Integrand size = 46, antiderivative size = 412 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^5 d^5 e g \sqrt {d+e x}}-\frac {16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^4 d^4 e}-\frac {4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt {d+e x}} \]
16/315*(-a*e*g+c*d*f)^2*(8*a*e^2*g+c*d*(-9*d*g+e*f))*(2*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)^(1/2)/c^5/d^5/e/g/(e*x+d)^(1/2)- 4/105*(-a*e*g+c*d*f)*(8*a*e^2*g+c*d*(-9*d*g+e*f))*(g*x+f)^2*(a*d*e+(a*e^2+ c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/g/(e*x+d)^(1/2)-2/63*(8*a*e^2*g+c*d*(-9* d*g+e*f))*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/g/(e*x +d)^(1/2)+2/9*e*(g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/g/(e *x+d)^(1/2)-16/315*(-a*e*g+c*d*f)^2*(8*a*e^2*g+c*d*(-9*d*g+e*f))*(e*x+d)^( 1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e
Time = 0.20 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (128 a^4 e^5 g^3-16 a^3 c d e^3 g^2 (27 e f+9 d g+4 e g x)+24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (105 f^3+126 f^2 g x+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt {d+e x}} \]
(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 = 0.72 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.88, 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 |
| \(\displaystyle \frac {2 e (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt {d+e x}}+\frac {1}{9} \left (-\frac {8 a e^2}{c d}+9 d-\frac {e f}{g}\right ) \left (\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}+\frac {6 (c d f-a e g) \left (\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}+\frac {4 (c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right )}{3 c d \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}\right )}{7 c d}\right )\) |
(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
3.8.84.3.1 Defintions of rubi rules used
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 = 0.55 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (35 e \,g^{3} x^{4} c^{4} d^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right )}{315 \sqrt {e x +d}\, c^{5} d^{5}}\) | \(407\) |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (35 e \,g^{3} x^{4} c^{4} d^{4}-40 a \,c^{3} d^{3} e^{2} g^{3} x^{3}+45 c^{4} d^{5} g^{3} x^{3}+135 c^{4} d^{4} e f \,g^{2} x^{3}+48 a^{2} c^{2} d^{2} e^{3} g^{3} x^{2}-54 a \,c^{3} d^{4} e \,g^{3} x^{2}-162 a \,c^{3} d^{3} e^{2} f \,g^{2} x^{2}+189 c^{4} d^{5} f \,g^{2} x^{2}+189 c^{4} d^{4} e \,f^{2} g \,x^{2}-64 a^{3} c d \,e^{4} g^{3} x +72 a^{2} c^{2} d^{3} e^{2} g^{3} x +216 a^{2} c^{2} d^{2} e^{3} f \,g^{2} x -252 a \,c^{3} d^{4} e f \,g^{2} x -252 a \,c^{3} d^{3} e^{2} f^{2} g x +315 c^{4} d^{5} f^{2} g x +105 c^{4} d^{4} e \,f^{3} x +128 a^{4} e^{5} g^{3}-144 a^{3} c \,d^{2} e^{3} g^{3}-432 a^{3} c d \,e^{4} f \,g^{2}+504 a^{2} c^{2} d^{3} e^{2} f \,g^{2}+504 a^{2} c^{2} d^{2} e^{3} f^{2} g -630 a \,c^{3} d^{4} e \,f^{2} g -210 a \,c^{3} d^{3} e^{2} f^{3}+315 d^{5} f^{3} c^{4}\right ) \sqrt {e x +d}}{315 c^{5} d^{5} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(425\) |
2/315/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(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)/c^5/d^5
Time = 0.31 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e g^{3} x^{4} + 105 \, {\left (3 \, c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{3} - 126 \, {\left (5 \, a c^{3} d^{4} e - 4 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g + 72 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} f g^{2} - 16 \, {\left (9 \, a^{3} c d^{2} e^{3} - 8 \, a^{4} e^{5}\right )} g^{3} + 5 \, {\left (27 \, c^{4} d^{4} e f g^{2} + {\left (9 \, c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} g^{3}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{4} e f^{2} g + 9 \, {\left (7 \, c^{4} d^{5} - 6 \, a c^{3} d^{3} e^{2}\right )} f g^{2} - 2 \, {\left (9 \, a c^{3} d^{4} e - 8 \, a^{2} c^{2} d^{2} e^{3}\right )} g^{3}\right )} x^{2} + {\left (105 \, c^{4} d^{4} e f^{3} + 63 \, {\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{2} g - 36 \, {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} f g^{2} + 8 \, {\left (9 \, a^{2} c^{2} d^{3} e^{2} - 8 \, a^{3} c d e^{4}\right )} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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")
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)
\[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
Time = 0.26 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{3}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{2} g}{5 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {2 \, {\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} \]
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")
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. 1185 vs. \(2 (382) = 764\).
