Integrand size = 46, antiderivative size = 352 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \left (c d^2-a e^2\right ) (c d f-a e g)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^5 d^5 \sqrt {d+e x}}-\frac {2 (c d f-a e g)^2 \left (4 a e^2 g-c d (e f+3 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^5 d^5 (d+e x)^{3/2}}-\frac {6 g (c d f-a e g) \left (2 a e^2 g-c d (e f+d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^5 d^5 (d+e x)^{5/2}}-\frac {2 g^2 \left (4 a e^2 g-c d (3 e f+d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^5 d^5 (d+e x)^{7/2}}+\frac {2 e g^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^5 d^5 (d+e x)^{9/2}} \] Output:
2*(-a*e^2+c*d^2)*(-a*e*g+c*d*f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ c^5/d^5/(e*x+d)^(1/2)-2/3*(-a*e*g+c*d*f)^2*(4*a*e^2*g-c*d*(3*d*g+e*f))*(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^5/d^5/(e*x+d)^(3/2)-6/5*g*(-a*e*g+c *d*f)*(2*a*e^2*g-c*d*(d*g+e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^ 5/d^5/(e*x+d)^(5/2)-2/7*g^2*(4*a*e^2*g-c*d*(d*g+3*e*f))*(a*d*e+(a*e^2+c*d^ 2)*x+c*d*e*x^2)^(7/2)/c^5/d^5/(e*x+d)^(7/2)+2/9*e*g^3*(a*d*e+(a*e^2+c*d^2) *x+c*d*e*x^2)^(9/2)/c^5/d^5/(e*x+d)^(9/2)
Time = 0.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (128 a^4 e^5 g^3-16 a^3 c d e^3 g^2 (27 e f+9 d g+4 e g x)+24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (105 f^3+126 f^2 g x+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt {d+e x}} \] Input:
Integrate[((d + e*x)^(3/2)*(f + g*x)^3)/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)]*(128*a^4*e^5*g^3 - 16*a^3*c*d*e^3*g^2*(27 *e*f + 9*d*g + 4*e*g*x) + 24*a^2*c^2*d^2*e^2*g*(3*d*g*(7*f + g*x) + e*(21* f^2 + 9*f*g*x + 2*g^2*x^2)) - 2*a*c^3*d^3*e*(9*d*g*(35*f^2 + 14*f*g*x + 3* g^2*x^2) + e*(105*f^3 + 126*f^2*g*x + 81*f*g^2*x^2 + 20*g^3*x^3)) + c^4*d^ 4*(9*d*(35*f^3 + 35*f^2*g*x + 21*f*g^2*x^2 + 5*g^3*x^3) + e*x*(105*f^3 + 1 89*f^2*g*x + 135*f*g^2*x^2 + 35*g^3*x^3))))/(315*c^5*d^5*Sqrt[d + e*x])
Time = 1.19 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1258, 1253, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
| \(\Big \downarrow \) 1258 |
| \(\displaystyle \frac {1}{9} \left (-\frac {8 a e^2}{c d}+9 d-\frac {e f}{g}\right ) \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {1}{9} \left (-\frac {8 a e^2}{c d}+9 d-\frac {e f}{g}\right ) \left (\frac {6 (c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{7 c d}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {1}{9} \left (-\frac {8 a e^2}{c d}+9 d-\frac {e f}{g}\right ) \left (\frac {6 (c d f-a e g) \left (\frac {4 (c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 c d}+\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}\right )}{7 c d}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {1}{9} \left (-\frac {8 a e^2}{c d}+9 d-\frac {e f}{g}\right ) \left (\frac {6 (c d f-a e g) \left (\frac {4 (c d f-a e g) \left (\frac {1}{3} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}+\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}\right )}{7 c d}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}}+\frac {1}{9} \left (-\frac {8 a e^2}{c d}+9 d-\frac {e f}{g}\right ) \left (\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}+\frac {6 (c d f-a e g) \left (\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}+\frac {4 (c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right )}{3 c d \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}\right )}{7 c d}\right )\) |
