Integrand size = 23, antiderivative size = 620 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=-\frac {16 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right ) x \sqrt {d+e x}}{1155 c^3 e^4 \sqrt {b x+c x^2}}+\frac {2 \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{1155 c^3 e^3}-\frac {4 (c d-3 b e) (3 c d-b e) (c d+b e) x \sqrt {d+e x} \sqrt {b x+c x^2}}{1155 c^2 e^2}+\frac {2}{231} \left (13 b d+\frac {c d^2}{e}-\frac {6 b^2 e}{c}\right ) x^2 \sqrt {d+e x} \sqrt {b x+c x^2}+\frac {2 (c d+b e) x \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}{33 c}+\frac {2}{11} x (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}+\frac {16 \sqrt {b} (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{1155 c^{7/2} e^3 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:
-16/1155*(-2*b*e+c*d)*(-b*e+2*c*d)*(b*e+c*d)*(b^2*e^2-b*c*d*e+c^2*d^2)*x*( e*x+d)^(1/2)/c^3/e^4/(c*x^2+b*x)^(1/2)+2/1155*(8*b^4*e^4-19*b^3*c*d*e^3+6* b^2*c^2*d^2*e^2-19*b*c^3*d^3*e+8*c^4*d^4)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/ c^3/e^3-4/1155*(-3*b*e+c*d)*(-b*e+3*c*d)*(b*e+c*d)*x*(e*x+d)^(1/2)*(c*x^2+ b*x)^(1/2)/c^2/e^2+2/231*(13*b*d+c*d^2/e-6*b^2*e/c)*x^2*(e*x+d)^(1/2)*(c*x ^2+b*x)^(1/2)+2/33*(b*e+c*d)*x*(e*x+d)^(1/2)*(c*x^2+b*x)^(3/2)/c+2/11*x*(e *x+d)^(3/2)*(c*x^2+b*x)^(3/2)+16/1155*b^(1/2)*(-2*b*e+c*d)*(-b*e+2*c*d)*(b *e+c*d)*(b^2*e^2-b*c*d*e+c^2*d^2)*x^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2)* x^(1/2)/b^(1/2)/(1+c*x/b)^(1/2),(1-b*e/c/d)^(1/2))/c^(7/2)/e^4/(b*(e*x+d)/ d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)-2/1155*b^(3/2)*(8*b^4*e^4-19*b^3*c*d*e^ 3+6*b^2*c^2*d^2*e^2-19*b*c^3*d^3*e+8*c^4*d^4)*x^(1/2)*(e*x+d)^(1/2)*Invers eJacobiAM(arctan(c^(1/2)*x^(1/2)/b^(1/2)),(1-b*e/c/d)^(1/2))/c^(7/2)/e^3/( b*(e*x+d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 22.67 (sec) , antiderivative size = 559, normalized size of antiderivative = 0.90 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (8 b^4 e^4-b^3 c e^3 (19 d+6 e x)+b^2 c^2 e^2 \left (6 d^2+14 d e x+5 e^2 x^2\right )+b c^3 e \left (-19 d^3+14 d^2 e x+205 d e^2 x^2+140 e^3 x^3\right )+c^4 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} \left (-8 \sqrt {\frac {b}{c}} \left (2 c^5 d^5-5 b c^4 d^4 e+2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5\right ) (b+c x) (d+e x)-8 i b e \left (2 c^5 d^5-5 b c^4 d^4 e+2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (8 c^5 d^5-21 b c^4 d^4 e+10 b^2 c^3 d^3 e^2+35 b^3 c^2 d^2 e^3-48 b^4 c d e^4+16 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{1155 b c^3 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \] Input:
Integrate[(d + e*x)^(3/2)*(b*x + c*x^2)^(3/2),x]
Output:
(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(8*b^4*e^4 - b^3*c*e^3*( 19*d + 6*e*x) + b^2*c^2*e^2*(6*d^2 + 14*d*e*x + 5*e^2*x^2) + b*c^3*e*(-19* d^3 + 14*d^2*e*x + 205*d*e^2*x^2 + 140*e^3*x^3) + c^4*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + Sqrt[b/c]*(-8*Sqrt[b/c]*( 2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4* c*d*e^4 + 2*b^5*e^5)*(b + c*x)*(d + e*x) - (8*I)*b*e*(2*c^5*d^5 - 5*b*c^4* d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5) *Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c] /Sqrt[x]], (c*d)/(b*e)] + I*b*e*(8*c^5*d^5 - 21*b*c^4*d^4*e + 10*b^2*c^3*d ^3*e^2 + 35*b^3*c^2*d^2*e^3 - 48*b^4*c*d*e^4 + 16*b^5*e^5)*Sqrt[1 + b/(c*x )]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d) /(b*e)])))/(1155*b*c^3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])
Time = 1.48 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.88, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {1166, 27, 1231, 27, 1231, 27, 1269, 1169, 122, 120, 127, 126}
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.
