Integrand size = 46, antiderivative size = 186 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (2 c d-b e) (c e f+c d g-b e g) (d+e x)^{3/2}}{3 c^3 e^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (c e f+3 c d g-2 b e g) \sqrt {d+e x}}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^2 \sqrt {d+e x}} \] Output:
2/3*(-b*e+2*c*d)*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/c^3/e^2/(d*(-b*e+c*d)- b*e^2*x-c*e^2*x^2)^(3/2)-2*(-2*b*e*g+3*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*g*(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)
Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.63 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (8 b^2 e^2 g-2 b c e (9 d g+e (f-6 g x))+c^2 \left (10 d^2 g+d e (f-15 g x)+3 e^2 x (-f+g x)\right )\right )}{3 c^3 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(7/2)*(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]*(8*b^2*e^2*g - 2*b*c*e*(9*d*g + e*(f - 6*g*x)) + c^2*(10* d^2*g + d*e*(f - 15*g*x) + 3*e^2*x*(-f + g*x))))/(3*c^3*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.72 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.17, 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.
| \(\displaystyle \int \frac {(d+e x)^{7/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-4 b e g+7 c d g+c e f) \int \frac {(d+e x)^{5/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-4 b e g+7 c d g+c e f) \left (\frac {2 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{c}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left (\frac {4 \sqrt {d+e x} (2 c d-b e)}{c^2 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-4 b e g+7 c d g+c e f)}{3 c e (2 c d-b e)}\) |
Input:
Int[((d + e*x)^(7/2)*(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)^(7/2))/(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)) - ((c*e*f + 7*c*d*g - 4*b*e*g)*((4*( 2*c*d - b*e)*Sqrt[d + e*x])/(c^2*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^ 2]) - (2*(d + e*x)^(3/2))/(c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])) )/(3*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 = 1.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (3 g \,x^{2} c^{2} e^{2}+12 b c \,e^{2} g x -15 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -18 b c d e g -2 b c \,e^{2} f +10 c^{2} d^{2} g +c^{2} d e f \right )}{3 \sqrt {e x +d}\, \left (c e x +b e -c d \right )^{2} c^{3} e^{2}}\) | \(132\) |
| gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (3 g \,x^{2} c^{2} e^{2}+12 b c \,e^{2} g x -15 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -18 b c d e g -2 b c \,e^{2} f +10 c^{2} d^{2} g +c^{2} d e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{3} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(138\) |
| orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (3 g \,x^{2} c^{2} e^{2}+12 b c \,e^{2} g x -15 c^{2} d e g x -3 c^{2} e^{2} f x +8 b^{2} e^{2} g -18 b c d e g -2 b c \,e^{2} f +10 c^{2} d^{2} g +c^{2} d e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{3} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(138\) |
Input:
int((e*x+d)^(7/2)*(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^2*e^2*g*x^2+12*b* c*e^2*g*x-15*c^2*d*e*g*x-3*c^2*e^2*f*x+8*b^2*e^2*g-18*b*c*d*e*g-2*b*c*e^2* f+10*c^2*d^2*g+c^2*d*e*f)/(c*e*x+b*e-c*d)^2/c^3/e^2
Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (3 \, c^{2} e^{2} g x^{2} + {\left (c^{2} d e - 2 \, b c e^{2}\right )} f + 2 \, {\left (5 \, c^{2} d^{2} - 9 \, b c d e + 4 \, b^{2} e^{2}\right )} g - 3 \, {\left (c^{2} e^{2} f + {\left (5 \, c^{2} d e - 4 \, b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3 \, {\left (c^{5} e^{5} x^{3} + c^{5} d^{3} e^{2} - 2 \, b c^{4} d^{2} e^{3} + b^{2} c^{3} d e^{4} - {\left (c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} x^{2} - {\left (c^{5} d^{2} e^{3} - b^{2} c^{3} e^{5}\right )} x\right )}} \] Input:
