Integrand size = 46, antiderivative size = 438 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \left (c d^2-a e^2\right ) (c d f-a e g)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^6 d^6 \sqrt {d+e x}}-\frac {2 (c d f-a e g)^3 \left (5 a e^2 g-c d (e f+4 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^6 d^6 (d+e x)^{3/2}}-\frac {4 g (c d f-a e g)^2 \left (5 a e^2 g-c d (2 e f+3 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^6 d^6 (d+e x)^{5/2}}-\frac {4 g^2 (c d f-a e g) \left (5 a e^2 g-c d (3 e f+2 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^6 d^6 (d+e x)^{7/2}}-\frac {2 g^3 \left (5 a e^2 g-c d (4 e f+d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^6 d^6 (d+e x)^{9/2}}+\frac {2 e g^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{11/2}}{11 c^6 d^6 (d+e x)^{11/2}} \] Output:
2*(-a*e^2+c*d^2)*(-a*e*g+c*d*f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ c^6/d^6/(e*x+d)^(1/2)-2/3*(-a*e*g+c*d*f)^3*(5*a*e^2*g-c*d*(4*d*g+e*f))*(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^6/d^6/(e*x+d)^(3/2)-4/5*g*(-a*e*g+c *d*f)^2*(5*a*e^2*g-c*d*(3*d*g+2*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2)/c^6/d^6/(e*x+d)^(5/2)-4/7*g^2*(-a*e*g+c*d*f)*(5*a*e^2*g-c*d*(2*d*g+3*e *f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^6/d^6/(e*x+d)^(7/2)-2/9*g^3 *(5*a*e^2*g-c*d*(d*g+4*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/c^6/d ^6/(e*x+d)^(9/2)+2/11*e*g^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(11/2)/c^6/d ^6/(e*x+d)^(11/2)
Time = 0.53 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-1280 a^5 e^6 g^4+128 a^4 c d e^4 g^3 (44 e f+11 d g+5 e g x)-32 a^3 c^2 d^2 e^3 g^2 \left (22 d g (9 f+g x)+e \left (297 f^2+88 f g x+15 g^2 x^2\right )\right )+16 a^2 c^3 d^3 e^2 g \left (33 d g \left (21 f^2+6 f g x+g^2 x^2\right )+e \left (462 f^3+297 f^2 g x+132 f g^2 x^2+25 g^3 x^3\right )\right )-2 a c^4 d^4 e \left (44 d g \left (105 f^3+63 f^2 g x+27 f g^2 x^2+5 g^3 x^3\right )+e \left (1155 f^4+1848 f^3 g x+1782 f^2 g^2 x^2+880 f g^3 x^3+175 g^4 x^4\right )\right )+c^5 d^5 \left (11 d \left (315 f^4+420 f^3 g x+378 f^2 g^2 x^2+180 f g^3 x^3+35 g^4 x^4\right )+e x \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )\right )}{3465 c^6 d^6 \sqrt {d+e x}} \] Input:
Integrate[((d + e*x)^(3/2)*(f + g*x)^4)/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)]*(-1280*a^5*e^6*g^4 + 128*a^4*c*d*e^4*g^3* (44*e*f + 11*d*g + 5*e*g*x) - 32*a^3*c^2*d^2*e^3*g^2*(22*d*g*(9*f + g*x) + e*(297*f^2 + 88*f*g*x + 15*g^2*x^2)) + 16*a^2*c^3*d^3*e^2*g*(33*d*g*(21*f ^2 + 6*f*g*x + g^2*x^2) + e*(462*f^3 + 297*f^2*g*x + 132*f*g^2*x^2 + 25*g^ 3*x^3)) - 2*a*c^4*d^4*e*(44*d*g*(105*f^3 + 63*f^2*g*x + 27*f*g^2*x^2 + 5*g ^3*x^3) + e*(1155*f^4 + 1848*f^3*g*x + 1782*f^2*g^2*x^2 + 880*f*g^3*x^3 + 175*g^4*x^4)) + c^5*d^5*(11*d*(315*f^4 + 420*f^3*g*x + 378*f^2*g^2*x^2 + 1 80*f*g^3*x^3 + 35*g^4*x^4) + e*x*(1155*f^4 + 2772*f^3*g*x + 2970*f^2*g^2*x ^2 + 1540*f*g^3*x^3 + 315*g^4*x^4))))/(3465*c^6*d^6*Sqrt[d + e*x])
Time = 1.52 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1258, 1253, 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.
