Integrand size = 46, antiderivative size = 186 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (2 c d-b e) (c e f+c d g-b e g) \sqrt {d+e x}}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^2 \sqrt {d+e x}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^3 e^2 (d+e x)^{3/2}} \] Output:
2*(-b*e+2*c*d)*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(1/2)/c^3/e^2/(d*(-b*e+c*d)-b* e^2*x-c*e^2*x^2)^(1/2)+2*(-2*b*e*g+3*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c* e^2*x^2)^(1/2)/c^3/e^2/(e*x+d)^(1/2)-2/3*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 )^(3/2)/c^3/e^2/(e*x+d)^(3/2)
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (-8 b^2 e^2 g+2 b c e (3 e f+11 d g-2 e g x)+c^2 \left (-14 d^2 g+e^2 x (3 f+g x)+d e (-9 f+7 g x)\right )\right )}{3 c^3 e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] 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 )^(3/2),x]
Output:
(-2*Sqrt[d + e*x]*(-8*b^2*e^2*g + 2*b*c*e*(3*e*f + 11*d*g - 2*e*g*x) + c^2 *(-14*d^2*g + e^2*x*(3*f + g*x) + d*e*(-9*f + 7*g*x))))/(3*c^3*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.71 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1218, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-4 b e g+5 c d g+3 c e f) \int \frac {(d+e x)^{3/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-4 b e g+5 c d g+3 c e f) \left (\frac {2 (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 c}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (-\frac {4 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right ) (-4 b e g+5 c d g+3 c e f)}{c e (2 c d-b e)}\) |
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)^(3/2 ),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(5/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c *d - b*e) - b*e^2*x - c*e^2*x^2]) - ((3*c*e*f + 5*c*d*g - 4*b*e*g)*((-4*(2 *c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c^2*e*Sqrt[d + e *x]) - (2*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c*e) ))/(c*e*(2*c*d - b*e))
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 = 1.67 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (-g \,x^{2} c^{2} e^{2}+4 b c \,e^{2} g x -7 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -22 b c d e g -6 b c \,e^{2} f +14 c^{2} d^{2} g +9 c^{2} d e f \right )}{3 \sqrt {e x +d}\, \left (c e x +b e -c d \right ) c^{3} e^{2}}\) | \(133\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-g \,x^{2} c^{2} e^{2}+4 b c \,e^{2} g x -7 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -22 b c d e g -6 b c \,e^{2} f +14 c^{2} d^{2} g +9 c^{2} d e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{3} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(139\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-g \,x^{2} c^{2} e^{2}+4 b c \,e^{2} g x -7 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -22 b c d e g -6 b c \,e^{2} f +14 c^{2} d^{2} g +9 c^{2} d e f \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{3} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(139\) |
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)^(3/2),x,method= _RETURNVERBOSE)
Output:
-2/3/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-c^2*e^2*g*x^2+4*b*c* e^2*g*x-7*c^2*d*e*g*x-3*c^2*e^2*f*x+8*b^2*e^2*g-22*b*c*d*e*g-6*b*c*e^2*f+1 4*c^2*d^2*g+9*c^2*d*e*f)/(c*e*x+b*e-c*d)/c^3/e^2
Time = 0.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (c^{2} e^{2} g x^{2} - 3 \, {\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} f - 2 \, {\left (7 \, c^{2} d^{2} - 11 \, b c d e + 4 \, b^{2} e^{2}\right )} g + {\left (3 \, c^{2} e^{2} f + {\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3 \, {\left (c^{4} e^{4} x^{2} + b c^{3} e^{4} x - c^{4} d^{2} e^{2} + b c^{3} d e^{3}\right )}} \] 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)^(3/2),x, algorithm="fricas")
