Integrand size = 44, antiderivative size = 108 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^2 d^2 (d+e x)^{7/2}}+\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^2 d^2 (d+e x)^{9/2}} \] Output:
2/7*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/(e*x+d) ^(7/2)+2/9*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/c^2/d^2/(e*x+d)^(9/2)
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} (-2 a e g+c d (9 f+7 g x))}{63 c^2 d^2 \sqrt {d+e x}} \] Input:
Integrate[((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e *x)^(5/2),x]
Output:
(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e*g + c*d*(9*f + 7* g*x)))/(63*c^2*d^2*Sqrt[d + e*x])
Time = 0.45 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {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.
\(\displaystyle \int \frac {(f+g x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 1221 |
\(\displaystyle \frac {1}{9} \left (-\frac {2 a e g}{c d}-\frac {7 d g}{e}+9 f\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2}}dx+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d e (d+e x)^{5/2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} \left (-\frac {2 a e g}{c d}-\frac {7 d g}{e}+9 f\right )}{63 c d (d+e x)^{7/2}}+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d e (d+e x)^{5/2}}\) |
Input:
Int[((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5 /2),x]
Output:
(2*(9*f - (7*d*g)/e - (2*a*e*g)/(c*d))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e* x^2)^(7/2))/(63*c*d*(d + e*x)^(7/2)) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c *d*e*x^2)^(7/2))/(9*c*d*e*(d + e*x)^(5/2))
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]
Time = 2.83 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{3} \left (-7 c d g x +2 a e g -9 d f c \right )}{63 \sqrt {e x +d}\, c^{2} d^{2}}\) | \(59\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-7 c d g x +2 a e g -9 d f c \right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(67\) |
orering | \(-\frac {2 \left (-7 c d g x +2 a e g -9 d f c \right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{63 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(68\) |
Input:
int((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(5/2),x,method =_RETURNVERBOSE)
Output:
-2/63*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(-7*c*d*g*x+ 2*a*e*g-9*c*d*f)/c^2/d^2
Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.60 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (7 \, c^{4} d^{4} g x^{4} + 9 \, a^{3} c d e^{3} f - 2 \, a^{4} e^{4} g + {\left (9 \, c^{4} d^{4} f + 19 \, a c^{3} d^{3} e g\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{3} e f + 5 \, a^{2} c^{2} d^{2} e^{2} g\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{2} e^{2} f + a^{3} c d e^{3} g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{63 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \] Input:
integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/63*(7*c^4*d^4*g*x^4 + 9*a^3*c*d*e^3*f - 2*a^4*e^4*g + (9*c^4*d^4*f + 19* a*c^3*d^3*e*g)*x^3 + 3*(9*a*c^3*d^3*e*f + 5*a^2*c^2*d^2*e^2*g)*x^2 + (27*a ^2*c^2*d^2*e^2*f + a^3*c*d*e^3*g)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e ^2)*x)*sqrt(e*x + d)/(c^2*d^2*e*x + c^2*d^3)
Timed out. \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/ 2),x)
Output:
Timed out
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.31 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f}{7 \, c d} + \frac {2 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} g}{63 \, c^{2} d^{2}} \] Input:
integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d *x + a*e)*f/(c*d) + 2/63*(7*c^4*d^4*x^4 + 19*a*c^3*d^3*e*x^3 + 15*a^2*c^2* d^2*e^2*x^2 + a^3*c*d*e^3*x - 2*a^4*e^4)*sqrt(c*d*x + a*e)*g/(c^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (96) = 192\).
Time = 0.15 (sec) , antiderivative size = 535, normalized size of antiderivative = 4.95 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} f {\left | e \right |}}{c d e} - \frac {42 \, {\left (5 \, {\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}}\right )} a f {\left | e \right |}}{c d e^{4}} - \frac {21 \, {\left (5 \, {\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}}\right )} a^{2} g {\left | e \right |}}{c^{2} d^{2} e^{3}} + \frac {3 \, {\left (35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} f {\left | e \right |}}{c d e^{7}} + \frac {6 \, {\left (35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} a g {\left | e \right |}}{c^{2} d^{2} e^{6}} - \frac {{\left (105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} g {\left | e \right |}}{c^{2} d^{2} e^{9}}\right )}}{315 \, e} \] Input:
integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")
Output:
2/315*(105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*f*abs(e)/(c*d*e) - 42*(5*((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))*a*f*abs(e)/(c*d*e^4) - 21*(5*((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))*a^2*g*abs(e)/(c^2*d^2*e^3) + 3*(35*((e*x + d)*c*d*e - c*d^2*e + a*e^3 )^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15* ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*f*abs(e)/(c*d*e^7) + 6*(35*((e* x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^ 2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*a *g*abs(e)/(c^2*d^2*e^6) - (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a ^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*g*abs(e)/(c^2*d^2*e^9))/e
Time = 6.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.24 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,c^2\,d^2\,g\,x^4}{9}+\frac {2\,a\,e\,x^2\,\left (5\,a\,e\,g+9\,c\,d\,f\right )}{21}+\frac {2\,c\,d\,x^3\,\left (19\,a\,e\,g+9\,c\,d\,f\right )}{63}-\frac {2\,a^3\,e^3\,\left (2\,a\,e\,g-9\,c\,d\,f\right )}{63\,c^2\,d^2}+\frac {2\,a^2\,e^2\,x\,\left (a\,e\,g+27\,c\,d\,f\right )}{63\,c\,d}\right )}{\sqrt {d+e\,x}} \] Input:
int(((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^(5 /2),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*c^2*d^2*g*x^4)/9 + (2*a *e*x^2*(5*a*e*g + 9*c*d*f))/21 + (2*c*d*x^3*(19*a*e*g + 9*c*d*f))/63 - (2* a^3*e^3*(2*a*e*g - 9*c*d*f))/(63*c^2*d^2) + (2*a^2*e^2*x*(a*e*g + 27*c*d*f ))/(63*c*d)))/(d + e*x)^(1/2)
Time = 0.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.25 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (7 c^{4} d^{4} g \,x^{4}+19 a \,c^{3} d^{3} e g \,x^{3}+9 c^{4} d^{4} f \,x^{3}+15 a^{2} c^{2} d^{2} e^{2} g \,x^{2}+27 a \,c^{3} d^{3} e f \,x^{2}+a^{3} c d \,e^{3} g x +27 a^{2} c^{2} d^{2} e^{2} f x -2 a^{4} e^{4} g +9 a^{3} c d \,e^{3} f \right )}{63 c^{2} d^{2}} \] Input:
int((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x)
Output:
(2*sqrt(a*e + c*d*x)*( - 2*a**4*e**4*g + 9*a**3*c*d*e**3*f + a**3*c*d*e**3 *g*x + 27*a**2*c**2*d**2*e**2*f*x + 15*a**2*c**2*d**2*e**2*g*x**2 + 27*a*c **3*d**3*e*f*x**2 + 19*a*c**3*d**3*e*g*x**3 + 9*c**4*d**4*f*x**3 + 7*c**4* d**4*g*x**4))/(63*c**2*d**2)