\(\displaystyle \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1258 |
\(\displaystyle \frac {1}{11} \left (-\frac {10 a e^2}{c d}+11 d-\frac {e f}{g}\right ) \int \frac {\sqrt {d+e x} (f+g x)^4}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1253 |
\(\displaystyle \frac {1}{11} \left (-\frac {10 a e^2}{c d}+11 d-\frac {e f}{g}\right ) \left (\frac {8 (c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{9 c d}+\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1253 |
\(\displaystyle \frac {1}{11} \left (-\frac {10 a e^2}{c d}+11 d-\frac {e f}{g}\right ) \left (\frac {8 (c d f-a e g) \left (\frac {6 (c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{7 c d}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}\right )}{9 c d}+\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1253 |
\(\displaystyle \frac {1}{11} \left (-\frac {10 a e^2}{c d}+11 d-\frac {e f}{g}\right ) \left (\frac {8 (c d f-a e g) \left (\frac {6 (c d f-a e g) \left (\frac {4 (c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 c d}+\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}\right )}{7 c d}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}\right )}{9 c d}+\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1221 |
\(\displaystyle \frac {1}{11} \left (-\frac {10 a e^2}{c d}+11 d-\frac {e f}{g}\right ) \left (\frac {8 (c d f-a e g) \left (\frac {6 (c d f-a e g) \left (\frac {4 (c d f-a e g) \left (\frac {1}{3} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}+\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}\right )}{7 c d}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}\right )}{9 c d}+\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d \sqrt {d+e x}}\right )+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}}+\frac {1}{11} \left (-\frac {10 a e^2}{c d}+11 d-\frac {e f}{g}\right ) \left (\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d \sqrt {d+e x}}+\frac {8 (c d f-a e g) \left (\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}}+\frac {6 (c d f-a e g) \left (\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt {d+e x}}+\frac {4 (c d f-a e g) \left (\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (-\frac {2 a e g}{c d}-\frac {d g}{e}+3 f\right )}{3 c d \sqrt {d+e x}}+\frac {2 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}\right )}{5 c d}\right )}{7 c d}\right )}{9 c d}\right )\) |
Input:
Int[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x ^2],x]
Output:
(2*e*(f + g*x)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(11*c*d*g*Sq rt[d + e*x]) + ((11*d - (10*a*e^2)/(c*d) - (e*f)/g)*((2*(f + g*x)^4*Sqrt[a *d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(9*c*d*Sqrt[d + e*x]) + (8*(c*d*f - a*e*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*Sqrt[d + e*x]) + (4*(c*d*f - a*e*g)*((2*(3 *f - (d*g)/e - (2*a*e*g)/(c*d))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 ])/(3*c*d*Sqrt[d + e*x]) + (2*g*Sqrt[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*c*d)))/11
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 = 3.43 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.42
method | result | size |
