*d*e*f^4 + c^2*f^5)*x^2 + 2*(d^2*e^3*f^2 - 2 *c*d*e^2*f^3 + c^2*e*f^4)*x), -2/3*(3*((b*c - a*d)*f^3*x^2 + 2*(b*c - a*d) *e*f^2*x + (b*c - a*d)*e^2*f)*sqrt(-d/(d*e - c*f))*arctan(sqrt(f*x + e)*sq rt(-d/(d*e - c*f))) + (b*d*e^2 + a*c*f^2 + 3*(b*c - a*d)*f^2*x + 2*(b*c - 2*a*d)*e*f)*sqrt(f*x + e))/(d^2*e^4*f - 2*c*d*e^3*f^2 + c^2*e^2*f^3 + (d^2 *e^2*f^3 - 2*c*d*e*f^4 + c^2*f^5)*x^2 + 2*(d^2*e^3*f^2 - 2*c*d*e^2*f^3 + c ^2*e*f^4)*x)]
Time = 4.88 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.16 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {f \left (a d - b c\right )}{\sqrt {e + f x} \left (c f - d e\right )^{2}} + \frac {f \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{2}} - \frac {a f - b e}{3 \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {b x}{d} + \frac {\left (a d - b c\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d}}{e^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)/(d*x+c)/(f*x+e)**(5/2),x)
Output:
Piecewise((2*(f*(a*d - b*c)/(sqrt(e + f*x)*(c*f - d*e)**2) + f*(a*d - b*c) *atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(sqrt((c*f - d*e)/d)*(c*f - d*e)* *2) - (a*f - b*e)/(3*(e + f*x)**(3/2)*(c*f - d*e)))/f, Ne(f, 0)), ((b*x/d + (a*d - b*c)*Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d)/e**(5/ 2), True))
Exception generated. \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)/(d*x+c)/(f*x+e)^(5/2),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 = 153, normalized size of antiderivative = 1.29 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=-\frac {2 \, {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (b d e^{2} + 3 \, {\left (f x + e\right )} b c f - 3 \, {\left (f x + e\right )} a d f - b c e f - a d e f + a c f^{2}\right )}}{3 \, {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} {\left (f x + e\right )}^{\frac {3}{2}}} \] Input:
integrate((b*x+a)/(d*x+c)/(f*x+e)^(5/2),x, algorithm="giac")
Output:
-2*(b*c*d - a*d^2)*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)) - 2/3*(b*d*e^2 + 3*(f*x + e)* b*c*f - 3*(f*x + e)*a*d*f - b*c*e*f - a*d*e*f + a*c*f^2)/((d^2*e^2*f - 2*c *d*e*f^2 + c^2*f^3)*(f*x + e)^(3/2))
Time = 1.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=\frac {2\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2\right )}{{\left (c\,f-d\,e\right )}^{5/2}}\right )\,\left (a\,d-b\,c\right )}{{\left (c\,f-d\,e\right )}^{5/2}}-\frac {\frac {2\,\left (a\,f-b\,e\right )}{3\,\left (c\,f-d\,e\right )}-\frac {2\,\left (e+f\,x\right )\,\left (a\,d\,f-b\,c\,f\right )}{{\left (c\,f-d\,e\right )}^2}}{f\,{\left (e+f\,x\right )}^{3/2}} \] Input:
int((a + b*x)/((e + f*x)^(5/2)*(c + d*x)),x)
Output:
(2*d^(1/2)*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))*(a*d - b*c))/(c*f - d*e)^(5/2) - ((2*(a*f - b*e))/(3*(c *f - d*e)) - (2*(e + f*x)*(a*d*f - b*c*f))/(c*f - d*e)^2)/(f*(e + f*x)^(3/ 2))
Time = 0.16 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.22 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx=\frac {2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) a d e f +2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) a d \,f^{2} x -2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) b c e f -2 \sqrt {d}\, \sqrt {f x +e}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) b c \,f^{2} x -\frac {2 a \,c^{2} f^{3}}{3}+\frac {10 a c d e \,f^{2}}{3}+2 a c d \,f^{3} x -\frac {8 a \,d^{2} e^{2} f}{3}-2 a \,d^{2} e \,f^{2} x -\frac {4 b \,c^{2} e \,f^{2}}{3}-2 b \,c^{2} f^{3} x +\frac {2 b c d \,e^{2} f}{3}+2 b c d e \,f^{2} x +\frac {2 b \,d^{2} e^{3}}{3}}{\sqrt {f x +e}\, f \left (c^{3} f^{4} x -3 c^{2} d e \,f^{3} x +3 c \,d^{2} e^{2} f^{2} x -d^{3} e^{3} f x +c^{3} e \,f^{3}-3 c^{2} d \,e^{2} f^{2}+3 c \,d^{2} e^{3} f -d^{3} e^{4}\right )} \] Input:
int((b*x+a)/(d*x+c)/(f*x+e)^(5/2),x)
Output:
(2*(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)))*a*d*e*f + 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)))*a*d*f**2*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))) *b*c*e*f - 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)))*b*c*f**2*x - a*c**2*f**3 + 5*a*c*d*e*f**2 + 3*a *c*d*f**3*x - 4*a*d**2*e**2*f - 3*a*d**2*e*f**2*x - 2*b*c**2*e*f**2 - 3*b* c**2*f**3*x + b*c*d*e**2*f + 3*b*c*d*e*f**2*x + b*d**2*e**3))/(3*sqrt(e + f*x)*f*(c**3*e*f**3 + c**3*f**4*x - 3*c**2*d*e**2*f**2 - 3*c**2*d*e*f**3*x + 3*c*d**2*e**3*f + 3*c*d**2*e**2*f**2*x - d**3*e**4 - d**3*e**3*f*x))