Integrand size = 24, antiderivative size = 128 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 b^2 (b B-A c) x^{3/2}}{3 c^4 \left (b x+c x^2\right )^{3/2}}-\frac {2 b (3 b B-2 A c) \sqrt {x}}{c^4 \sqrt {b x+c x^2}}-\frac {2 (3 b B-A c) \sqrt {b x+c x^2}}{c^4 \sqrt {x}}+\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 c^4 x^{3/2}} \] Output:
2/3*b^2*(-A*c+B*b)*x^(3/2)/c^4/(c*x^2+b*x)^(3/2)-2*b*(-2*A*c+3*B*b)*x^(1/2 )/c^4/(c*x^2+b*x)^(1/2)-2*(-A*c+3*B*b)*(c*x^2+b*x)^(1/2)/c^4/x^(1/2)+2/3*B *(c*x^2+b*x)^(3/2)/c^4/x^(3/2)
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 x^{3/2} \left (-16 b^3 B+8 b^2 c (A-3 B x)-6 b c^2 x (-2 A+B x)+c^3 x^2 (3 A+B x)\right )}{3 c^4 (x (b+c x))^{3/2}} \] Input:
Integrate[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x]
Output:
(2*x^(3/2)*(-16*b^3*B + 8*b^2*c*(A - 3*B*x) - 6*b*c^2*x*(-2*A + B*x) + c^3 *x^2*(3*A + B*x)))/(3*c^4*(x*(b + c*x))^(3/2))
Time = 0.47 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1218, 1128, 1128, 1122}
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.
\(\displaystyle \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle \frac {(2 b B-A c) \int \frac {x^{7/2}}{\left (c x^2+b x\right )^{3/2}}dx}{b c}-\frac {2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {(2 b B-A c) \left (\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}}-\frac {4 b \int \frac {x^{5/2}}{\left (c x^2+b x\right )^{3/2}}dx}{3 c}\right )}{b c}-\frac {2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {(2 b B-A c) \left (\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}}-\frac {4 b \left (\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}}-\frac {2 b \int \frac {x^{3/2}}{\left (c x^2+b x\right )^{3/2}}dx}{c}\right )}{3 c}\right )}{b c}-\frac {2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {\left (\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}}-\frac {4 b \left (\frac {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}}\right )}{3 c}\right ) (2 b B-A c)}{b c}-\frac {2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}\) |
Input:
Int[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x]
Output:
(-2*(b*B - A*c)*x^(9/2))/(3*b*c*(b*x + c*x^2)^(3/2)) + ((2*b*B - A*c)*((2* x^(5/2))/(3*c*Sqrt[b*x + c*x^2]) - (4*b*((4*b*Sqrt[x])/(c^2*Sqrt[b*x + c*x ^2]) + (2*x^(3/2))/(c*Sqrt[b*x + c*x^2])))/(3*c)))/(b*c)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 1.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {2 \left (c x +b \right ) \left (B \,c^{3} x^{3}+3 A \,c^{3} x^{2}-6 B b \,c^{2} x^{2}+12 A b \,c^{2} x -24 B \,b^{2} c x +8 A \,b^{2} c -16 B \,b^{3}\right ) x^{\frac {5}{2}}}{3 c^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(82\) |
default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (B \,c^{3} x^{3}+3 A \,c^{3} x^{2}-6 B b \,c^{2} x^{2}+12 A b \,c^{2} x -24 B \,b^{2} c x +8 A \,b^{2} c -16 B \,b^{3}\right )}{3 \sqrt {x}\, \left (c x +b \right )^{2} c^{4}}\) | \(82\) |
orering | \(\frac {2 \left (c x +b \right ) \left (B \,c^{3} x^{3}+3 A \,c^{3} x^{2}-6 B b \,c^{2} x^{2}+12 A b \,c^{2} x -24 B \,b^{2} c x +8 A \,b^{2} c -16 B \,b^{3}\right ) x^{\frac {5}{2}}}{3 c^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(82\) |
risch | \(\frac {2 \left (B c x +3 A c -8 B b \right ) \left (c x +b \right ) \sqrt {x}}{3 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {2 b \left (6 A \,c^{2} x -9 B b c x +5 A b c -8 B \,b^{2}\right ) \sqrt {x}}{3 c^{4} \left (c x +b \right ) \sqrt {x \left (c x +b \right )}}\) | \(87\) |
Input:
int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3*(c*x+b)*(B*c^3*x^3+3*A*c^3*x^2-6*B*b*c^2*x^2+12*A*b*c^2*x-24*B*b^2*c*x +8*A*b^2*c-16*B*b^3)*x^(5/2)/c^4/(c*x^2+b*x)^(5/2)
Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (B c^{3} x^{3} - 16 \, B b^{3} + 8 \, A b^{2} c - 3 \, {\left (2 \, B b c^{2} - A c^{3}\right )} x^{2} - 12 \, {\left (2 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (c^{6} x^{3} + 2 \, b c^{5} x^{2} + b^{2} c^{4} x\right )}} \] Input:
integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
Output:
2/3*(B*c^3*x^3 - 16*B*b^3 + 8*A*b^2*c - 3*(2*B*b*c^2 - A*c^3)*x^2 - 12*(2* B*b^2*c - A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(x)/(c^6*x^3 + 2*b*c^5*x^2 + b ^2*c^4*x)
Timed out. \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x**(9/2)*(B*x+A)/(c*x**2+b*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} x^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")
Output:
2/3*(B*c*x + B*b)*sqrt(c*x + b)*x^3/(c^4*x^3 + 3*b*c^3*x^2 + 3*b^2*c^2*x + b^3*c) + integrate((A*b*c*x^3 - (2*B*b^2 + (2*B*b*c - A*c^2)*x)*x^3)*sqrt (c*x + b)/(c^5*x^5 + 4*b*c^4*x^4 + 6*b^2*c^3*x^3 + 4*b^3*c^2*x^2 + b^4*c*x ), x)
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.72 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (9 \, {\left (c x + b\right )} B b^{2} - B b^{3} - 6 \, {\left (c x + b\right )} A b c + A b^{2} c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{4}} + \frac {2 \, {\left ({\left (c x + b\right )}^{\frac {3}{2}} B c^{8} - 9 \, \sqrt {c x + b} B b c^{8} + 3 \, \sqrt {c x + b} A c^{9}\right )}}{3 \, c^{12}} \] Input:
integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="giac")
Output:
-2/3*(9*(c*x + b)*B*b^2 - B*b^3 - 6*(c*x + b)*A*b*c + A*b^2*c)/((c*x + b)^ (3/2)*c^4) + 2/3*((c*x + b)^(3/2)*B*c^8 - 9*sqrt(c*x + b)*B*b*c^8 + 3*sqrt (c*x + b)*A*c^9)/c^12
Timed out. \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {x^{9/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \] Input:
int((x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x)
Output:
int((x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2), x)
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int \frac {x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {\frac {2}{3} b \,c^{3} x^{3}+2 a \,c^{3} x^{2}-4 b^{2} c^{2} x^{2}+8 a b \,c^{2} x -16 b^{3} c x +\frac {16}{3} a \,b^{2} c -\frac {32}{3} b^{4}}{\sqrt {c x +b}\, c^{4} \left (c x +b \right )} \] Input:
int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x)
Output:
(2*(8*a*b**2*c + 12*a*b*c**2*x + 3*a*c**3*x**2 - 16*b**4 - 24*b**3*c*x - 6 *b**2*c**2*x**2 + b*c**3*x**3))/(3*sqrt(b + c*x)*c**4*(b + c*x))