default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right )}{3465 \sqrt {e x +d}\, c^{6} d^{6}}\) | \(623\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right ) \sqrt {e x +d}}{3465 c^{6} d^{6} \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(641\) |
orering | \(-\frac {2 \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right ) \left (c d x +a e \right ) \sqrt {e x +d}}{3465 c^{6} d^{6} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\) | \(642\) |
Input:
int((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,meth od=_RETURNVERBOSE)
Output:
-2/3465/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-315*c^5*d^5*e*g^4*x^5+ 350*a*c^4*d^4*e^2*g^4*x^4-385*c^5*d^6*g^4*x^4-1540*c^5*d^5*e*f*g^3*x^4-400 *a^2*c^3*d^3*e^3*g^4*x^3+440*a*c^4*d^5*e*g^4*x^3+1760*a*c^4*d^4*e^2*f*g^3* x^3-1980*c^5*d^6*f*g^3*x^3-2970*c^5*d^5*e*f^2*g^2*x^3+480*a^3*c^2*d^2*e^4* g^4*x^2-528*a^2*c^3*d^4*e^2*g^4*x^2-2112*a^2*c^3*d^3*e^3*f*g^3*x^2+2376*a* c^4*d^5*e*f*g^3*x^2+3564*a*c^4*d^4*e^2*f^2*g^2*x^2-4158*c^5*d^6*f^2*g^2*x^ 2-2772*c^5*d^5*e*f^3*g*x^2-640*a^4*c*d*e^5*g^4*x+704*a^3*c^2*d^3*e^3*g^4*x +2816*a^3*c^2*d^2*e^4*f*g^3*x-3168*a^2*c^3*d^4*e^2*f*g^3*x-4752*a^2*c^3*d^ 3*e^3*f^2*g^2*x+5544*a*c^4*d^5*e*f^2*g^2*x+3696*a*c^4*d^4*e^2*f^3*g*x-4620 *c^5*d^6*f^3*g*x-1155*c^5*d^5*e*f^4*x+1280*a^5*e^6*g^4-1408*a^4*c*d^2*e^4* g^4-5632*a^4*c*d*e^5*f*g^3+6336*a^3*c^2*d^3*e^3*f*g^3+9504*a^3*c^2*d^2*e^4 *f^2*g^2-11088*a^2*c^3*d^4*e^2*f^2*g^2-7392*a^2*c^3*d^3*e^3*f^3*g+9240*a*c ^4*d^5*e*f^3*g+2310*a*c^4*d^4*e^2*f^4-3465*c^5*d^6*f^4)/c^6/d^6
Time = 0.10 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (315 \, c^{5} d^{5} e g^{4} x^{5} + 1155 \, {\left (3 \, c^{5} d^{6} - 2 \, a c^{4} d^{4} e^{2}\right )} f^{4} - 1848 \, {\left (5 \, a c^{4} d^{5} e - 4 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g + 1584 \, {\left (7 \, a^{2} c^{3} d^{4} e^{2} - 6 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{2} - 704 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} f g^{3} + 128 \, {\left (11 \, a^{4} c d^{2} e^{4} - 10 \, a^{5} e^{6}\right )} g^{4} + 35 \, {\left (44 \, c^{5} d^{5} e f g^{3} + {\left (11 \, c^{5} d^{6} - 10 \, a c^{4} d^{4} e^{2}\right )} g^{4}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{5} e f^{2} g^{2} + 22 \, {\left (9 \, c^{5} d^{6} - 8 \, a c^{4} d^{4} e^{2}\right )} f g^{3} - 4 \, {\left (11 \, a c^{4} d^{5} e - 10 \, a^{2} c^{3} d^{3} e^{3}\right )} g^{4}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{5} e f^{3} g + 99 \, {\left (7 \, c^{5} d^{6} - 6 \, a c^{4} d^{4} e^{2}\right )} f^{2} g^{2} - 44 \, {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} f g^{3} + 8 \, {\left (11 \, a^{2} c^{3} d^{4} e^{2} - 10 \, a^{3} c^{2} d^{2} e^{4}\right )} g^{4}\right )} x^{2} + {\left (1155 \, c^{5} d^{5} e f^{4} + 924 \, {\left (5 \, c^{5} d^{6} - 4 \, a c^{4} d^{4} e^{2}\right )} f^{3} g - 792 \, {\left (7 \, a c^{4} d^{5} e - 6 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g^{2} + 352 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{3} - 64 \, {\left (11 \, a^{3} c^{2} d^{3} e^{3} - 10 \, a^{4} c d e^{5}\right )} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{6} d^{6} e x + c^{6} d^{7}\right )}} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="fricas")
