Integrand size = 22, antiderivative size = 98 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=\frac {(b B-A c) \sqrt {x}}{2 c^2 (b+c x)^2}-\frac {(5 b B-A c) \sqrt {x}}{4 b c^2 (b+c x)}+\frac {(3 b B+A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{3/2} c^{5/2}} \] Output:
1/2*(-A*c+B*b)*x^(1/2)/c^2/(c*x+b)^2-1/4*(-A*c+5*B*b)*x^(1/2)/b/c^2/(c*x+b )+1/4*(A*c+3*B*b)*arctan(c^(1/2)*x^(1/2)/b^(1/2))/b^(3/2)/c^(5/2)
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {\sqrt {x} \left (3 b^2 B+A b c+5 b B c x-A c^2 x\right )}{4 b c^2 (b+c x)^2}+\frac {(3 b B+A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{3/2} c^{5/2}} \] Input:
Integrate[(x^(7/2)*(A + B*x))/(b*x + c*x^2)^3,x]
Output:
-1/4*(Sqrt[x]*(3*b^2*B + A*b*c + 5*b*B*c*x - A*c^2*x))/(b*c^2*(b + c*x)^2) + ((3*b*B + A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(3/2)*c^(5/2))
Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {9, 87, 51, 73, 218}
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.
\(\displaystyle \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {\sqrt {x} (A+B x)}{(b+c x)^3}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(A c+3 b B) \int \frac {\sqrt {x}}{(b+c x)^2}dx}{4 b c}-\frac {x^{3/2} (b B-A c)}{2 b c (b+c x)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {(A c+3 b B) \left (\frac {\int \frac {1}{\sqrt {x} (b+c x)}dx}{2 c}-\frac {\sqrt {x}}{c (b+c x)}\right )}{4 b c}-\frac {x^{3/2} (b B-A c)}{2 b c (b+c x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(A c+3 b B) \left (\frac {\int \frac {1}{b+c x}d\sqrt {x}}{c}-\frac {\sqrt {x}}{c (b+c x)}\right )}{4 b c}-\frac {x^{3/2} (b B-A c)}{2 b c (b+c x)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(A c+3 b B) \left (\frac {\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}}-\frac {\sqrt {x}}{c (b+c x)}\right )}{4 b c}-\frac {x^{3/2} (b B-A c)}{2 b c (b+c x)^2}\) |
Input:
Int[(x^(7/2)*(A + B*x))/(b*x + c*x^2)^3,x]
Output:
-1/2*((b*B - A*c)*x^(3/2))/(b*c*(b + c*x)^2) + ((3*b*B + A*c)*(-(Sqrt[x]/( c*(b + c*x))) + ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]]/(Sqrt[b]*c^(3/2))))/(4*b *c)
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.83 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\frac {\left (A c -5 B b \right ) x^{\frac {3}{2}}}{4 b c}-\frac {\left (A c +3 B b \right ) \sqrt {x}}{4 c^{2}}}{\left (c x +b \right )^{2}}+\frac {\left (A c +3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 c^{2} b \sqrt {b c}}\) | \(79\) |
default | \(\frac {\frac {\left (A c -5 B b \right ) x^{\frac {3}{2}}}{4 b c}-\frac {\left (A c +3 B b \right ) \sqrt {x}}{4 c^{2}}}{\left (c x +b \right )^{2}}+\frac {\left (A c +3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 c^{2} b \sqrt {b c}}\) | \(79\) |
Input:
int(x^(7/2)*(B*x+A)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
Output:
2*(1/8*(A*c-5*B*b)/b/c*x^(3/2)-1/8*(A*c+3*B*b)/c^2*x^(1/2))/(c*x+b)^2+1/4* (A*c+3*B*b)/c^2/b/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))