integrate((e*x+d)^(7/2)*(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*(3*c^2*e^2*g*x^2 + (c^2*d*e - 2*b*c*e^2)*f + 2*(5*c^2*d^2 - 9*b*c*d*e + 4*b^2*e^2)*g - 3*(c^2*e^2*f + (5*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^5*e^5*x^3 + c^5*d^3*e^2 - 2*b*c^4*d^2*e^3 + b^2*c^3*d*e^4 - (c^5*d*e^4 - 2*b*c^4*e^5)*x^2 - (c^5*d^ 2*e^3 - b^2*c^3*e^5)*x)
Timed out. \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(7/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/ 2),x)
Output:
Timed out
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (3 \, c e x - c d + 2 \, b e\right )} f}{3 \, {\left (c^{3} e^{2} x - c^{3} d e + b c^{2} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} + 10 \, c^{2} d^{2} - 18 \, b c d e + 8 \, b^{2} e^{2} - 3 \, {\left (5 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \, {\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )} \sqrt {-c e x + c d - b e}} \] Input:
integrate((e*x+d)^(7/2)*(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 - c*d + 2*b*e)*f/((c^3*e^2*x - c^3*d*e + b*c^2*e^2)*sqrt(-c* e*x + c*d - b*e)) + 2/3*(3*c^2*e^2*x^2 + 10*c^2*d^2 - 18*b*c*d*e + 8*b^2*e ^2 - 3*(5*c^2*d*e - 4*b*c*e^2)*x)*g/((c^4*e^3*x - c^4*d*e^2 + b*c^3*e^3)*s qrt(-c*e*x + c*d - b*e))
Time = 0.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} g}{c^{3} e} + \frac {2 \, c^{2} d e f - b c e^{2} f + 2 \, c^{2} d^{2} g - 3 \, b c d e g + b^{2} e^{2} g + 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c e f + 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c d g - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b e g}{{\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} e}\right )}}{3 \, e} \] Input:
integrate((e*x+d)^(7/2)*(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*(3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*g/(c^3*e) + (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 + 3*((e*x + d)*c - 2*c*d + b*e )*c*e*f + 9*((e*x + d)*c - 2*c*d + b*e)*c*d*g - 6*((e*x + d)*c - 2*c*d + b *e)*b*e*g)/(((e*x + d)*c - 2*c*d + b*e)*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c ^3*e))/e
Time = 11.97 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {\left (\frac {\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-36\,g\,b\,c\,d\,e-4\,f\,b\,c\,e^2+20\,g\,c^2\,d^2+2\,f\,c^2\,d\,e\right )}{3\,c^5\,e^5}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}}{c^3\,e^3}-\frac {2\,x\,\sqrt {d+e\,x}\,\left (5\,c\,d\,g-4\,b\,e\,g+c\,e\,f\right )}{c^4\,e^4}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x^3+\frac {x\,\left (3\,b^2\,c^3\,e^5-3\,c^5\,d^2\,e^3\right )}{3\,c^5\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \] Input:
int(((f + g*x)*(d + e*x)^(7/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2 ),x)
Output:
-((((d + e*x)^(1/2)*(16*b^2*e^2*g + 20*c^2*d^2*g - 4*b*c*e^2*f + 2*c^2*d*e *f - 36*b*c*d*e*g))/(3*c^5*e^5) + (2*g*x^2*(d + e*x)^(1/2))/(c^3*e^3) - (2 *x*(d + e*x)^(1/2)*(5*c*d*g - 4*b*e*g + c*e*f))/(c^4*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^3*e^5 - 3*c^5*d^2*e^3))/( 3*c^5*e^5) + (d*(b*e - c*d)^2)/(c^2*e^3) + (x^2*(2*b*e - c*d))/(c*e))
Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 c^{2} e^{2} g \,x^{2}+8 b c \,e^{2} g x -10 c^{2} d e g x -2 c^{2} e^{2} f x +\frac {16}{3} b^{2} e^{2} g -12 b c d e g -\frac {4}{3} b c \,e^{2} f +\frac {20}{3} c^{2} d^{2} g +\frac {2}{3} c^{2} d e f}{\sqrt {-c e x -b e +c d}\, c^{3} e^{2} \left (c e x +b e -c d \right )} \] Input:
int((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*(8*b**2*e**2*g - 18*b*c*d*e*g - 2*b*c*e**2*f + 12*b*c*e**2*g*x + 10*c** 2*d**2*g + c**2*d*e*f - 15*c**2*d*e*g*x - 3*c**2*e**2*f*x + 3*c**2*e**2*g* x**2))/(3*sqrt( - b*e + c*d - c*e*x)*c**3*e**2*(b*e - c*d + c*e*x))