Integrand size = 44, antiderivative size = 104 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (c d f-a e g) \sqrt {d+e x}}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt {d+e x}} \] Output:
-2*(-a*e*g+c*d*f)*(e*x+d)^(1/2)/c^2/d^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ (1/2)+2*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/(e*x+d)^(1/2)
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x} (2 a e g+c d (-f+g x))}{c^2 d^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[((d + e*x)^(3/2)*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x ^2)^(3/2),x]
Output:
(2*Sqrt[d + e*x]*(2*a*e*g + c*d*(-f + g*x)))/(c^2*d^2*Sqrt[(a*e + c*d*x)*( d + e*x)])
Time = 0.55 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, 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.
\(\displaystyle \int \frac {(d+e x)^{3/2} (f+g x)}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle -\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d \left (c d^2-a e^2\right )}-\frac {2 (d+e x)^{3/2} (c d f-a e g)}{c d \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle -\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{c^2 d^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 (d+e x)^{3/2} (c d f-a e g)}{c d \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
Input:
Int[((d + e*x)^(3/2)*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3 /2),x]
Output:
(-2*(c*d*f - a*e*g)*(d + e*x)^(3/2))/(c*d*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c* d^2 + a*e^2)*x + c*d*e*x^2]) - (2*(2*a*e^2*g - c*d*(e*f + d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^2*d^2*(c*d^2 - a*e^2)*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 = 2.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d g x +2 a e g -d f c \right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) c^{2} d^{2}}\) | \(58\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (c d g x +2 a e g -d f c \right ) \left (e x +d \right )^{\frac {3}{2}}}{c^{2} d^{2} \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(66\) |
orering | \(\frac {2 \left (c d g x +2 a e g -d f c \right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{c^{2} d^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(67\) |
Input:
int((e*x+d)^(3/2)*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method =_RETURNVERBOSE)
Output:
2/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(c*d*g*x+2*a*e*g-c*d*f)/(c*d*x +a*e)/c^2/d^2
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - c d f + 2 \, a e g\right )} \sqrt {e x + d}}{c^{3} d^{3} e x^{2} + a c^{2} d^{3} e + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")
Output:
2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x - c*d*f + 2*a*e*g)* sqrt(e*x + d)/(c^3*d^3*e*x^2 + a*c^2*d^3*e + (c^3*d^4 + a*c^2*d^2*e^2)*x)
\[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(g*x+f)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/ 2),x)
Output:
Integral((d + e*x)**(3/2)*(f + g*x)/((d + e*x)*(a*e + c*d*x))**(3/2), x)
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, f}{\sqrt {c d x + a e} c d} + \frac {2 \, {\left (c d x + 2 \, a e\right )} g}{\sqrt {c d x + a e} c^{2} d^{2}} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")
Output:
-2*f/(sqrt(c*d*x + a*e)*c*d) + 2*(c*d*x + 2*a*e)*g/(sqrt(c*d*x + a*e)*c^2* d^2)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {c d x + a e} g}{c^{2} d^{2}} - \frac {2 \, {\left (c d f - a e g\right )}}{\sqrt {c d x + a e} c^{2} d^{2}} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
Output:
2*sqrt(c*d*x + a*e)*g/(c^2*d^2) - 2*(c*d*f - a*e*g)/(sqrt(c*d*x + a*e)*c^2 *d^2)
Time = 6.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\left (\frac {\left (4\,a\,e\,g-2\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c^3\,d^3\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{c^2\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (c^3\,d^4+a\,c^2\,d^2\,e^2\right )}{c^3\,d^3\,e}} \] Input:
int(((f + g*x)*(d + e*x)^(3/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3 /2),x)
Output:
((((4*a*e*g - 2*c*d*f)*(d + e*x)^(1/2))/(c^3*d^3*e) + (2*g*x*(d + e*x)^(1/ 2))/(c^2*d^2*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(a/c + x^2 + (x*(c^3*d^4 + a*c^2*d^2*e^2))/(c^3*d^3*e))
Time = 0.54 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.34 \[ \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c d g x +4 a e g -2 c d f}{\sqrt {c d x +a e}\, c^{2} d^{2}} \] Input:
int((e*x+d)^(3/2)*(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*(2*a*e*g - c*d*f + c*d*g*x))/(sqrt(a*e + c*d*x)*c**2*d**2)