Integrand size = 39, antiderivative size = 175 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^3 d^3 (d+e x)^{3/2}}+\frac {4 e \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^3 d^3 (d+e x)^{5/2}}+\frac {2 e^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^3 d^3 (d+e x)^{7/2}} \] Output:
2/3*(-a*e^2+c*d^2)^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)+4/5*e*(-a*e^2+c*d^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)+2/7*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/ (e*x+d)^(7/2)
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.50 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (8 a^2 e^4-4 a c d e^2 (7 d+3 e x)+c^2 d^2 \left (35 d^2+42 d e x+15 e^2 x^2\right )\right )}{105 c^3 d^3 (d+e x)^{3/2}} \] Input:
Integrate[(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(8*a^2*e^4 - 4*a*c*d*e^2*(7*d + 3*e*x) + c^2*d^2*(35*d^2 + 42*d*e*x + 15*e^2*x^2)))/(105*c^3*d^3*(d + e*x)^(3/2))
Time = 0.54 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{7 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{5 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}\right )}{7 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c d^2 (d+e x)^{3/2}}\right )}{7 d}\) |
Input:
Int[(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
(2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*c*d) + (4*(d^2 - (a*e^2)/c)*((4*(d^2 - (a*e^2)/c)*(a*d*e + (c*d^2 + a*e^2)*x + c* d*e*x^2)^(3/2))/(15*c*d^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*Sqrt[d + e*x])))/(7*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 = 1.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.57
method | result | size |
default | \(\frac {2 \left (c d x +a e \right ) \left (15 x^{2} c^{2} d^{2} e^{2}-12 x a c d \,e^{3}+42 x \,c^{2} d^{3} e +8 a^{2} e^{4}-28 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{105 d^{3} c^{3} \sqrt {e x +d}}\) | \(100\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (15 x^{2} c^{2} d^{2} e^{2}-12 x a c d \,e^{3}+42 x \,c^{2} d^{3} e +8 a^{2} e^{4}-28 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{105 d^{3} c^{3} \sqrt {e x +d}}\) | \(110\) |
orering | \(\frac {2 \left (15 x^{2} c^{2} d^{2} e^{2}-12 x a c d \,e^{3}+42 x \,c^{2} d^{3} e +8 a^{2} e^{4}-28 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \left (c d x +a e \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{105 d^{3} c^{3} \sqrt {e x +d}}\) | \(111\) |
Input:
int((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURN VERBOSE)
Output:
2/105*(c*d*x+a*e)*(15*c^2*d^2*e^2*x^2-12*a*c*d*e^3*x+42*c^2*d^3*e*x+8*a^2* e^4-28*a*c*d^2*e^2+35*c^2*d^4)*((e*x+d)*(c*d*x+a*e))^(1/2)/d^3/c^3/(e*x+d) ^(1/2)
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.91 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (15 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 28 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + 3 \, {\left (14 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (35 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - 4 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{105 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \] Input:
integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit hm="fricas")
Output:
2/105*(15*c^3*d^3*e^2*x^3 + 35*a*c^2*d^4*e - 28*a^2*c*d^2*e^3 + 8*a^3*e^5 + 3*(14*c^3*d^4*e + a*c^2*d^2*e^3)*x^2 + (35*c^3*d^5 + 14*a*c^2*d^3*e^2 - 4*a^2*c*d*e^4)*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)
\[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\int \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
Output:
Integral(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**(3/2), x)
Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.80 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (15 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 28 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + 3 \, {\left (14 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (35 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - 4 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{105 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \] Input:
integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit hm="maxima")
Output:
2/105*(15*c^3*d^3*e^2*x^3 + 35*a*c^2*d^4*e - 28*a^2*c*d^2*e^3 + 8*a^3*e^5 + 3*(14*c^3*d^4*e + a*c^2*d^2*e^3)*x^2 + (35*c^3*d^5 + 14*a*c^2*d^3*e^2 - 4*a^2*c*d*e^4)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^3*d^3*e*x + c^3*d^4)
Time = 0.18 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.58 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (105 \, \sqrt {c d x + a e} a d^{2} e - 35 \, {\left (3 \, \sqrt {c d x + a e} a e - {\left (c d x + a e\right )}^{\frac {3}{2}}\right )} d^{2} - \frac {70 \, {\left (3 \, \sqrt {c d x + a e} a e - {\left (c d x + a e\right )}^{\frac {3}{2}}\right )} a e^{2}}{c} + \frac {14 \, {\left (15 \, \sqrt {c d x + a e} a^{2} e^{2} - 10 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a e + 3 \, {\left (c d x + a e\right )}^{\frac {5}{2}}\right )} e}{c} + \frac {7 \, {\left (15 \, \sqrt {c d x + a e} a^{2} e^{2} - 10 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a e + 3 \, {\left (c d x + a e\right )}^{\frac {5}{2}}\right )} a e^{3}}{c^{2} d^{2}} - \frac {3 \, {\left (35 \, \sqrt {c d x + a e} a^{3} e^{3} - 35 \, {\left (c d x + a e\right )}^{\frac {3}{2}} a^{2} e^{2} + 21 \, {\left (c d x + a e\right )}^{\frac {5}{2}} a e - 5 \, {\left (c d x + a e\right )}^{\frac {7}{2}}\right )} e^{2}}{c^{2} d^{2}}\right )}}{105 \, c d} \] Input:
integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit hm="giac")
Output:
2/105*(105*sqrt(c*d*x + a*e)*a*d^2*e - 35*(3*sqrt(c*d*x + a*e)*a*e - (c*d* x + a*e)^(3/2))*d^2 - 70*(3*sqrt(c*d*x + a*e)*a*e - (c*d*x + a*e)^(3/2))*a *e^2/c + 14*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3 *(c*d*x + a*e)^(5/2))*e/c + 7*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a*e^3/(c^2*d^2) - 3*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)* a*e - 5*(c*d*x + a*e)^(7/2))*e^2/(c^2*d^2))/(c*d)
Time = 5.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.03 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^3\,\sqrt {d+e\,x}}{7}+\frac {\sqrt {d+e\,x}\,\left (16\,a^3\,e^5-56\,a^2\,c\,d^2\,e^3+70\,a\,c^2\,d^4\,e\right )}{105\,c^3\,d^3\,e}+\frac {2\,x^2\,\left (14\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{35\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-8\,a^2\,c\,d\,e^4+28\,a\,c^2\,d^3\,e^2+70\,c^3\,d^5\right )}{105\,c^3\,d^3\,e}\right )}{x+\frac {d}{e}} \] Input:
int((d + e*x)^(3/2)*(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^3*(d + e*x)^(1/2))/ 7 + ((d + e*x)^(1/2)*(16*a^3*e^5 - 56*a^2*c*d^2*e^3 + 70*a*c^2*d^4*e))/(10 5*c^3*d^3*e) + (2*x^2*(a*e^2 + 14*c*d^2)*(d + e*x)^(1/2))/(35*c*d) + (x*(d + e*x)^(1/2)*(70*c^3*d^5 + 28*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(105*c^3*d^ 3*e)))/(x + d/e)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70 \[ \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (15 c^{3} d^{3} e^{2} x^{3}+3 a \,c^{2} d^{2} e^{3} x^{2}+42 c^{3} d^{4} e \,x^{2}-4 a^{2} c d \,e^{4} x +14 a \,c^{2} d^{3} e^{2} x +35 c^{3} d^{5} x +8 a^{3} e^{5}-28 a^{2} c \,d^{2} e^{3}+35 a \,c^{2} d^{4} e \right )}{105 c^{3} d^{3}} \] Input:
int((e*x+d)^(3/2)*(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**3*e**5 - 28*a**2*c*d**2*e**3 - 4*a**2*c*d*e**4* x + 35*a*c**2*d**4*e + 14*a*c**2*d**3*e**2*x + 3*a*c**2*d**2*e**3*x**2 + 3 5*c**3*d**5*x + 42*c**3*d**4*e*x**2 + 15*c**3*d**3*e**2*x**3))/(105*c**3*d **3)