Output:
2/3465*(315*c^5*d^5*e*g^4*x^5 + 1155*(3*c^5*d^6 - 2*a*c^4*d^4*e^2)*f^4 - 1 848*(5*a*c^4*d^5*e - 4*a^2*c^3*d^3*e^3)*f^3*g + 1584*(7*a^2*c^3*d^4*e^2 - 6*a^3*c^2*d^2*e^4)*f^2*g^2 - 704*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*f*g^3 + 128*(11*a^4*c*d^2*e^4 - 10*a^5*e^6)*g^4 + 35*(44*c^5*d^5*e*f*g^3 + (11* c^5*d^6 - 10*a*c^4*d^4*e^2)*g^4)*x^4 + 10*(297*c^5*d^5*e*f^2*g^2 + 22*(9*c ^5*d^6 - 8*a*c^4*d^4*e^2)*f*g^3 - 4*(11*a*c^4*d^5*e - 10*a^2*c^3*d^3*e^3)* g^4)*x^3 + 6*(462*c^5*d^5*e*f^3*g + 99*(7*c^5*d^6 - 6*a*c^4*d^4*e^2)*f^2*g ^2 - 44*(9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*f*g^3 + 8*(11*a^2*c^3*d^4*e^2 - 10*a^3*c^2*d^2*e^4)*g^4)*x^2 + (1155*c^5*d^5*e*f^4 + 924*(5*c^5*d^6 - 4* a*c^4*d^4*e^2)*f^3*g - 792*(7*a*c^4*d^5*e - 6*a^2*c^3*d^3*e^3)*f^2*g^2 + 3 52*(9*a^2*c^3*d^4*e^2 - 8*a^3*c^2*d^2*e^4)*f*g^3 - 64*(11*a^3*c^2*d^3*e^3 - 10*a^4*c*d*e^5)*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt (e*x + d)/(c^6*d^6*e*x + c^6*d^7)
\[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{4}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(g*x+f)**4/(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)**4/sqrt((d + e*x)*(a*e + c*d*x)), x)
Time = 0.12 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{4}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {8 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{3} g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {4 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f^{2} g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {8 \, {\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} f g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} + \frac {2 \, {\left (315 \, c^{6} d^{6} e x^{6} + 1408 \, a^{5} c d^{2} e^{5} - 1280 \, a^{6} e^{7} + 35 \, {\left (11 \, c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{5} - 5 \, {\left (11 \, a c^{5} d^{6} e - 10 \, a^{2} c^{4} d^{4} e^{3}\right )} x^{4} + 8 \, {\left (11 \, a^{2} c^{4} d^{5} e^{2} - 10 \, a^{3} c^{3} d^{3} e^{4}\right )} x^{3} - 16 \, {\left (11 \, a^{3} c^{3} d^{4} e^{3} - 10 \, a^{4} c^{2} d^{2} e^{5}\right )} x^{2} + 64 \, {\left (11 \, a^{4} c^{2} d^{3} e^{4} - 10 \, a^{5} c d e^{6}\right )} x\right )} g^{4}}{3465 \, \sqrt {c d x + a e} c^{6} d^{6}} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)^4/(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^4/(sqrt(c*d*x + a*e)*c^2*d^2) + 8/15*(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^3*g/(sqrt(c*d*x + a*e)*c^3*d^3) + 4/35*(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^2*g^2/(sqrt(c*d*x + a*e)*c^4*d^4) + 8/315*(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)*f*g^3/(sqrt(c*d*x + a*e)*c^5*d^ 5) + 2/3465*(315*c^6*d^6*e*x^6 + 1408*a^5*c*d^2*e^5 - 1280*a^6*e^7 + 35*(1 1*c^6*d^7 - a*c^5*d^5*e^2)*x^5 - 5*(11*a*c^5*d^6*e - 10*a^2*c^4*d^4*e^3)*x ^4 + 8*(11*a^2*c^4*d^5*e^2 - 10*a^3*c^3*d^3*e^4)*x^3 - 16*(11*a^3*c^3*d^4* e^3 - 10*a^4*c^2*d^2*e^5)*x^2 + 64*(11*a^4*c^2*d^3*e^4 - 10*a^5*c*d*e^6)*x )*g^4/(sqrt(c*d*x + a*e)*c^6*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (404) = 808\).
