Integrand size = 46, antiderivative size = 263 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (2 c d-b e)^2 (c e f+c d g-b e g) \sqrt {d+e x}}{c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (2 c d-b e) (2 c e f+4 c d g-3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^4 e^2 \sqrt {d+e x}}-\frac {2 (c e f+5 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^4 e^2 (d+e x)^{3/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^4 e^2 (d+e x)^{5/2}} \] Output:
2*(-b*e+2*c*d)^2*(-b*e*g+c*d*g+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*(-b*e+2*c*d)*(-3*b*e*g+4*c*d*g+2*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*(-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)^(3/2)/c^4/e^2/(e*x+d)^(3/2)+2/5* g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^4/e^2/(e*x+d)^(5/2)
Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (48 b^3 e^3 g-8 b^2 c e^2 (5 e f+28 d g-3 e g x)+2 b c^2 e \left (167 d^2 g+d e (70 f-44 g x)-e^2 x (10 f+3 g x)\right )+c^3 \left (-158 d^3 g+e^3 x^2 (5 f+3 g x)+2 d e^2 x (25 f+8 g x)+d^2 e (-115 f+79 g x)\right )\right )}{15 c^4 e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] 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 )^(3/2),x]
Output:
(-2*Sqrt[d + e*x]*(48*b^3*e^3*g - 8*b^2*c*e^2*(5*e*f + 28*d*g - 3*e*g*x) + 2*b*c^2*e*(167*d^2*g + d*e*(70*f - 44*g*x) - e^2*x*(10*f + 3*g*x)) + c^3* (-158*d^3*g + e^3*x^2*(5*f + 3*g*x) + 2*d*e^2*x*(25*f + 8*g*x) + d^2*e*(-1 15*f + 79*g*x))))/(15*c^4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.88 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.09, 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.
| \(\displaystyle \int \frac {(d+e x)^{7/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-6 b e g+7 c d g+5 c e f) \int \frac {(d+e x)^{5/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-6 b e g+7 c d g+5 c e f) \left (\frac {4 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-6 b e g+7 c d g+5 c e f) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 c}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right ) (-6 b e g+7 c d g+5 c e f)}{c e (2 c d-b e)}\) |
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)^(3/2 ),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(7/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c *d - b*e) - b*e^2*x - c*e^2*x^2]) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*((-2*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*c*e) + (4*(2*c *d - b*e)*((-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)))/(5*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 = 1.52 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (3 e^{3} g \,x^{3} c^{3}-6 b \,c^{2} e^{3} g \,x^{2}+16 c^{3} d \,e^{2} g \,x^{2}+5 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -88 b \,c^{2} d \,e^{2} g x -20 b \,c^{2} e^{3} f x +79 c^{3} d^{2} e g x +50 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -224 b^{2} c d \,e^{2} g -40 b^{2} c \,e^{3} f +334 b \,c^{2} d^{2} e g +140 b \,c^{2} d \,e^{2} f -158 c^{3} d^{3} g -115 d^{2} f \,c^{3} e \right )}{15 \sqrt {e x +d}\, \left (c e x +b e -c d \right ) c^{4} e^{2}}\) | \(229\) |
| gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (3 e^{3} g \,x^{3} c^{3}-6 b \,c^{2} e^{3} g \,x^{2}+16 c^{3} d \,e^{2} g \,x^{2}+5 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -88 b \,c^{2} d \,e^{2} g x -20 b \,c^{2} e^{3} f x +79 c^{3} d^{2} e g x +50 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -224 b^{2} c d \,e^{2} g -40 b^{2} c \,e^{3} f +334 b \,c^{2} d^{2} e g +140 b \,c^{2} d \,e^{2} f -158 c^{3} d^{3} g -115 d^{2} f \,c^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{15 c^{4} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(235\) |
| orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (3 e^{3} g \,x^{3} c^{3}-6 b \,c^{2} e^{3} g \,x^{2}+16 c^{3} d \,e^{2} g \,x^{2}+5 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -88 b \,c^{2} d \,e^{2} g x -20 b \,c^{2} e^{3} f x +79 c^{3} d^{2} e g x +50 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -224 b^{2} c d \,e^{2} g -40 b^{2} c \,e^{3} f +334 b \,c^{2} d^{2} e g +140 b \,c^{2} d \,e^{2} f -158 c^{3} d^{3} g -115 d^{2} f \,c^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{15 c^{4} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(235\) |
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)^(3/2),x,method= _RETURNVERBOSE)
Output:
2/15/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(3*c^3*e^3*g*x^3-6*b*c ^2*e^3*g*x^2+16*c^3*d*e^2*g*x^2+5*c^3*e^3*f*x^2+24*b^2*c*e^3*g*x-88*b*c^2* d*e^2*g*x-20*b*c^2*e^3*f*x+79*c^3*d^2*e*g*x+50*c^3*d*e^2*f*x+48*b^3*e^3*g- 224*b^2*c*d*e^2*g-40*b^2*c*e^3*f+334*b*c^2*d^2*e*g+140*b*c^2*d*e^2*f-158*c ^3*d^3*g-115*c^3*d^2*e*f)/(c*e*x+b*e-c*d)/c^4/e^2
Time = 0.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (3 \, c^{3} e^{3} g x^{3} + {\left (5 \, c^{3} e^{3} f + 2 \, {\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} - 5 \, {\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f - 2 \, {\left (79 \, c^{3} d^{3} - 167 \, b c^{2} d^{2} e + 112 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g + {\left (10 \, {\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (c^{5} e^{4} x^{2} + b c^{4} e^{4} x - c^{5} d^{2} e^{2} + b c^{4} d e^{3}\right )}} \] 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)^(3/2),x, algorithm="fricas")
Output:
2/15*(3*c^3*e^3*g*x^3 + (5*c^3*e^3*f + 2*(8*c^3*d*e^2 - 3*b*c^2*e^3)*g)*x^ 2 - 5*(23*c^3*d^2*e - 28*b*c^2*d*e^2 + 8*b^2*c*e^3)*f - 2*(79*c^3*d^3 - 16 7*b*c^2*d^2*e + 112*b^2*c*d*e^2 - 24*b^3*e^3)*g + (10*(5*c^3*d*e^2 - 2*b*c ^2*e^3)*f + (79*c^3*d^2*e - 88*b*c^2*d*e^2 + 24*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^5*e^4*x^2 + b*c^4*e^4*x - c^5*d^2*e^2 + b*c^4*d*e^3)
Timed out. \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \] 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)**(3/ 2),x)
Output:
Timed out
Time = 0.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} e^{2} x^{2} - 23 \, c^{2} d^{2} + 28 \, b c d e - 8 \, b^{2} e^{2} + 2 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, \sqrt {-c e x + c d - b e} c^{3} e} - \frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 158 \, c^{3} d^{3} + 334 \, b c^{2} d^{2} e - 224 \, b^{2} c d e^{2} + 48 \, b^{3} e^{3} + 2 \, {\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} x^{2} + {\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt {-c e x + c d - b e} c^{4} e^{2}} \] 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)^(3/2),x, algorithm="maxima")
Output:
-2/3*(c^2*e^2*x^2 - 23*c^2*d^2 + 28*b*c*d*e - 8*b^2*e^2 + 2*(5*c^2*d*e - 2 *b*c*e^2)*x)*f/(sqrt(-c*e*x + c*d - b*e)*c^3*e) - 2/15*(3*c^3*e^3*x^3 - 15 8*c^3*d^3 + 334*b*c^2*d^2*e - 224*b^2*c*d*e^2 + 48*b^3*e^3 + 2*(8*c^3*d*e^ 2 - 3*b*c^2*e^3)*x^2 + (79*c^3*d^2*e - 88*b*c^2*d*e^2 + 24*b^2*c*e^3)*x)*g /(sqrt(-c*e*x + c*d - b*e)*c^4*e^2)