| \(\displaystyle \int \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {2 \int \frac {(d (11 c d-5 b e)+6 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}}{2 \sqrt {d+e x}}dx}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d (11 c d-5 b e)+6 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}}{\sqrt {d+e x}}dx}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {2 \int \frac {3 e \left (b d \left (c^2 d^2+13 b c e d-6 b^2 e^2\right )+(2 c d-b e) \left (c^2 d^2-b c e d+8 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 \sqrt {d+e x}}dx}{21 c e^2}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\int \frac {\left (b d \left (c^2 d^2+13 b c e d-6 b^2 e^2\right )+(2 c d-b e) \left (c^2 d^2-b c e d+8 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{\sqrt {d+e x}}dx}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {-\frac {2 \int -\frac {b d \left (8 c^4 d^4-19 b c^3 e d^3+6 b^2 c^2 e^2 d^2-19 b^3 c e^3 d+8 b^4 e^4\right )+8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\int \frac {b d \left (8 c^4 d^4-19 b c^3 e d^3+6 b^2 c^2 e^2 d^2-19 b^3 c e^3 d+8 b^4 e^4\right )+8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {8 (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {8 \sqrt {x} \sqrt {b+c x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {8 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
Input:
Int[(d + e*x)^(3/2)*(b*x + c*x^2)^(3/2),x]
Output:
(2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(5/2))/(11*c) + ((2*Sqrt[d + e*x]*(c^2*d^ 2 + 13*b*c*d*e - 6*b^2*e^2 + 14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/ (21*c*e) - ((-2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2* e^2 - 19*b^3*c*d*e^3 + 8*b^4*e^4 - 3*c*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(15*c*e^2) + ((16*Sqrt[-b]*(c*d - 2*b*e )*(2*c*d - b*e)*(c*d + b*e)*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e )/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d* (c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3*e + 3*b^2*c^2*d^2*e^2 + 13*b^3*c*d* e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Arc Sin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sq rt[b*x + c*x^2]))/(15*c*e^2))/(7*c*e))/(11*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 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[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e 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] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1358\) vs. \(2(557)=1114\).
Time = 1.62 (sec) , antiderivative size = 1359, normalized size of antiderivative = 2.19
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1359\) |
| elliptic | \(\text {Expression too large to display}\) | \(1623\) |
Input:
int((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/1155*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(105*c^6*e^7*x^7+16*((e*x+d)/d)^(1/ 2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2), (-d*c/(b*e-c*d))^(1/2))*b^6*d*e^6-16*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d ))^(1/2)*(-e*x/d)^(1/2)*EllipticE(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2) )*b^6*d*e^6-48*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2 )*EllipticF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b^5*c*d^2*e^5+35*((e *x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticF(((e*x+ d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b^4*c^2*d^3*e^4+10*((e*x+d)/d)^(1/2)*( e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2),(-d* c/(b*e-c*d))^(1/2))*b^3*c^3*d^4*e^3-21*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c *d))^(1/2)*(-e*x/d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/ 2))*b^2*c^4*d^5*e^2+8*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/ d)^(1/2)*EllipticF(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b*c^5*d^6*e+5 6*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticE(( (e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b^5*c*d^2*e^5-56*((e*x+d)/d)^(1/2 )*(e*(c*x+b)/(b*e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticE(((e*x+d)/d)^(1/2),( -d*c/(b*e-c*d))^(1/2))*b^4*c^2*d^3*e^4+56*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b* e-c*d))^(1/2)*(-e*x/d)^(1/2)*EllipticE(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^ (1/2))*b^2*c^4*d^5*e^2-56*((e*x+d)/d)^(1/2)*(e*(c*x+b)/(b*e-c*d))^(1/2)*(- e*x/d)^(1/2)*EllipticE(((e*x+d)/d)^(1/2),(-d*c/(b*e-c*d))^(1/2))*b*c^5*...