Time = 0.14 (sec) , antiderivative size = 1159, normalized size of antiderivative = 2.65 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2), x, algorithm="giac")
Output:
2/3465*e*(3465*(c^5*d^6*f^4 - a*c^4*d^4*e^2*f^4 - 4*a*c^4*d^5*e*f^3*g + 4* a^2*c^3*d^3*e^3*f^3*g + 6*a^2*c^3*d^4*e^2*f^2*g^2 - 6*a^3*c^2*d^2*e^4*f^2* g^2 - 4*a^3*c^2*d^3*e^3*f*g^3 + 4*a^4*c*d*e^5*f*g^3 + a^4*c*d^2*e^4*g^4 - a^5*e^6*g^4)*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/(c^6*d^6*e) + (1155*( (e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^4*d^4*e^8*f^4 + 4620*((e*x + d) *c*d*e - c*d^2*e + a*e^3)^(3/2)*c^4*d^5*e^7*f^3*g - 9240*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^3*d^3*e^9*f^3*g - 13860*((e*x + d)*c*d*e - c* d^2*e + a*e^3)^(3/2)*a*c^3*d^4*e^8*f^2*g^2 + 20790*((e*x + d)*c*d*e - c*d^ 2*e + a*e^3)^(3/2)*a^2*c^2*d^2*e^10*f^2*g^2 + 13860*((e*x + d)*c*d*e - c*d ^2*e + a*e^3)^(3/2)*a^2*c^2*d^3*e^9*f*g^3 - 18480*((e*x + d)*c*d*e - c*d^2 *e + a*e^3)^(3/2)*a^3*c*d*e^11*f*g^3 - 4620*((e*x + d)*c*d*e - c*d^2*e + a *e^3)^(3/2)*a^3*c*d^2*e^10*g^4 + 5775*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^ (3/2)*a^4*e^12*g^4 + 2772*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^3*d^ 3*e^6*f^3*g + 4158*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^3*d^4*e^5*f ^2*g^2 - 12474*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^2*d^2*e^7*f^2 *g^2 - 8316*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^2*d^3*e^6*f*g^3 + 16632*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c*d*e^8*f*g^3 + 4158 *((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c*d^2*e^7*g^4 - 6930*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9*g^4 + 2970*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^2*d^2*e^4*f^2*g^2 + 1980*((e*x + d)*c*d*e - c*...
