Integrand size = 46, antiderivative size = 263 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (2 c d-b e)^2 (c e f+c d g-b e g) (d+e x)^{3/2}}{3 c^4 e^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c d-b e) (2 c e f+4 c d g-3 b e g) \sqrt {d+e x}}{c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (c e f+5 c d g-3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^4 e^2 \sqrt {d+e x}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^4 e^2 (d+e x)^{3/2}} \] Output:
2/3*(-b*e+2*c*d)^2*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/c^4/e^2/(d*(-b*e+c*d )-b*e^2*x-c*e^2*x^2)^(3/2)-2*(-b*e+2*c*d)*(-3*b*e*g+4*c*d*g+2*c*e*f)*(e*x+ d)^(1/2)/c^4/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-2*(-3*b*e*g+5*c*d* g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^4/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^4/e^2/(e*x+d)^(3/2)
Time = 0.22 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (-16 b^3 e^3 g+8 b^2 c e^2 (8 d g+e (f-3 g x))-2 b c^2 e \left (41 d^2 g+2 d e (5 f-18 g x)+3 e^2 x (-2 f+g x)\right )+c^3 \left (34 d^3 g+d^2 e (11 f-51 g x)+e^3 x^2 (3 f+g x)+6 d e^2 x (-3 f+2 g x)\right )\right )}{3 c^4 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(9/2)*(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]*(-16*b^3*e^3*g + 8*b^2*c*e^2*(8*d*g + e*(f - 3*g*x)) - 2* b*c^2*e*(41*d^2*g + 2*d*e*(5*f - 18*g*x) + 3*e^2*x*(-2*f + g*x)) + c^3*(34 *d^3*g + d^2*e*(11*f - 51*g*x) + e^3*x^2*(3*f + g*x) + 6*d*e^2*x*(-3*f + 2 *g*x))))/(3*c^4*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.88 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {(d+e x)^{9/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-2 b e g+3 c d g+c e f) \int \frac {(d+e x)^{7/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-2 b e g+3 c d g+c e f) \left (\frac {4 (2 c d-b e) \int \frac {(d+e x)^{5/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-2 b e g+3 c d g+c e f) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{c}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left (\frac {4 (2 c d-b e) \left (\frac {4 \sqrt {d+e x} (2 c d-b e)}{c^2 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-2 b e g+3 c d g+c e f)}{c e (2 c d-b e)}\) |
Input:
Int[((d + e*x)^(9/2)*(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)^(9/2))/(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 + 3*c*d*g - 2*b*e*g)*((-2* (d + e*x)^(5/2))/(3*c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (4*(2 *c*d - b*e)*((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)))/(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 = 1.68 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (-e^{3} g \,x^{3} c^{3}+6 b \,c^{2} e^{3} g \,x^{2}-12 c^{3} d \,e^{2} g \,x^{2}-3 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -72 b \,c^{2} d \,e^{2} g x -12 b \,c^{2} e^{3} f x +51 c^{3} d^{2} e g x +18 c^{3} d \,e^{2} f x +16 b^{3} e^{3} g -64 b^{2} c d \,e^{2} g -8 b^{2} c \,e^{3} f +82 b \,c^{2} d^{2} e g +20 b \,c^{2} d \,e^{2} f -34 c^{3} d^{3} g -11 d^{2} f \,c^{3} e \right )}{3 \sqrt {e x +d}\, \left (c e x +b e -c d \right )^{2} c^{4} e^{2}}\) | \(229\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-e^{3} g \,x^{3} c^{3}+6 b \,c^{2} e^{3} g \,x^{2}-12 c^{3} d \,e^{2} g \,x^{2}-3 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -72 b \,c^{2} d \,e^{2} g x -12 b \,c^{2} e^{3} f x +51 c^{3} d^{2} e g x +18 c^{3} d \,e^{2} f x +16 b^{3} e^{3} g -64 b^{2} c d \,e^{2} g -8 b^{2} c \,e^{3} f +82 b \,c^{2} d^{2} e g +20 b \,c^{2} d \,e^{2} f -34 c^{3} d^{3} g -11 d^{2} f \,c^{3} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{4} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(235\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-e^{3} g \,x^{3} c^{3}+6 b \,c^{2} e^{3} g \,x^{2}-12 c^{3} d \,e^{2} g \,x^{2}-3 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -72 b \,c^{2} d \,e^{2} g x -12 b \,c^{2} e^{3} f x +51 c^{3} d^{2} e g x +18 c^{3} d \,e^{2} f x +16 b^{3} e^{3} g -64 b^{2} c d \,e^{2} g -8 b^{2} c \,e^{3} f +82 b \,c^{2} d^{2} e g +20 b \,c^{2} d \,e^{2} f -34 c^{3} d^{3} g -11 d^{2} f \,c^{3} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{4} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(235\) |