Time = 0.36 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (4 \, c^{3} d^{2} e f - 4 \, b c^{2} d e^{2} f + b^{2} c e^{3} f + 4 \, c^{3} d^{3} g - 8 \, b c^{2} d^{2} e g + 5 \, b^{2} c d e^{2} g - b^{3} e^{3} g\right )}}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} e} + \frac {60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{18} d e^{5} f - 30 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{17} e^{6} f + 120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{18} d^{2} e^{4} g - 150 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{17} d e^{5} g + 45 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{16} e^{6} g - 5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{17} e^{5} f - 25 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{17} d e^{4} g + 15 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{16} e^{5} g + 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{16} e^{4} g}{c^{20} e^{5}}\right )}}{15 \, e} \] 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)^(3/2),x, algorithm="giac")
Output:
2/15*(15*(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)/(sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*e) + (60*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^18*d*e^5*f - 30*sqrt( -(e*x + d)*c + 2*c*d - b*e)*b*c^17*e^6*f + 120*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^18*d^2*e^4*g - 150*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^17*d*e^5*g + 45*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^16*e^6*g - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^17*e^5*f - 25*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^17 *d*e^4*g + 15*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^16*e^5*g + 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^16*e^4*g)/(c^20*e ^5))/e
Time = 11.05 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (16\,c\,d\,g-6\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^3\,e^2}-\frac {\sqrt {d+e\,x}\,\left (-96\,g\,b^3\,e^3+448\,g\,b^2\,c\,d\,e^2+80\,f\,b^2\,c\,e^3-668\,g\,b\,c^2\,d^2\,e-280\,f\,b\,c^2\,d\,e^2+316\,g\,c^3\,d^3+230\,f\,c^3\,d^2\,e\right )}{15\,c^5\,e^4}+\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{5\,c^2\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-176\,g\,b\,c^2\,d\,e^2-40\,f\,b\,c^2\,e^3+158\,g\,c^3\,d^2\,e+100\,f\,c^3\,d\,e^2\right )}{15\,c^5\,e^4}\right )}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \] Input:
int(((f + g*x)*(d + e*x)^(7/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2 ),x)
Output:
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*x^2*(d + e*x)^(1/2)*(16*c *d*g - 6*b*e*g + 5*c*e*f))/(15*c^3*e^2) - ((d + e*x)^(1/2)*(316*c^3*d^3*g - 96*b^3*e^3*g + 80*b^2*c*e^3*f + 230*c^3*d^2*e*f - 280*b*c^2*d*e^2*f - 66 8*b*c^2*d^2*e*g + 448*b^2*c*d*e^2*g))/(15*c^5*e^4) + (2*g*x^3*(d + e*x)^(1 /2))/(5*c^2*e) + (x*(d + e*x)^(1/2)*(48*b^2*c*e^3*g - 40*b*c^2*e^3*f + 100 *c^3*d*e^2*f + 158*c^3*d^2*e*g - 176*b*c^2*d*e^2*g))/(15*c^5*e^4)))/(x^2 + (b*x)/c + (d*(b*e - c*d))/(c*e^2))
Time = 0.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {-\frac {2}{5} c^{3} e^{3} g \,x^{3}+\frac {4}{5} b \,c^{2} e^{3} g \,x^{2}-\frac {32}{15} c^{3} d \,e^{2} g \,x^{2}-\frac {2}{3} c^{3} e^{3} f \,x^{2}-\frac {16}{5} b^{2} c \,e^{3} g x +\frac {176}{15} b \,c^{2} d \,e^{2} g x +\frac {8}{3} b \,c^{2} e^{3} f x -\frac {158}{15} c^{3} d^{2} e g x -\frac {20}{3} c^{3} d \,e^{2} f x -\frac {32}{5} b^{3} e^{3} g +\frac {448}{15} b^{2} c d \,e^{2} g +\frac {16}{3} b^{2} c \,e^{3} f -\frac {668}{15} b \,c^{2} d^{2} e g -\frac {56}{3} b \,c^{2} d \,e^{2} f +\frac {316}{15} c^{3} d^{3} g +\frac {46}{3} c^{3} d^{2} e f}{\sqrt {-c e x -b e +c d}\, c^{4} e^{2}} \] 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)^(3/2),x)
Output:
(2*( - 48*b**3*e**3*g + 224*b**2*c*d*e**2*g + 40*b**2*c*e**3*f - 24*b**2*c *e**3*g*x - 334*b*c**2*d**2*e*g - 140*b*c**2*d*e**2*f + 88*b*c**2*d*e**2*g *x + 20*b*c**2*e**3*f*x + 6*b*c**2*e**3*g*x**2 + 158*c**3*d**3*g + 115*c** 3*d**2*e*f - 79*c**3*d**2*e*g*x - 50*c**3*d*e**2*f*x - 16*c**3*d*e**2*g*x* *2 - 5*c**3*e**3*f*x**2 - 3*c**3*e**3*g*x**3))/(15*sqrt( - b*e + c*d - c*e *x)*c**4*e**2)