Time = 0.21 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.03 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 33 \, b^{2} c^{4} d^{4} e^{2} + 14 \, b^{3} c^{3} d^{3} e^{3} + 33 \, b^{4} c^{2} d^{2} e^{4} - 48 \, b^{5} c d e^{5} + 16 \, b^{6} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, b^{2} c^{4} d^{3} e^{3} + 2 \, b^{3} c^{3} d^{2} e^{4} - 5 \, b^{4} c^{2} d e^{5} + 2 \, b^{5} c e^{6}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (105 \, c^{6} e^{6} x^{4} + 8 \, c^{6} d^{4} e^{2} - 19 \, b c^{5} d^{3} e^{3} + 6 \, b^{2} c^{4} d^{2} e^{4} - 19 \, b^{3} c^{3} d e^{5} + 8 \, b^{4} c^{2} e^{6} + 140 \, {\left (c^{6} d e^{5} + b c^{5} e^{6}\right )} x^{3} + 5 \, {\left (c^{6} d^{2} e^{4} + 41 \, b c^{5} d e^{5} + b^{2} c^{4} e^{6}\right )} x^{2} - 2 \, {\left (3 \, c^{6} d^{3} e^{3} - 7 \, b c^{5} d^{2} e^{4} - 7 \, b^{2} c^{4} d e^{5} + 3 \, b^{3} c^{3} e^{6}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3465 \, c^{5} e^{5}} \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x, algorithm="fricas")
Output:
2/3465*((16*c^6*d^6 - 48*b*c^5*d^5*e + 33*b^2*c^4*d^4*e^2 + 14*b^3*c^3*d^3 *e^3 + 33*b^4*c^2*d^2*e^4 - 48*b^5*c*d*e^5 + 16*b^6*e^6)*sqrt(c*e)*weierst rassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c* d + b*e)/(c*e)) + 24*(2*c^6*d^5*e - 5*b*c^5*d^4*e^2 + 2*b^2*c^4*d^3*e^3 + 2*b^3*c^3*d^2*e^4 - 5*b^4*c^2*d*e^5 + 2*b^5*c*e^6)*sqrt(c*e)*weierstrassZe ta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2 *d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^ 2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3 *b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3 *(105*c^6*e^6*x^4 + 8*c^6*d^4*e^2 - 19*b*c^5*d^3*e^3 + 6*b^2*c^4*d^2*e^4 - 19*b^3*c^3*d*e^5 + 8*b^4*c^2*e^6 + 140*(c^6*d*e^5 + b*c^5*e^6)*x^3 + 5*(c ^6*d^2*e^4 + 41*b*c^5*d*e^5 + b^2*c^4*e^6)*x^2 - 2*(3*c^6*d^3*e^3 - 7*b*c^ 5*d^2*e^4 - 7*b^2*c^4*d*e^5 + 3*b^3*c^3*e^6)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^5*e^5)
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(c*x**2+b*x)**(3/2),x)
Output:
Integral((x*(b + c*x))**(3/2)*(d + e*x)**(3/2), x)
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x)^(3/2)*(e*x + d)^(3/2), x)
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x)^(3/2)*(e*x + d)^(3/2), x)
Timed out. \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \] Input:
int((b*x + c*x^2)^(3/2)*(d + e*x)^(3/2),x)
Output:
int((b*x + c*x^2)^(3/2)*(d + e*x)^(3/2), x)
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {18 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b^{3} d \,e^{2}-12 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b^{3} e^{3} x -60 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b^{2} c \,d^{2} e +28 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b^{2} c d \,e^{2} x +10 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b^{2} c \,e^{3} x^{2}+18 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b \,c^{2} d^{3}+28 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b \,c^{2} d^{2} e x +410 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b \,c^{2} d \,e^{2} x^{2}+280 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b \,c^{2} e^{3} x^{3}-12 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, c^{3} d^{3} x +10 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, c^{3} d^{2} e \,x^{2}+280 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, c^{3} d \,e^{2} x^{3}+210 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, c^{3} e^{3} x^{4}+24 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c e \,x^{2}+b e x +c d x +b d}d x \right ) b^{4} e^{4}-84 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c e \,x^{2}+b e x +c d x +b d}d x \right ) b^{3} c d \,e^{3}+108 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c e \,x^{2}+b e x +c d x +b d}d x \right ) b^{2} c^{2} d^{2} e^{2}-84 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c e \,x^{2}+b e x +c d x +b d}d x \right ) b \,c^{3} d^{3} e +24 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c e \,x^{2}+b e x +c d x +b d}d x \right ) c^{4} d^{4}-9 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}d x \right ) b^{4} d^{2} e^{2}+30 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}d x \right ) b^{3} c \,d^{3} e -9 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}d x \right ) b^{2} c^{2} d^{4}}{1155 c^{2} e^{2}} \] Input:
int((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x)
Output:
(18*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**3*d*e**2 - 12*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**3*e**3*x - 60*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b* *2*c*d**2*e + 28*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c*d*e**2*x + 10* sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c*e**3*x**2 + 18*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**2*d**3 + 28*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b *c**2*d**2*e*x + 410*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**2*d*e**2*x** 2 + 280*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**2*e**3*x**3 - 12*sqrt(x)* sqrt(d + e*x)*sqrt(b + c*x)*c**3*d**3*x + 10*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**3*d**2*e*x**2 + 280*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**3*d*e **2*x**3 + 210*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**3*e**3*x**4 + 24*int ((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b*d + b*e*x + c*d*x + c*e*x**2), x)*b**4*e**4 - 84*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b*d + b*e*x + c*d*x + c*e*x**2),x)*b**3*c*d*e**3 + 108*int((sqrt(x)*sqrt(d + e*x)*sqr t(b + c*x)*x)/(b*d + b*e*x + c*d*x + c*e*x**2),x)*b**2*c**2*d**2*e**2 - 84 *int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b*d + b*e*x + c*d*x + c*e*x* *2),x)*b*c**3*d**3*e + 24*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b*d + b*e*x + c*d*x + c*e*x**2),x)*c**4*d**4 - 9*int((sqrt(x)*sqrt(d + e*x)*s qrt(b + c*x))/(b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3),x)*b**4*d**2*e**2 + 30*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x))/(b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3),x)*b**3*c*d**3*e - 9*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c...