Time = 6.60 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5\,\sqrt {d+e\,x}}{11\,c\,d}-\frac {\sqrt {d+e\,x}\,\left (2560\,a^5\,e^6\,g^4-2816\,a^4\,c\,d^2\,e^4\,g^4-11264\,a^4\,c\,d\,e^5\,f\,g^3+12672\,a^3\,c^2\,d^3\,e^3\,f\,g^3+19008\,a^3\,c^2\,d^2\,e^4\,f^2\,g^2-22176\,a^2\,c^3\,d^4\,e^2\,f^2\,g^2-14784\,a^2\,c^3\,d^3\,e^3\,f^3\,g+18480\,a\,c^4\,d^5\,e\,f^3\,g+4620\,a\,c^4\,d^4\,e^2\,f^4-6930\,c^5\,d^6\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (1280\,a^4\,c\,d\,e^5\,g^4-1408\,a^3\,c^2\,d^3\,e^3\,g^4-5632\,a^3\,c^2\,d^2\,e^4\,f\,g^3+6336\,a^2\,c^3\,d^4\,e^2\,f\,g^3+9504\,a^2\,c^3\,d^3\,e^3\,f^2\,g^2-11088\,a\,c^4\,d^5\,e\,f^2\,g^2-7392\,a\,c^4\,d^4\,e^2\,f^3\,g+9240\,c^5\,d^6\,f^3\,g+2310\,c^5\,d^5\,e\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,a^3\,c^2\,d^2\,e^4\,g^4+1056\,a^2\,c^3\,d^4\,e^2\,g^4+4224\,a^2\,c^3\,d^3\,e^3\,f\,g^3-4752\,a\,c^4\,d^5\,e\,f\,g^3-7128\,a\,c^4\,d^4\,e^2\,f^2\,g^2+8316\,c^5\,d^6\,f^2\,g^2+5544\,c^5\,d^5\,e\,f^3\,g\right )}{3465\,c^6\,d^6\,e}+\frac {4\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (40\,a^2\,e^3\,g^2-44\,a\,c\,d^2\,e\,g^2-176\,a\,c\,d\,e^2\,f\,g+198\,c^2\,d^3\,f\,g+297\,c^2\,d^2\,e\,f^2\right )}{693\,c^3\,d^3\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}\,\left (11\,c\,g\,d^2+44\,c\,f\,d\,e-10\,a\,g\,e^2\right )}{99\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \] Input:
int(((f + g*x)^4*(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*g^4*x^5*(d + e*x)^(1/2) )/(11*c*d) - ((d + e*x)^(1/2)*(2560*a^5*e^6*g^4 - 6930*c^5*d^6*f^4 + 4620* a*c^4*d^4*e^2*f^4 - 2816*a^4*c*d^2*e^4*g^4 - 14784*a^2*c^3*d^3*e^3*f^3*g + 12672*a^3*c^2*d^3*e^3*f*g^3 + 18480*a*c^4*d^5*e*f^3*g - 11264*a^4*c*d*e^5 *f*g^3 - 22176*a^2*c^3*d^4*e^2*f^2*g^2 + 19008*a^3*c^2*d^2*e^4*f^2*g^2))/( 3465*c^6*d^6*e) + (x*(d + e*x)^(1/2)*(2310*c^5*d^5*e*f^4 + 9240*c^5*d^6*f^ 3*g - 1408*a^3*c^2*d^3*e^3*g^4 + 1280*a^4*c*d*e^5*g^4 - 7392*a*c^4*d^4*e^2 *f^3*g - 11088*a*c^4*d^5*e*f^2*g^2 + 6336*a^2*c^3*d^4*e^2*f*g^3 - 5632*a^3 *c^2*d^2*e^4*f*g^3 + 9504*a^2*c^3*d^3*e^3*f^2*g^2))/(3465*c^6*d^6*e) + (x^ 2*(d + e*x)^(1/2)*(8316*c^5*d^6*f^2*g^2 + 1056*a^2*c^3*d^4*e^2*g^4 - 960*a ^3*c^2*d^2*e^4*g^4 + 5544*c^5*d^5*e*f^3*g - 7128*a*c^4*d^4*e^2*f^2*g^2 + 4 224*a^2*c^3*d^3*e^3*f*g^3 - 4752*a*c^4*d^5*e*f*g^3))/(3465*c^6*d^6*e) + (4 *g^2*x^3*(d + e*x)^(1/2)*(40*a^2*e^3*g^2 + 297*c^2*d^2*e*f^2 + 198*c^2*d^3 *f*g - 44*a*c*d^2*e*g^2 - 176*a*c*d*e^2*f*g))/(693*c^3*d^3*e) + (2*g^3*x^4 *(d + e*x)^(1/2)*(11*c*d^2*g - 10*a*e^2*g + 44*c*d*e*f))/(99*c^2*d^2*e)))/ (x + d/e)