Output:
2/3*(c^2*e^2*g*x^2 - 3*(3*c^2*d*e - 2*b*c*e^2)*f - 2*(7*c^2*d^2 - 11*b*c*d *e + 4*b^2*e^2)*g + (3*c^2*e^2*f + (7*c^2*d*e - 4*b*c*e^2)*g)*x)*sqrt(-c*e ^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^4*e^4*x^2 + b*c^3*e^4*x - c^4*d^2*e^2 + b*c^3*d*e^3)
\[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \] 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)**(3/ 2),x)
Output:
Integral((d + e*x)**(5/2)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(3/2 ), x)
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.60 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (c e x - 3 \, c d + 2 \, b e\right )} f}{\sqrt {-c e x + c d - b e} c^{2} e} - \frac {2 \, {\left (c^{2} e^{2} x^{2} - 14 \, c^{2} d^{2} + 22 \, b c d e - 8 \, b^{2} e^{2} + {\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \, \sqrt {-c e x + c d - b e} c^{3} e^{2}} \] 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)^(3/2),x, algorithm="maxima")
Output:
-2*(c*e*x - 3*c*d + 2*b*e)*f/(sqrt(-c*e*x + c*d - b*e)*c^2*e) - 2/3*(c^2*e ^2*x^2 - 14*c^2*d^2 + 22*b*c*d*e - 8*b^2*e^2 + (7*c^2*d*e - 4*b*c*e^2)*x)* g/(sqrt(-c*e*x + c*d - b*e)*c^3*e^2)
Time = 0.30 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (2 \, c^{2} d e f - b c e^{2} f + 2 \, c^{2} d^{2} g - 3 \, b c d e g + b^{2} e^{2} g\right )}}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} e} + \frac {3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{7} e^{3} f + 9 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{7} d e^{2} g - 6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{6} e^{3} g - {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{6} e^{2} g}{c^{9} e^{3}}\right )}}{3 \, e} \] 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)^(3/2),x, algorithm="giac")
Output:
2/3*(3*(2*c^2*d*e*f - b*c*e^2*f + 2*c^2*d^2*g - 3*b*c*d*e*g + b^2*e^2*g)/( sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*e) + (3*sqrt(-(e*x + d)*c + 2*c*d - b *e)*c^7*e^3*f + 9*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^7*d*e^2*g - 6*sqrt(-( e*x + d)*c + 2*c*d - b*e)*b*c^6*e^3*g - (-(e*x + d)*c + 2*c*d - b*e)^(3/2) *c^6*e^2*g)/(c^9*e^3))/e
Time = 10.81 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {2\,g\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,e^2}-\frac {\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-44\,g\,b\,c\,d\,e-12\,f\,b\,c\,e^2+28\,g\,c^2\,d^2+18\,f\,c^2\,d\,e\right )}{3\,c^4\,e^4}+\frac {2\,x\,\sqrt {d+e\,x}\,\left (7\,c\,d\,g-4\,b\,e\,g+3\,c\,e\,f\right )}{3\,c^3\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \] 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)^(3/2 ),x)
Output:
(((2*g*x^2*(d + e*x)^(1/2))/(3*c^2*e^2) - ((d + e*x)^(1/2)*(16*b^2*e^2*g + 28*c^2*d^2*g - 12*b*c*e^2*f + 18*c^2*d*e*f - 44*b*c*d*e*g))/(3*c^4*e^4) + (2*x*(d + e*x)^(1/2)*(7*c*d*g - 4*b*e*g + 3*c*e*f))/(3*c^3*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(x^2 + (b*x)/c + (d*(b*e - c*d))/(c*e ^2))
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.57 \[ \int \frac {(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {-\frac {2}{3} c^{2} e^{2} g \,x^{2}+\frac {8}{3} b c \,e^{2} g x -\frac {14}{3} c^{2} d e g x -2 c^{2} e^{2} f x +\frac {16}{3} b^{2} e^{2} g -\frac {44}{3} b c d e g -4 b c \,e^{2} f +\frac {28}{3} c^{2} d^{2} g +6 c^{2} d e f}{\sqrt {-c e x -b e +c d}\, c^{3} e^{2}} \] 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)^(3/2),x)
Output:
(2*(8*b**2*e**2*g - 22*b*c*d*e*g - 6*b*c*e**2*f + 4*b*c*e**2*g*x + 14*c**2 *d**2*g + 9*c**2*d*e*f - 7*c**2*d*e*g*x - 3*c**2*e**2*f*x - c**2*e**2*g*x* *2))/(3*sqrt( - b*e + c*d - c*e*x)*c**3*e**2)