Input:
int((e*x+d)^(9/2)*(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)*(-c^3*e^3*g*x^3+6*b*c^2 *e^3*g*x^2-12*c^3*d*e^2*g*x^2-3*c^3*e^3*f*x^2+24*b^2*c*e^3*g*x-72*b*c^2*d* e^2*g*x-12*b*c^2*e^3*f*x+51*c^3*d^2*e*g*x+18*c^3*d*e^2*f*x+16*b^3*e^3*g-64 *b^2*c*d*e^2*g-8*b^2*c*e^3*f+82*b*c^2*d^2*e*g+20*b*c^2*d*e^2*f-34*c^3*d^3* g-11*c^3*d^2*e*f)/(c*e*x+b*e-c*d)^2/c^4/e^2
Time = 0.14 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (c^{3} e^{3} g x^{3} + 3 \, {\left (c^{3} e^{3} f + 2 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + {\left (11 \, c^{3} d^{2} e - 20 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \, {\left (17 \, c^{3} d^{3} - 41 \, b c^{2} d^{2} e + 32 \, b^{2} c d e^{2} - 8 \, b^{3} e^{3}\right )} g - 3 \, {\left (2 \, {\left (3 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3 \, {\left (c^{6} e^{5} x^{3} + c^{6} d^{3} e^{2} - 2 \, b c^{5} d^{2} e^{3} + b^{2} c^{4} d e^{4} - {\left (c^{6} d e^{4} - 2 \, b c^{5} e^{5}\right )} x^{2} - {\left (c^{6} d^{2} e^{3} - b^{2} c^{4} e^{5}\right )} x\right )}} \] Input:
integrate((e*x+d)^(9/2)*(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*(c^3*e^3*g*x^3 + 3*(c^3*e^3*f + 2*(2*c^3*d*e^2 - b*c^2*e^3)*g)*x^2 + (11*c^3*d^2*e - 20*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + 2*(17*c^3*d^3 - 41*b*c^2 *d^2*e + 32*b^2*c*d*e^2 - 8*b^3*e^3)*g - 3*(2*(3*c^3*d*e^2 - 2*b*c^2*e^3)* f + (17*c^3*d^2*e - 24*b*c^2*d*e^2 + 8*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^6*e^5*x^3 + c^6*d^3*e^2 - 2*b*c^ 5*d^2*e^3 + b^2*c^4*d*e^4 - (c^6*d*e^4 - 2*b*c^5*e^5)*x^2 - (c^6*d^2*e^3 - b^2*c^4*e^5)*x)
Timed out. \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(9/2)*(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 = 246, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} + 11 \, c^{2} d^{2} - 20 \, b c d e + 8 \, b^{2} e^{2} - 6 \, {\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, {\left (c^{4} e^{2} x - c^{4} d e + b c^{3} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (c^{3} e^{3} x^{3} + 34 \, c^{3} d^{3} - 82 \, b c^{2} d^{2} e + 64 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 6 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} g}{3 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )} \sqrt {-c e x + c d - b e}} \] Input:
integrate((e*x+d)^(9/2)*(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^2*e^2*x^2 + 11*c^2*d^2 - 20*b*c*d*e + 8*b^2*e^2 - 6*(3*c^2*d*e - 2*b*c*e^2)*x)*f/((c^4*e^2*x - c^4*d*e + b*c^3*e^2)*sqrt(-c*e*x + c*d - b*e )) + 2/3*(c^3*e^3*x^3 + 34*c^3*d^3 - 82*b*c^2*d^2*e + 64*b^2*c*d*e^2 - 16* b^3*e^3 + 6*(2*c^3*d*e^2 - b*c^2*e^3)*x^2 - 3*(17*c^3*d^2*e - 24*b*c^2*d*e ^2 + 8*b^2*c*e^3)*x)*g/((c^5*e^3*x - c^5*d*e^2 + b*c^4*e^3)*sqrt(-c*e*x + c*d - b*e))
