Integrand size = 46, antiderivative size = 169 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 (c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {4 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^3 d^3 (d+e x)^{3/2}}+\frac {2 g^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^3 d^3 (d+e x)^{5/2}} \] Output:
2*(-a*e*g+c*d*f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/(e*x+d) ^(1/2)+4/3*g*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^ 3/(e*x+d)^(3/2)+2/5*g^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/(e *x+d)^(5/2)
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (8 a^2 e^2 g^2-4 a c d e g (5 f+g x)+c^2 d^2 \left (15 f^2+10 f g x+3 g^2 x^2\right )\right )}{15 c^3 d^3 \sqrt {d+e x}} \] Input:
Integrate[(Sqrt[d + e*x]*(f + g*x)^2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d *e*x^2],x]
Output:
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(5*f + g*x) + c^2*d^2*(15*f^2 + 10*f*g*x + 3*g^2*x^2)))/(15*c^3*d^3*Sqrt[d + e*x])
Time = 0.71 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1253, 1221, 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.
\(\displaystyle \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1253 |
\(\displaystyle \frac {4 (c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 c d}+\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1221 |
\(\displaystyle \frac {4 (c d f-a e g) \left (\frac {1}{3} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}+\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}+\frac {4 (c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right )}{3 c d \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}\) |
Input:
Int[(Sqrt[d + e*x]*(f + g*x)^2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 ],x]
Output:
(2*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d*Sqrt[d + e*x]) + (4*(c*d*f - a*e*g)*((2*(3*f - (d*g)/e - (2*a*e*g)/(c*d))*Sqrt[a* d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d*Sqrt[d + e*x]) + (2*g*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d*e)))/(5*c*d)
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*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n* ((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Simp[n*((c*e*f + c*d*g - b*e*g)/(c*e*(m - n - 1))) Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c* x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d* e + a*e^2, 0] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (Intege rQ[2*p] || IntegerQ[n])
Time = 2.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x +10 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-20 a c d e f g +15 f^{2} c^{2} d^{2}\right )}{15 \sqrt {e x +d}\, d^{3} c^{3}}\) | \(98\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (3 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x +10 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-20 a c d e f g +15 f^{2} c^{2} d^{2}\right ) \sqrt {e x +d}}{15 d^{3} c^{3} \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(116\) |
orering | \(\frac {2 \left (3 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x +10 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-20 a c d e f g +15 f^{2} c^{2} d^{2}\right ) \left (c d x +a e \right ) \sqrt {e x +d}}{15 d^{3} c^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\) | \(117\) |
Input:
int((e*x+d)^(1/2)*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,meth od=_RETURNVERBOSE)
Output:
2/15/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(3*c^2*d^2*g^2*x^2-4*a*c*d* e*g^2*x+10*c^2*d^2*f*g*x+8*a^2*e^2*g^2-20*a*c*d*e*f*g+15*c^2*d^2*f^2)/d^3/ c^3
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} g^{2} x^{2} + 15 \, c^{2} d^{2} f^{2} - 20 \, a c d e f g + 8 \, a^{2} e^{2} g^{2} + 2 \, {\left (5 \, c^{2} d^{2} f g - 2 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="fricas")
Output:
2/15*(3*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2 - 20*a*c*d*e*f*g + 8*a^2*e^2*g^2 + 2*(5*c^2*d^2*f*g - 2*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a *e^2)*x)*sqrt(e*x + d)/(c^3*d^3*e*x + c^3*d^4)
\[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x} \left (f + g x\right )^{2}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \] Input:
integrate((e*x+d)**(1/2)*(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)** (1/2),x)
Output:
Integral(sqrt(d + e*x)*(f + g*x)**2/sqrt((d + e*x)*(a*e + c*d*x)), x)
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d x + a e} f^{2}}{c d} + \frac {4 \, {\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} f g}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} x^{3} - a c^{2} d^{2} e x^{2} + 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} g^{2}}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="maxima")
Output:
2*sqrt(c*d*x + a*e)*f^2/(c*d) + 4/3*(c^2*d^2*x^2 - a*c*d*e*x - 2*a^2*e^2)* f*g/(sqrt(c*d*x + a*e)*c^2*d^2) + 2/15*(3*c^3*d^3*x^3 - a*c^2*d^2*e*x^2 + 4*a^2*c*d*e^2*x + 8*a^3*e^3)*g^2/(sqrt(c*d*x + a*e)*c^3*d^3)
Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e {\left (\frac {15 \, {\left (c^{2} d^{2} f^{2} - 2 \, a c d e f g + a^{2} e^{2} g^{2}\right )} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{c^{3} d^{3} e} + \frac {10 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d e^{2} f g - 10 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} g^{2} + 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} g^{2}}{c^{3} d^{3} e^{5}}\right )}}{15 \, {\left | e \right |}} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="giac")
Output:
2/15*e*(15*(c^2*d^2*f^2 - 2*a*c*d*e*f*g + a^2*e^2*g^2)*sqrt((e*x + d)*c*d* e - c*d^2*e + a*e^3)/(c^3*d^3*e) + (10*((e*x + d)*c*d*e - c*d^2*e + a*e^3) ^(3/2)*c*d*e^2*f*g - 10*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3*g^ 2 + 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*g^2)/(c^3*d^3*e^5))/abs(e)
Time = 6.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^2\,g^2-40\,a\,c\,d\,e\,f\,g+30\,c^2\,d^2\,f^2\right )}{15\,c^3\,d^3\,e}+\frac {2\,g^2\,x^2\,\sqrt {d+e\,x}}{5\,c\,d\,e}-\frac {4\,g\,x\,\left (2\,a\,e\,g-5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \] Input:
int(((f + g*x)^2*(d + e*x)^(1/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^ (1/2),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(((d + e*x)^(1/2)*(16*a^2*e ^2*g^2 + 30*c^2*d^2*f^2 - 40*a*c*d*e*f*g))/(15*c^3*d^3*e) + (2*g^2*x^2*(d + e*x)^(1/2))/(5*c*d*e) - (4*g*x*(2*a*e*g - 5*c*d*f)*(d + e*x)^(1/2))/(15* c^2*d^2*e)))/(x + d/e)
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (3 c^{2} d^{2} g^{2} x^{2}-4 a c d e \,g^{2} x +10 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-20 a c d e f g +15 c^{2} d^{2} f^{2}\right )}{15 c^{3} d^{3}} \] Input:
int((e*x+d)^(1/2)*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
(2*sqrt(a*e + c*d*x)*(8*a**2*e**2*g**2 - 20*a*c*d*e*f*g - 4*a*c*d*e*g**2*x + 15*c**2*d**2*f**2 + 10*c**2*d**2*f*g*x + 3*c**2*d**2*g**2*x**2))/(15*c* *3*d**3)