Integrand size = 46, antiderivative size = 112 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^2 e^2 \sqrt {d+e x}} \] Output:
2*(-b*e*g+c*d*g+c*e*f)*(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)+2*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^2/e^2/(e*x+d)^(1/2)
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.54 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x} (2 c d g-2 b e g+c e (f-g x))}{c^2 e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(3/2)*(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]*(2*c*d*g - 2*b*e*g + c*e*(f - g*x)))/(c^2*e^2*Sqrt[(d + e *x)*(-(b*e) + c*(d - e*x))])
Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32, 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.
\(\displaystyle \int \frac {(d+e x)^{3/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle \frac {2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-2 b e g+3 c d g+c e f) \int \frac {\sqrt {d+e x}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e (2 c d-b e)}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{c^2 e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
Input:
Int[((d + e*x)^(3/2)*(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)^(3/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c *d - b*e) - b*e^2*x - c*e^2*x^2]) + (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^2*e^2*(2*c*d - b*e)*Sqrt[d + e*x])
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 = 1.62 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e g x +2 b e g -2 c d g -f c e \right )}{\sqrt {e x +d}\, \left (c e x +b e -c d \right ) c^{2} e^{2}}\) | \(72\) |
gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (c e g x +2 b e g -2 c d g -f c e \right ) \left (e x +d \right )^{\frac {3}{2}}}{c^{2} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(78\) |
orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (c e g x +2 b e g -2 c d g -f c e \right ) \left (e x +d \right )^{\frac {3}{2}}}{c^{2} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(78\) |
Input:
int((e*x+d)^(3/2)*(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/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*g*x+2*b*e*g-2*c*d*g- c*e*f)/(c*e*x+b*e-c*d)/c^2/e^2
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e g x - c e f - 2 \, {\left (c d - b e\right )} g\right )} \sqrt {e x + d}}{c^{3} e^{4} x^{2} + b c^{2} e^{4} x - c^{3} d^{2} e^{2} + b c^{2} d e^{3}} \] Input:
integrate((e*x+d)^(3/2)*(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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*g*x - c*e*f - 2*(c*d - b *e)*g)*sqrt(e*x + d)/(c^3*e^4*x^2 + b*c^2*e^4*x - c^3*d^2*e^2 + b*c^2*d*e^ 3)
\[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(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)**(3/2)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(3/2 ), x)
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.55 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, f}{\sqrt {-c e x + c d - b e} c e} - \frac {2 \, {\left (c e x - 2 \, c d + 2 \, b e\right )} g}{\sqrt {-c e x + c d - b e} c^{2} e^{2}} \] Input:
integrate((e*x+d)^(3/2)*(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*f/(sqrt(-c*e*x + c*d - b*e)*c*e) - 2*(c*e*x - 2*c*d + 2*b*e)*g/(sqrt(-c* e*x + c*d - b*e)*c^2*e^2)
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} g}{c^{2} e} + \frac {c e f + c d g - b e g}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} e}\right )}}{e} \] Input:
integrate((e*x+d)^(3/2)*(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*(sqrt(-(e*x + d)*c + 2*c*d - b*e)*g/(c^2*e) + (c*e*f + c*d*g - b*e*g)/(s qrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*e))/e
Time = 10.87 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {\left (\frac {\sqrt {d+e\,x}\,\left (4\,c\,d\,g-4\,b\,e\,g+2\,c\,e\,f\right )}{c^3\,e^4}-\frac {2\,g\,x\,\sqrt {d+e\,x}}{c^2\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \] Input:
int(((f + g*x)*(d + e*x)^(3/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2 ),x)
Output:
-((((d + e*x)^(1/2)*(4*c*d*g - 4*b*e*g + 2*c*e*f))/(c^3*e^4) - (2*g*x*(d + e*x)^(1/2))/(c^2*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 = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.40 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {-2 c e g x -4 b e g +4 c d g +2 c e f}{\sqrt {-c e x -b e +c d}\, c^{2} e^{2}} \] Input:
int((e*x+d)^(3/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(2*( - 2*b*e*g + 2*c*d*g + c*e*f - c*e*g*x))/(sqrt( - b*e + c*d - c*e*x)*c **2*e**2)