Time = 0.36 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {4 \, c^{3} d^{2} e f - 4 \, b c^{2} d e^{2} f + b^{2} c e^{3} f + 4 \, c^{3} d^{3} g - 8 \, b c^{2} d^{2} e g + 5 \, b^{2} c d e^{2} g - b^{3} e^{3} g + 12 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} d e f - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c e^{2} f + 24 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} d^{2} g - 30 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c d e g + 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b^{2} e^{2} g}{{\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} e} + \frac {3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{9} e^{3} f + 15 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{9} d e^{2} g - 9 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{8} e^{3} g - {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{8} e^{2} g}{c^{12} e^{3}}\right )}}{3 \, e} \] Input:
integrate((e*x+d)^(9/2)*(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*((4*c^3*d^2*e*f - 4*b*c^2*d*e^2*f + b^2*c*e^3*f + 4*c^3*d^3*g - 8*b*c ^2*d^2*e*g + 5*b^2*c*d*e^2*g - b^3*e^3*g + 12*((e*x + d)*c - 2*c*d + b*e)* c^2*d*e*f - 6*((e*x + d)*c - 2*c*d + b*e)*b*c*e^2*f + 24*((e*x + d)*c - 2* c*d + b*e)*c^2*d^2*g - 30*((e*x + d)*c - 2*c*d + b*e)*b*c*d*e*g + 9*((e*x + d)*c - 2*c*d + b*e)*b^2*e^2*g)/(((e*x + d)*c - 2*c*d + b*e)*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*e) + (3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^9*e^3* f + 15*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^9*d*e^2*g - 9*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^8*e^3*g - (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^8*e^2*g) /(c^12*e^3))/e
Time = 12.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (-32\,g\,b^3\,e^3+128\,g\,b^2\,c\,d\,e^2+16\,f\,b^2\,c\,e^3-164\,g\,b\,c^2\,d^2\,e-40\,f\,b\,c^2\,d\,e^2+68\,g\,c^3\,d^3+22\,f\,c^3\,d^2\,e\right )}{3\,c^6\,e^5}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (4\,c\,d\,g-2\,b\,e\,g+c\,e\,f\right )}{c^4\,e^3}+\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,e^2}-\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-144\,g\,b\,c^2\,d\,e^2-24\,f\,b\,c^2\,e^3+102\,g\,c^3\,d^2\,e+36\,f\,c^3\,d\,e^2\right )}{3\,c^6\,e^5}\right )}{x^3+\frac {x\,\left (3\,b^2\,c^4\,e^5-3\,c^6\,d^2\,e^3\right )}{3\,c^6\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \] Input:
int(((f + g*x)*(d + e*x)^(9/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2 ),x)
Output:
-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(((d + e*x)^(1/2)*(68*c^3*d^ 3*g - 32*b^3*e^3*g + 16*b^2*c*e^3*f + 22*c^3*d^2*e*f - 40*b*c^2*d*e^2*f - 164*b*c^2*d^2*e*g + 128*b^2*c*d*e^2*g))/(3*c^6*e^5) + (2*x^2*(d + e*x)^(1/ 2)*(4*c*d*g - 2*b*e*g + c*e*f))/(c^4*e^3) + (2*g*x^3*(d + e*x)^(1/2))/(3*c ^3*e^2) - (x*(d + e*x)^(1/2)*(48*b^2*c*e^3*g - 24*b*c^2*e^3*f + 36*c^3*d*e ^2*f + 102*c^3*d^2*e*g - 144*b*c^2*d*e^2*g))/(3*c^6*e^5)))/(x^3 + (x*(3*b^ 2*c^4*e^5 - 3*c^6*d^2*e^3))/(3*c^6*e^5) + (d*(b*e - c*d)^2)/(c^2*e^3) + (x ^2*(2*b*e - c*d))/(c*e))
Time = 0.23 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {2}{3} c^{3} e^{3} g \,x^{3}-4 b \,c^{2} e^{3} g \,x^{2}+8 c^{3} d \,e^{2} g \,x^{2}+2 c^{3} e^{3} f \,x^{2}-16 b^{2} c \,e^{3} g x +48 b \,c^{2} d \,e^{2} g x +8 b \,c^{2} e^{3} f x -34 c^{3} d^{2} e g x -12 c^{3} d \,e^{2} f x -\frac {32}{3} b^{3} e^{3} g +\frac {128}{3} b^{2} c d \,e^{2} g +\frac {16}{3} b^{2} c \,e^{3} f -\frac {164}{3} b \,c^{2} d^{2} e g -\frac {40}{3} b \,c^{2} d \,e^{2} f +\frac {68}{3} c^{3} d^{3} g +\frac {22}{3} c^{3} d^{2} e f}{\sqrt {-c e x -b e +c d}\, c^{4} e^{2} \left (c e x +b e -c d \right )} \] Input:
int((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*( - 16*b**3*e**3*g + 64*b**2*c*d*e**2*g + 8*b**2*c*e**3*f - 24*b**2*c*e **3*g*x - 82*b*c**2*d**2*e*g - 20*b*c**2*d*e**2*f + 72*b*c**2*d*e**2*g*x + 12*b*c**2*e**3*f*x - 6*b*c**2*e**3*g*x**2 + 34*c**3*d**3*g + 11*c**3*d**2 *e*f - 51*c**3*d**2*e*g*x - 18*c**3*d*e**2*f*x + 12*c**3*d*e**2*g*x**2 + 3 *c**3*e**3*f*x**2 + c**3*e**3*g*x**3))/(3*sqrt( - b*e + c*d - c*e*x)*c**4* e**2*(b*e - c*d + c*e*x))