Time = 0.10 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.97 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=\left [-\frac {{\left (3 \, B b^{3} + A b^{2} c + {\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} c + A b c^{2}\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x - b - 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right ) + 2 \, {\left (3 \, B b^{3} c + A b^{2} c^{2} + {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (b^{2} c^{5} x^{2} + 2 \, b^{3} c^{4} x + b^{4} c^{3}\right )}}, -\frac {{\left (3 \, B b^{3} + A b^{2} c + {\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} c + A b c^{2}\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (3 \, B b^{3} c + A b^{2} c^{2} + {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (b^{2} c^{5} x^{2} + 2 \, b^{3} c^{4} x + b^{4} c^{3}\right )}}\right ] \] Input:
integrate(x^(7/2)*(B*x+A)/(c*x^2+b*x)^3,x, algorithm="fricas")
Output:
[-1/8*((3*B*b^3 + A*b^2*c + (3*B*b*c^2 + A*c^3)*x^2 + 2*(3*B*b^2*c + A*b*c ^2)*x)*sqrt(-b*c)*log((c*x - b - 2*sqrt(-b*c)*sqrt(x))/(c*x + b)) + 2*(3*B *b^3*c + A*b^2*c^2 + (5*B*b^2*c^2 - A*b*c^3)*x)*sqrt(x))/(b^2*c^5*x^2 + 2* b^3*c^4*x + b^4*c^3), -1/4*((3*B*b^3 + A*b^2*c + (3*B*b*c^2 + A*c^3)*x^2 + 2*(3*B*b^2*c + A*b*c^2)*x)*sqrt(b*c)*arctan(sqrt(b*c)/(c*sqrt(x))) + (3*B *b^3*c + A*b^2*c^2 + (5*B*b^2*c^2 - A*b*c^3)*x)*sqrt(x))/(b^2*c^5*x^2 + 2* b^3*c^4*x + b^4*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 1316 vs. \(2 (90) = 180\).
Time = 166.17 (sec) , antiderivative size = 1316, normalized size of antiderivative = 13.43 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x**(7/2)*(B*x+A)/(c*x**2+b*x)**3,x)
Output:
Piecewise((zoo*(-2*A/(3*x**(3/2)) - 2*B/sqrt(x)), Eq(b, 0) & Eq(c, 0)), (( 2*A*x**(3/2)/3 + 2*B*x**(5/2)/5)/b**3, Eq(c, 0)), ((-2*A/(3*x**(3/2)) - 2* B/sqrt(x))/c**3, Eq(b, 0)), (A*b**2*c*log(sqrt(x) - sqrt(-b/c))/(8*b**3*c* *3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c)) - A* b**2*c*log(sqrt(x) + sqrt(-b/c))/(8*b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x* sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c)) - 2*A*b*c**2*sqrt(x)*sqrt(-b/c)/(8* b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c )) + 2*A*b*c**2*x*log(sqrt(x) - sqrt(-b/c))/(8*b**3*c**3*sqrt(-b/c) + 16*b **2*c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c)) - 2*A*b*c**2*x*log(sqrt( x) + sqrt(-b/c))/(8*b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b *c**5*x**2*sqrt(-b/c)) + 2*A*c**3*x**(3/2)*sqrt(-b/c)/(8*b**3*c**3*sqrt(-b /c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c)) + A*c**3*x**2* log(sqrt(x) - sqrt(-b/c))/(8*b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b /c) + 8*b*c**5*x**2*sqrt(-b/c)) - A*c**3*x**2*log(sqrt(x) + sqrt(-b/c))/(8 *b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/ c)) + 3*B*b**3*log(sqrt(x) - sqrt(-b/c))/(8*b**3*c**3*sqrt(-b/c) + 16*b**2 *c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c)) - 3*B*b**3*log(sqrt(x) + sq rt(-b/c))/(8*b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b*c**5*x **2*sqrt(-b/c)) - 6*B*b**2*c*sqrt(x)*sqrt(-b/c)/(8*b**3*c**3*sqrt(-b/c) + 16*b**2*c**4*x*sqrt(-b/c) + 8*b*c**5*x**2*sqrt(-b/c)) + 6*B*b**2*c*x*lo...