Time = 0.34 (sec) , antiderivative size = 1185, normalized size of antiderivative = 2.88 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \]
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")
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) - 2*(105*sqrt(-c*d^2*e + a*e^3)*c^4*d^5*e^3*f^3 - 105*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^3*e^5*f^3 - 63*sqrt(-c*d^2*e + a*e^3)*c^4*d^6*e^2*f^2*g - 189*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^4*e^4*f^2*g + 252*sqrt(-c*d^2*e + a*e ^3)*a^2*c^2*d^2*e^6*f^2*g + 27*sqrt(-c*d^2*e + a*e^3)*c^4*d^7*e*f*g^2 + 45 *sqrt(-c*d^2*e + a*e^3)*a*c^3*d^5*e^3*f*g^2 + 144*sqrt(-c*d^2*e + a*e^3)*a ^2*c^2*d^3*e^5*f*g^2 - 216*sqrt(-c*d^2*e + a*e^3)*a^3*c*d*e^7*f*g^2 - 5*sq rt(-c*d^2*e + a*e^3)*c^4*d^8*g^3 - 7*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2* g^3 - 12*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4*g^3 - 40*sqrt(-c*d^2*e + a *e^3)*a^3*c*d^2*e^6*g^3 + 64*sqrt(-c*d^2*e + a*e^3)*a^4*e^8*g^3)/(c^5*d^5* e^4) + (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^3*e^6*f^3 + 31 5*((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...
Time = 12.40 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^{3/2} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,a^4\,e^5\,g^3-288\,a^3\,c\,d^2\,e^3\,g^3-864\,a^3\,c\,d\,e^4\,f\,g^2+1008\,a^2\,c^2\,d^3\,e^2\,f\,g^2+1008\,a^2\,c^2\,d^2\,e^3\,f^2\,g-1260\,a\,c^3\,d^4\,e\,f^2\,g-420\,a\,c^3\,d^3\,e^2\,f^3+630\,c^4\,d^5\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}}{9\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^3\,c\,d\,e^4\,g^3+144\,a^2\,c^2\,d^3\,e^2\,g^3+432\,a^2\,c^2\,d^2\,e^3\,f\,g^2-504\,a\,c^3\,d^4\,e\,f\,g^2-504\,a\,c^3\,d^3\,e^2\,f^2\,g+630\,c^4\,d^5\,f^2\,g+210\,c^4\,d^4\,e\,f^3\right )}{315\,c^5\,d^5\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}\,\left (16\,a^2\,e^3\,g^2-18\,a\,c\,d^2\,e\,g^2-54\,a\,c\,d\,e^2\,f\,g+63\,c^2\,d^3\,f\,g+63\,c^2\,d^2\,e\,f^2\right )}{105\,c^3\,d^3\,e}+\frac {2\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (9\,c\,g\,d^2+27\,c\,f\,d\,e-8\,a\,g\,e^2\right )}{63\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \]
((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)