Input:
Int[((d + e*x)^(3/2)*(f + g*x)^3)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x ^2],x]
Output:
(2*e*(f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(9*c*d*g*Sqr t[d + e*x]) + ((9*d - (8*a*e^2)/(c*d) - (e*f)/g)*((2*(f + g*x)^3*Sqrt[a*d* e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*d*Sqrt[d + e*x]) + (6*(c*d*f - a* e*g)*((2*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d*S qrt[d + e*x]) + (4*(c*d*f - a*e*g)*((2*(3*f - (d*g)/e - (2*a*e*g)/(c*d))*S qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d*Sqrt[d + e*x]) + (2*g*S qrt[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)))/(7*c*d)))/9
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])
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(c*g*(n + p + 2))), x] - Simp[(b*e*g*(n + 1 ) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2)) Int[(d + e*x)^ (m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p - 1, 0] && !LtQ[n, -1] && IntegerQ[2*p]
Time = 3.56 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (35 e \,g^{3} x^{4} d^{4} c^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right )}{315 \sqrt {e x +d}\, d^{5} c^{5}}\) | \(407\) |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (35 e \,g^{3} x^{4} d^{4} c^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right ) \sqrt {e x +d}}{315 d^{5} c^{5} \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(425\) |
| orering | \(\frac {2 \left (35 e \,g^{3} x^{4} d^{4} c^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right ) \left (c d x +a e \right ) \sqrt {e x +d}}{315 d^{5} c^{5} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\) | \(426\) |
Input:
int((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,meth od=_RETURNVERBOSE)
Output:
2/315/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(35*c^4*d^4*e*g^3*x^4-40*a *c^3*d^3*e^2*g^3*x^3+45*c^4*d^5*g^3*x^3+135*c^4*d^4*e*f*g^2*x^3+48*a^2*c^2 *d^2*e^3*g^3*x^2-54*a*c^3*d^4*e*g^3*x^2-162*a*c^3*d^3*e^2*f*g^2*x^2+189*c^ 4*d^5*f*g^2*x^2+189*c^4*d^4*e*f^2*g*x^2-64*a^3*c*d*e^4*g^3*x+72*a^2*c^2*d^ 3*e^2*g^3*x+216*a^2*c^2*d^2*e^3*f*g^2*x-252*a*c^3*d^4*e*f*g^2*x-252*a*c^3* d^3*e^2*f^2*g*x+315*c^4*d^5*f^2*g*x+105*c^4*d^4*e*f^3*x+128*a^4*e^5*g^3-14 4*a^3*c*d^2*e^3*g^3-432*a^3*c*d*e^4*f*g^2+504*a^2*c^2*d^3*e^2*f*g^2+504*a^ 2*c^2*d^2*e^3*f^2*g-630*a*c^3*d^4*e*f^2*g-210*a*c^3*d^3*e^2*f^3+315*c^4*d^ 5*f^3)/d^5/c^5
Time = 0.09 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e g^{3} x^{4} + 105 \, {\left (3 \, c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{3} - 126 \, {\left (5 \, a c^{3} d^{4} e - 4 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g + 72 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} f g^{2} - 16 \, {\left (9 \, a^{3} c d^{2} e^{3} - 8 \, a^{4} e^{5}\right )} g^{3} + 5 \, {\left (27 \, c^{4} d^{4} e f g^{2} + {\left (9 \, c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} g^{3}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{4} e f^{2} g + 9 \, {\left (7 \, c^{4} d^{5} - 6 \, a c^{3} d^{3} e^{2}\right )} f g^{2} - 2 \, {\left (9 \, a c^{3} d^{4} e - 8 \, a^{2} c^{2} d^{2} e^{3}\right )} g^{3}\right )} x^{2} + {\left (105 \, c^{4} d^{4} e f^{3} + 63 \, {\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{2} g - 36 \, {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} f g^{2} + 8 \, {\left (9 \, a^{2} c^{2} d^{3} e^{2} - 8 \, a^{3} c d e^{4}\right )} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="fricas")