Time = 0.29 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (315 c^{5} d^{5} e \,g^{4} x^{5}-350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}+385 c^{5} d^{6} g^{4} x^{4}+1540 c^{5} d^{5} e f \,g^{3} x^{4}+400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}-440 a \,c^{4} d^{5} e \,g^{4} x^{3}-1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}+1980 c^{5} d^{6} f \,g^{3} x^{3}+2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}-480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}+528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}+2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}-2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}-3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}+4158 c^{5} d^{6} f^{2} g^{2} x^{2}+2772 c^{5} d^{5} e \,f^{3} g \,x^{2}+640 a^{4} c d \,e^{5} g^{4} x -704 a^{3} c^{2} d^{3} e^{3} g^{4} x -2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x +3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x +4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x -5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x -3696 a \,c^{4} d^{4} e^{2} f^{3} g x +4620 c^{5} d^{6} f^{3} g x +1155 c^{5} d^{5} e \,f^{4} x -1280 a^{5} e^{6} g^{4}+1408 a^{4} c \,d^{2} e^{4} g^{4}+5632 a^{4} c d \,e^{5} f \,g^{3}-6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}-9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}+11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}+7392 a^{2} c^{3} d^{3} e^{3} f^{3} g -9240 a \,c^{4} d^{5} e \,f^{3} g -2310 a \,c^{4} d^{4} e^{2} f^{4}+3465 c^{5} d^{6} f^{4}\right )}{3465 c^{6} d^{6}} \] Input:
int((e*x+d)^(3/2)*(g*x+f)^4/(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)*( - 1280*a**5*e**6*g**4 + 1408*a**4*c*d**2*e**4*g**4 + 5632*a**4*c*d*e**5*f*g**3 + 640*a**4*c*d*e**5*g**4*x - 6336*a**3*c**2*d* *3*e**3*f*g**3 - 704*a**3*c**2*d**3*e**3*g**4*x - 9504*a**3*c**2*d**2*e**4 *f**2*g**2 - 2816*a**3*c**2*d**2*e**4*f*g**3*x - 480*a**3*c**2*d**2*e**4*g **4*x**2 + 11088*a**2*c**3*d**4*e**2*f**2*g**2 + 3168*a**2*c**3*d**4*e**2* f*g**3*x + 528*a**2*c**3*d**4*e**2*g**4*x**2 + 7392*a**2*c**3*d**3*e**3*f* *3*g + 4752*a**2*c**3*d**3*e**3*f**2*g**2*x + 2112*a**2*c**3*d**3*e**3*f*g **3*x**2 + 400*a**2*c**3*d**3*e**3*g**4*x**3 - 9240*a*c**4*d**5*e*f**3*g - 5544*a*c**4*d**5*e*f**2*g**2*x - 2376*a*c**4*d**5*e*f*g**3*x**2 - 440*a*c **4*d**5*e*g**4*x**3 - 2310*a*c**4*d**4*e**2*f**4 - 3696*a*c**4*d**4*e**2* f**3*g*x - 3564*a*c**4*d**4*e**2*f**2*g**2*x**2 - 1760*a*c**4*d**4*e**2*f* g**3*x**3 - 350*a*c**4*d**4*e**2*g**4*x**4 + 3465*c**5*d**6*f**4 + 4620*c* *5*d**6*f**3*g*x + 4158*c**5*d**6*f**2*g**2*x**2 + 1980*c**5*d**6*f*g**3*x **3 + 385*c**5*d**6*g**4*x**4 + 1155*c**5*d**5*e*f**4*x + 2772*c**5*d**5*e *f**3*g*x**2 + 2970*c**5*d**5*e*f**2*g**2*x**3 + 1540*c**5*d**5*e*f*g**3*x **4 + 315*c**5*d**5*e*g**4*x**5))/(3465*c**6*d**6)