Integrand size = 46, antiderivative size = 114 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^{3/2}}{3 c^2 e^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 g \sqrt {d+e x}}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \] Output:
2/3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/c^2/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2 *x^2)^(3/2)-2*g*(e*x+d)^(1/2)/c^2/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/ 2)
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (-2 c d g+2 b e g+c e (f+3 g x))}{3 c^2 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(5/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2 )^(5/2),x]
Output:
(-2*Sqrt[d + e*x]*(-2*c*d*g + 2*b*e*g + c*e*(f + 3*g*x)))/(3*c^2*e^2*(-(c* d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.56 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1218, 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.
\(\displaystyle \int \frac {(d+e x)^{5/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle \frac {(2 b e g-5 c d g+c e f) \int \frac {(d+e x)^{3/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}+\frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 \sqrt {d+e x} (2 b e g-5 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
Input:
Int[((d + e*x)^(5/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2 ),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(5/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (2*(c*e*f - 5*c*d*g + 2*b*e*g)*Sqr t[d + e*x])/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2* x^2])
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_.)*((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 = 1.56 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (3 c e g x +2 b e g -2 c d g +f c e \right )}{3 \sqrt {e x +d}\, \left (c e x +b e -c d \right )^{2} c^{2} e^{2}}\) | \(72\) |
gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (3 c e g x +2 b e g -2 c d g +f c e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{2} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(78\) |
orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (3 c e g x +2 b e g -2 c d g +f c e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{2} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(78\) |
Input:
int((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method= _RETURNVERBOSE)
Output:
2/3/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(3*c*e*g*x+2*b*e*g-2*c* d*g+c*e*f)/(c*e*x+b*e-c*d)^2/c^2/e^2
Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (3 \, c e g x + c e f - 2 \, {\left (c d - b e\right )} g\right )} \sqrt {e x + d}}{3 \, {\left (c^{4} e^{5} x^{3} + c^{4} d^{3} e^{2} - 2 \, b c^{3} d^{2} e^{3} + b^{2} c^{2} d e^{4} - {\left (c^{4} d e^{4} - 2 \, b c^{3} e^{5}\right )} x^{2} - {\left (c^{4} d^{2} e^{3} - b^{2} c^{2} e^{5}\right )} x\right )}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
Output:
2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*c*e*g*x + c*e*f - 2*(c*d - b*e)*g)*sqrt(e*x + d)/(c^4*e^5*x^3 + c^4*d^3*e^2 - 2*b*c^3*d^2*e^3 + b^ 2*c^2*d*e^4 - (c^4*d*e^4 - 2*b*c^3*e^5)*x^2 - (c^4*d^2*e^3 - b^2*c^2*e^5)* x)
\[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/ 2),x)
Output:
Integral((d + e*x)**(5/2)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2 ), x)
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (3 \, c e x - 2 \, c d + 2 \, b e\right )} g}{3 \, {\left (c^{3} e^{3} x - c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt {-c e x + c d - b e}} - \frac {2 \, f}{3 \, {\left (c^{2} e^{2} x - c^{2} d e + b c e^{2}\right )} \sqrt {-c e x + c d - b e}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")
Output:
-2/3*(3*c*e*x - 2*c*d + 2*b*e)*g/((c^3*e^3*x - c^3*d*e^2 + b*c^2*e^3)*sqrt (-c*e*x + c*d - b*e)) - 2/3*f/((c^2*e^2*x - c^2*d*e + b*c*e^2)*sqrt(-c*e*x + c*d - b*e))
Time = 0.36 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (c e f + c d g - b e g + 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} g\right )}}{3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} e^{2}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
Output:
-2/3*(c*e*f + c*d*g - b*e*g + 3*((e*x + d)*c - 2*c*d + b*e)*g)/(((e*x + d) *c - 2*c*d + b*e)*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*e^2)
Time = 11.92 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {\left (\frac {\sqrt {d+e\,x}\,\left (4\,b\,e\,g-4\,c\,d\,g+2\,c\,e\,f\right )}{3\,c^4\,e^5}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{c^3\,e^4}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x^3+\frac {x\,\left (3\,b^2\,c^2\,e^5-3\,c^4\,d^2\,e^3\right )}{3\,c^4\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \] Input:
int(((f + g*x)*(d + e*x)^(5/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2 ),x)
Output:
((((d + e*x)^(1/2)*(4*b*e*g - 4*c*d*g + 2*c*e*f))/(3*c^4*e^5) + (2*g*x*(d + e*x)^(1/2))/(c^3*e^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(x^3 + (x*(3*b^2*c^2*e^5 - 3*c^4*d^2*e^3))/(3*c^4*e^5) + (d*(b*e - c*d)^2)/(c^ 2*e^3) + (x^2*(2*b*e - c*d))/(c*e))
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.53 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {-2 c e g x -\frac {4}{3} b e g +\frac {4}{3} c d g -\frac {2}{3} c e f}{\sqrt {-c e x -b e +c d}\, c^{2} e^{2} \left (c e x +b e -c d \right )} \] Input:
int((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*( - 2*b*e*g + 2*c*d*g - c*e*f - 3*c*e*g*x))/(3*sqrt( - b*e + c*d - c*e* x)*c**2*e**2*(b*e - c*d + c*e*x))