Output:
2/315*(35*c^4*d^4*e*g^3*x^4 + 105*(3*c^4*d^5 - 2*a*c^3*d^3*e^2)*f^3 - 126* (5*a*c^3*d^4*e - 4*a^2*c^2*d^2*e^3)*f^2*g + 72*(7*a^2*c^2*d^3*e^2 - 6*a^3* c*d*e^4)*f*g^2 - 16*(9*a^3*c*d^2*e^3 - 8*a^4*e^5)*g^3 + 5*(27*c^4*d^4*e*f* g^2 + (9*c^4*d^5 - 8*a*c^3*d^3*e^2)*g^3)*x^3 + 3*(63*c^4*d^4*e*f^2*g + 9*( 7*c^4*d^5 - 6*a*c^3*d^3*e^2)*f*g^2 - 2*(9*a*c^3*d^4*e - 8*a^2*c^2*d^2*e^3) *g^3)*x^2 + (105*c^4*d^4*e*f^3 + 63*(5*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^2*g - 36*(7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*f*g^2 + 8*(9*a^2*c^2*d^3*e^2 - 8*a^ 3*c*d*e^4)*g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5*d^6)
\[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)** (1/2),x)
Output:
Integral((d + e*x)**(3/2)*(f + g*x)**3/sqrt((d + e*x)*(a*e + c*d*x)), x)
Time = 0.09 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{3}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{2} g}{5 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {2 \, {\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="maxima")
Output:
2/3*(c^2*d^2*e*x^2 + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)* f^3/(sqrt(c*d*x + a*e)*c^2*d^2) + 2/5*(3*c^3*d^3*e*x^3 - 10*a^2*c*d^2*e^2 + 8*a^3*e^4 + (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*c*d *e^3)*x)*f^2*g/(sqrt(c*d*x + a*e)*c^3*d^3) + 2/35*(15*c^4*d^4*e*x^4 + 56*a ^3*c*d^2*e^3 - 48*a^4*e^5 + 3*(7*c^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d ^4*e - 6*a^2*c^2*d^2*e^3)*x^2 + 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)*x)*f *g^2/(sqrt(c*d*x + a*e)*c^4*d^4) + 2/315*(35*c^5*d^5*e*x^5 - 144*a^4*c*d^2 *e^4 + 128*a^5*e^6 + 5*(9*c^5*d^6 - a*c^4*d^4*e^2)*x^4 - (9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*x^3 + 2*(9*a^2*c^3*d^4*e^2 - 8*a^3*c^2*d^2*e^4)*x^2 - 8 *(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*x)*g^3/(sqrt(c*d*x + a*e)*c^5*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (324) = 648\).
Time = 0.14 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.14 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="giac")
Output:
2/315*e*(315*(c^4*d^5*f^3 - a*c^3*d^3*e^2*f^3 - 3*a*c^3*d^4*e*f^2*g + 3*a^ 2*c^2*d^2*e^3*f^2*g + 3*a^2*c^2*d^3*e^2*f*g^2 - 3*a^3*c*d*e^4*f*g^2 - a^3* c*d^2*e^3*g^3 + a^4*e^5*g^3)*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/(c^5* d^5*e) + (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^3*e^6*f^3 + 315*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^4*e^5*f^2*g - 630*((e* x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^2*d^2*e^7*f^2*g - 630*((e*x + d) *c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^2*d^3*e^6*f*g^2 + 945*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c*d*e^8*f*g^2 + 315*((e*x + d)*c*d*e - c*d^2 *e + a*e^3)^(3/2)*a^2*c*d^2*e^7*g^3 - 420*((e*x + d)*c*d*e - c*d^2*e + a*e ^3)^(3/2)*a^3*e^9*g^3 + 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^2* d^2*e^4*f^2*g + 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^2*d^3*e^3* f*g^2 - 567*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c*d*e^5*f*g^2 - 18 9*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c*d^2*e^4*g^3 + 378*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6*g^3 + 135*((e*x + d)*c*d*e - c*d ^2*e + a*e^3)^(7/2)*c*d*e^2*f*g^2 + 45*((e*x + d)*c*d*e - c*d^2*e + a*e^3) ^(7/2)*c*d^2*e*g^3 - 180*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3*g ^3 + 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*g^3)/(c^5*d^5*e^8))/abs( e)
Time = 6.79 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,a^4\,e^5\,g^3-288\,a^3\,c\,d^2\,e^3\,g^3-864\,a^3\,c\,d\,e^4\,f\,g^2+1008\,a^2\,c^2\,d^3\,e^2\,f\,g^2+1008\,a^2\,c^2\,d^2\,e^3\,f^2\,g-1260\,a\,c^3\,d^4\,e\,f^2\,g-420\,a\,c^3\,d^3\,e^2\,f^3+630\,c^4\,d^5\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}}{9\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^3\,c\,d\,e^4\,g^3+144\,a^2\,c^2\,d^3\,e^2\,g^3+432\,a^2\,c^2\,d^2\,e^3\,f\,g^2-504\,a\,c^3\,d^4\,e\,f\,g^2-504\,a\,c^3\,d^3\,e^2\,f^2\,g+630\,c^4\,d^5\,f^2\,g+210\,c^4\,d^4\,e\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}\,\left (16\,a^2\,e^3\,g^2-18\,a\,c\,d^2\,e\,g^2-54\,a\,c\,d\,e^2\,f\,g+63\,c^2\,d^3\,f\,g+63\,c^2\,d^2\,e\,f^2\right )}{105\,c^3\,d^3\,e}+\frac {2\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (9\,c\,g\,d^2+27\,c\,f\,d\,e-8\,a\,g\,e^2\right )}{63\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \] Input:
int(((f + g*x)^3*(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)*(((d + e*x)^(1/2)*(256*a^4* e^5*g^3 + 630*c^4*d^5*f^3 - 420*a*c^3*d^3*e^2*f^3 - 288*a^3*c*d^2*e^3*g^3 + 1008*a^2*c^2*d^2*e^3*f^2*g + 1008*a^2*c^2*d^3*e^2*f*g^2 - 1260*a*c^3*d^4 *e*f^2*g - 864*a^3*c*d*e^4*f*g^2))/(315*c^5*d^5*e) + (2*g^3*x^4*(d + e*x)^ (1/2))/(9*c*d) + (x*(d + e*x)^(1/2)*(210*c^4*d^4*e*f^3 + 630*c^4*d^5*f^2*g + 144*a^2*c^2*d^3*e^2*g^3 - 128*a^3*c*d*e^4*g^3 - 504*a*c^3*d^3*e^2*f^2*g + 432*a^2*c^2*d^2*e^3*f*g^2 - 504*a*c^3*d^4*e*f*g^2))/(315*c^5*d^5*e) + ( 2*g*x^2*(d + e*x)^(1/2)*(16*a^2*e^3*g^2 + 63*c^2*d^2*e*f^2 + 63*c^2*d^3*f* g - 18*a*c*d^2*e*g^2 - 54*a*c*d*e^2*f*g))/(105*c^3*d^3*e) + (2*g^2*x^3*(d + e*x)^(1/2)*(9*c*d^2*g - 8*a*e^2*g + 27*c*d*e*f))/(63*c^2*d^2*e)))/(x + d /e)
Time = 0.23 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (35 c^{4} d^{4} e \,g^{3} x^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 c^{4} d^{5} f^{3}\right )}{315 c^{5} d^{5}} \] Input:
int((e*x+d)^(3/2)*(g*x+f)^3/(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)*(128*a**4*e**5*g**3 - 144*a**3*c*d**2*e**3*g**3 - 432 *a**3*c*d*e**4*f*g**2 - 64*a**3*c*d*e**4*g**3*x + 504*a**2*c**2*d**3*e**2* f*g**2 + 72*a**2*c**2*d**3*e**2*g**3*x + 504*a**2*c**2*d**2*e**3*f**2*g + 216*a**2*c**2*d**2*e**3*f*g**2*x + 48*a**2*c**2*d**2*e**3*g**3*x**2 - 630* a*c**3*d**4*e*f**2*g - 252*a*c**3*d**4*e*f*g**2*x - 54*a*c**3*d**4*e*g**3* x**2 - 210*a*c**3*d**3*e**2*f**3 - 252*a*c**3*d**3*e**2*f**2*g*x - 162*a*c **3*d**3*e**2*f*g**2*x**2 - 40*a*c**3*d**3*e**2*g**3*x**3 + 315*c**4*d**5* f**3 + 315*c**4*d**5*f**2*g*x + 189*c**4*d**5*f*g**2*x**2 + 45*c**4*d**5*g **3*x**3 + 105*c**4*d**4*e*f**3*x + 189*c**4*d**4*e*f**2*g*x**2 + 135*c**4 *d**4*e*f*g**2*x**3 + 35*c**4*d**4*e*g**3*x**4))/(315*c**5*d**5)