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=-\frac {{\left (5 \, B b c - A c^{2}\right )} x^{\frac {3}{2}} + {\left (3 \, B b^{2} + A b c\right )} \sqrt {x}}{4 \, {\left (b c^{4} x^{2} + 2 \, b^{2} c^{3} x + b^{3} c^{2}\right )}} + \frac {{\left (3 \, B b + A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b c^{2}} \] Input:
integrate(x^(7/2)*(B*x+A)/(c*x^2+b*x)^3,x, algorithm="maxima")
Output:
-1/4*((5*B*b*c - A*c^2)*x^(3/2) + (3*B*b^2 + A*b*c)*sqrt(x))/(b*c^4*x^2 + 2*b^2*c^3*x + b^3*c^2) + 1/4*(3*B*b + A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sq rt(b*c)*b*c^2)
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=\frac {{\left (3 \, B b + A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b c^{2}} - \frac {5 \, B b c x^{\frac {3}{2}} - A c^{2} x^{\frac {3}{2}} + 3 \, B b^{2} \sqrt {x} + A b c \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} b c^{2}} \] Input:
integrate(x^(7/2)*(B*x+A)/(c*x^2+b*x)^3,x, algorithm="giac")
Output:
1/4*(3*B*b + A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b*c^2) - 1/4*(5*B *b*c*x^(3/2) - A*c^2*x^(3/2) + 3*B*b^2*sqrt(x) + A*b*c*sqrt(x))/((c*x + b) ^2*b*c^2)
Time = 5.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c+3\,B\,b\right )}{4\,b^{3/2}\,c^{5/2}}-\frac {\frac {\sqrt {x}\,\left (A\,c+3\,B\,b\right )}{4\,c^2}-\frac {x^{3/2}\,\left (A\,c-5\,B\,b\right )}{4\,b\,c}}{b^2+2\,b\,c\,x+c^2\,x^2} \] Input:
int((x^(7/2)*(A + B*x))/(b*x + c*x^2)^3,x)
Output:
(atan((c^(1/2)*x^(1/2))/b^(1/2))*(A*c + 3*B*b))/(4*b^(3/2)*c^(5/2)) - ((x^ (1/2)*(A*c + 3*B*b))/(4*c^2) - (x^(3/2)*(A*c - 5*B*b))/(4*b*c))/(b^2 + c^2 *x^2 + 2*b*c*x)
Time = 0.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.17 \[ \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx=\frac {\sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c +2 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) a b \,c^{2} x +\sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) a \,c^{3} x^{2}+3 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) b^{4}+6 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c x +3 \sqrt {c}\, \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {x}\, c}{\sqrt {c}\, \sqrt {b}}\right ) b^{2} c^{2} x^{2}-\sqrt {x}\, a \,b^{2} c^{2}+\sqrt {x}\, a b \,c^{3} x -3 \sqrt {x}\, b^{4} c -5 \sqrt {x}\, b^{3} c^{2} x}{4 b^{2} c^{3} \left (c^{2} x^{2}+2 b c x +b^{2}\right )} \] Input:
int(x^(7/2)*(B*x+A)/(c*x^2+b*x)^3,x)
Output:
(sqrt(c)*sqrt(b)*atan((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*a*b**2*c + 2*sqrt(c)* sqrt(b)*atan((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*a*b*c**2*x + sqrt(c)*sqrt(b)*a tan((sqrt(x)*c)/(sqrt(c)*sqrt(b)))*a*c**3*x**2 + 3*sqrt(c)*sqrt(b)*atan((s qrt(x)*c)/(sqrt(c)*sqrt(b)))*b**4 + 6*sqrt(c)*sqrt(b)*atan((sqrt(x)*c)/(sq rt(c)*sqrt(b)))*b**3*c*x + 3*sqrt(c)*sqrt(b)*atan((sqrt(x)*c)/(sqrt(c)*sqr t(b)))*b**2*c**2*x**2 - sqrt(x)*a*b**2*c**2 + sqrt(x)*a*b*c**3*x - 3*sqrt( x)*b**4*c - 5*sqrt(x)*b**3*c**2*x)/(4*b**2*c**3*(b**2 + 2*b*c*x + c**2*x** 2))