Integrand size = 39, antiderivative size = 173 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {4 e \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^3 d^3 (d+e x)^{3/2}}+\frac {2 e^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^3 d^3 (d+e x)^{5/2}} \] Output:
2*(-a*e^2+c*d^2)^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*e*(-a*e^2+c*d^2)*(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*e^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 = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.50 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (8 a^2 e^4-4 a c d e^2 (5 d+e x)+c^2 d^2 \left (15 d^2+10 d e x+3 e^2 x^2\right )\right )}{15 c^3 d^3 \sqrt {d+e x}} \] Input:
Integrate[(d + e*x)^(5/2)/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^4 - 4*a*c*d*e^2*(5*d + e*x) + c^ 2*d^2*(15*d^2 + 10*d*e*x + 3*e^2*x^2)))/(15*c^3*d^3*Sqrt[d + e*x])
Time = 0.51 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {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.
\(\displaystyle \int \frac {(d+e x)^{5/2}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 d}+\frac {2 (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 d}+\frac {2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{5 d}+\frac {2 (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{5 d}\) |
Input:
Int[(d + e*x)^(5/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
(2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d) + (4*(d^2 - (a*e^2)/c)*((4*(d^2 - (a*e^2)/c)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d ^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d)))/(5*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_)*((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]
Time = 1.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.53
method | result | size |
default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 x^{2} c^{2} d^{2} e^{2}-4 x a c d \,e^{3}+10 x \,c^{2} d^{3} e +8 a^{2} e^{4}-20 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right )}{15 \sqrt {e x +d}\, d^{3} c^{3}}\) | \(92\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (3 x^{2} c^{2} d^{2} e^{2}-4 x a c d \,e^{3}+10 x \,c^{2} d^{3} e +8 a^{2} e^{4}-20 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) \sqrt {e x +d}}{15 d^{3} c^{3} \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(110\) |
orering | \(\frac {2 \left (3 x^{2} c^{2} d^{2} e^{2}-4 x a c d \,e^{3}+10 x \,c^{2} d^{3} e +8 a^{2} e^{4}-20 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) \left (c d x +a e \right ) \sqrt {e x +d}}{15 d^{3} c^{3} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\) | \(111\) |
Input:
int((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURN VERBOSE)
Output:
2/15/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(3*c^2*d^2*e^2*x^2-4*a*c*d* e^3*x+10*c^2*d^3*e*x+8*a^2*e^4-20*a*c*d^2*e^2+15*c^2*d^4)/d^3/c^3
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 20 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} + 2 \, {\left (5 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \] Input:
integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit hm="fricas")
Output:
2/15*(3*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 20*a*c*d^2*e^2 + 8*a^2*e^4 + 2*(5*c ^2*d^3*e - 2*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr t(e*x + d)/(c^3*d^3*e*x + c^3*d^4)
\[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \] Input:
integrate((e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
Output:
Integral((d + e*x)**(5/2)/sqrt((d + e*x)*(a*e + c*d*x)), x)
Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.71 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (3 \, c^{3} d^{3} e^{2} x^{3} + 15 \, a c^{2} d^{4} e - 20 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + {\left (10 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (15 \, c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 4 \, a^{2} c d e^{4}\right )} x\right )}}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} \] Input:
integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit hm="maxima")
Output:
2/15*(3*c^3*d^3*e^2*x^3 + 15*a*c^2*d^4*e - 20*a^2*c*d^2*e^3 + 8*a^3*e^5 + (10*c^3*d^4*e - a*c^2*d^2*e^3)*x^2 + (15*c^3*d^5 - 10*a*c^2*d^3*e^2 + 4*a^ 2*c*d*e^4)*x)/(sqrt(c*d*x + a*e)*c^3*d^3)
Time = 0.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e {\left (\frac {15 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{c^{3} d^{3} e} + \frac {10 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} e - 10 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} + 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}}{c^{3} d^{3} e^{3}}\right )}}{15 \, {\left | e \right |}} \] Input:
integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit hm="giac")
Output:
2/15*e*(15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*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^2*e - 10*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 + 3*((e*x + d) *c*d*e - c*d^2*e + a*e^3)^(5/2))/(c^3*d^3*e^3))/abs(e)
Time = 5.95 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^2\,\sqrt {d+e\,x}}{5\,c\,d}-\frac {4\,x\,\left (2\,a\,e^2-5\,c\,d^2\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2}+\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^4-40\,a\,c\,d^2\,e^2+30\,c^2\,d^4\right )}{15\,c^3\,d^3\,e}\right )}{x+\frac {d}{e}} \] Input:
int((d + e*x)^(5/2)/(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)*((2*e*x^2*(d + e*x)^(1/2))/ (5*c*d) - (4*x*(2*a*e^2 - 5*c*d^2)*(d + e*x)^(1/2))/(15*c^2*d^2) + ((d + e *x)^(1/2)*(16*a^2*e^4 + 30*c^2*d^4 - 40*a*c*d^2*e^2))/(15*c^3*d^3*e)))/(x + d/e)
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.45 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (3 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x +10 c^{2} d^{3} e x +8 a^{2} e^{4}-20 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right )}{15 c^{3} d^{3}} \] Input:
int((e*x+d)^(5/2)/(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**4 - 20*a*c*d**2*e**2 - 4*a*c*d*e**3*x + 15 *c**2*d**4 + 10*c**2*d**3*e*x + 3*c**2*d**2*e**2*x**2))/(15*c**3*d**3)