Integrand size = 22, antiderivative size = 307 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^4 \left (c d^2-b d e+a e^2\right )^{3/2}} \]
-1/3*(c*x^2+b*x+a)^(3/2)/e/(e*x+d)^3+c^(3/2)*arctanh(1/2*(2*c*x+b)/c^(1/2) /(c*x^2+b*x+a)^(1/2))/e^4-1/16*(-b*e+2*c*d)*(8*c^2*d^2-b^2*e^2-4*c*e*(-3*a *e+2*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2 )/(c*x^2+b*x+a)^(1/2))/e^4/(a*e^2-b*d*e+c*d^2)^(3/2)-1/8*(8*c^2*d^3-b*e^2* (-2*a*e+b*d)-2*c*d*e*(-2*a*e+3*b*d)+e*(12*c^2*d^2+b^2*e^2-4*c*e*(-2*a*e+3* b*d))*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2
Time = 11.03 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {-\frac {2 (a+x (b+c x))^{3/2}}{(d+e x)^3}+\frac {3 (2 c d-b e) (a+x (b+c x))^{3/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {3 \left (\frac {2 \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) (a+x (b+c x))^{3/2}}{d+e x}+\frac {2 \sqrt {a+x (b+c x)} \left (-b^3 e^3+4 c^3 d^2 (-2 d+e x)-b c e^2 (5 b d-10 a e+b e x)+2 c^2 e (b d (7 d-2 e x)+2 a e (-3 d+2 e x))\right )}{e^2}+\frac {16 c^{3/2} \left (c d^2+e (-b d+a e)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+(2 c d-b e) \sqrt {c d^2+e (-b d+a e)} \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{e^3}\right )}{8 \left (c d^2+e (-b d+a e)\right )^2}}{6 e} \]
((-2*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + (3*(2*c*d - b*e)*(a + x*(b + c *x))^(3/2))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (3*((2*(-4*c^2*d^ 2 + b^2*e^2 + 4*c*e*(b*d - 2*a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x) + (2 *Sqrt[a + x*(b + c*x)]*(-(b^3*e^3) + 4*c^3*d^2*(-2*d + e*x) - b*c*e^2*(5*b *d - 10*a*e + b*e*x) + 2*c^2*e*(b*d*(7*d - 2*e*x) + 2*a*e*(-3*d + 2*e*x))) )/e^2 + (16*c^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTanh[(b + 2*c*x)/(2*Sq rt[c]*Sqrt[a + x*(b + c*x)])] + (2*c*d - b*e)*Sqrt[c*d^2 + e*(-(b*d) + a*e )]*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)] )])/e^3))/(8*(c*d^2 + e*(-(b*d) + a*e))^2))/(6*e)
Time = 0.64 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1161, 1229, 27, 1269, 1092, 219, 1154, 219}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {\int \frac {(b+2 c x) \sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {-\frac {\int \frac {e^2 b^3+6 c d e b^2-4 c \left (2 c d^2+3 a e^2\right ) b+8 a c^2 d e-16 c^2 \left (c d^2-b e d+a e^2\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {e^2 b^3+6 c d e b^2-4 c \left (2 c d^2+3 a e^2\right ) b+8 a c^2 d e-16 c^2 \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-\frac {\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {16 c^2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {-\frac {\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {32 c^2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {16 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {-\frac {2 (2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {16 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}-\frac {16 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\) |
-1/3*(a + b*x + c*x^2)^(3/2)/(e*(d + e*x)^3) + (-1/4*((8*c^2*d^3 - b*e^2*( b*d - 2*a*e) - 2*c*d*e*(3*b*d - 2*a*e) + e*(12*c^2*d^2 + b^2*e^2 - 4*c*e*( 3*b*d - 2*a*e))*x)*Sqrt[a + b*x + c*x^2])/(e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - ((-16*c^(3/2)*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(b + 2*c*x)/(2*S qrt[c]*Sqrt[a + b*x + c*x^2])])/e + ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d ^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e*Sqrt[c*d^2 - b*d*e + a*e^2 ]))/(8*e^2*(c*d^2 - b*d*e + a*e^2)))/(2*e)
3.24.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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. \(3084\) vs. \(2(281)=562\).
Time = 0.45 (sec) , antiderivative size = 3085, normalized size of antiderivative = 10.05
1/e^4*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*((x+d/e)^2*c+(b*e-2*c*d)/e*( x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)-1/6*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2 )*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/ e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+1/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(- 1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^ 2-b*d*e+c*d^2)/e^2)^(5/2)+3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/3*((x+d /e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c* d)/e*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e) +(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c* d)^2/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b *e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e ^2*(((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2* (b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2 *c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2 )/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2 *c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d) /e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))+4*c/(a*e^2-b*d*e+c*d ^2)*e^2*(1/8*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d /e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e- 2*c*d)^2/e^2)/c*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*...
Leaf count of result is larger than twice the leaf count of optimal. 1206 vs. \(2 (281) = 562\).
Time = 186.22 (sec) , antiderivative size = 4911, normalized size of antiderivative = 16.00 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
[1/96*(48*(c^3*d^7 - 2*b*c^2*d^6*e - 2*a*b*c*d^4*e^3 + a^2*c*d^3*e^4 + (b^ 2*c + 2*a*c^2)*d^5*e^2 + (c^3*d^4*e^3 - 2*b*c^2*d^3*e^4 - 2*a*b*c*d*e^6 + a^2*c*e^7 + (b^2*c + 2*a*c^2)*d^2*e^5)*x^3 + 3*(c^3*d^5*e^2 - 2*b*c^2*d^4* e^3 - 2*a*b*c*d^2*e^5 + a^2*c*d*e^6 + (b^2*c + 2*a*c^2)*d^3*e^4)*x^2 + 3*( c^3*d^6*e - 2*b*c^2*d^5*e^2 - 2*a*b*c*d^3*e^4 + a^2*c*d^2*e^5 + (b^2*c + 2 *a*c^2)*d^4*e^3)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(b^2*c + 4*a*c^2)*d^4*e^2 + (b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(b^2*c + 4*a*c^2)*d*e^5 + (b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(b^2*c + 4*a*c^2)*d^2*e^4 + (b ^3 - 12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(b^2*c + 4*a*c^2)*d^3*e^3 + (b^3 - 12*a*b*c)*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e ^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d* e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b* x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(24*c^3*d^6*e - 42*b*c^2* d^5*e^2 - 10*a^2*b*d*e^6 + 8*a^3*e^7 + (15*b^2*c + 44*a*c^2)*d^4*e^3 + (3* b^3 - 40*a*b*c)*d^3*e^4 - (a*b^2 - 28*a^2*c)*d^2*e^5 + (44*c^3*d^4*e^3 - 8 8*b*c^2*d^3*e^4 + (47*b^2*c + 76*a*c^2)*d^2*e^5 - (3*b^3 + 76*a*b*c)*d*e^6 + (3*a*b^2 + 32*a^2*c)*e^7)*x^2 + 2*(30*c^3*d^5*e^2 - 53*b*c^2*d^4*e^3...
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 2059 vs. \(2 (281) = 562\).
Time = 4.24 (sec) , antiderivative size = 2059, normalized size of antiderivative = 6.71 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
-1/8*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + 24*a*c^2*d*e^2 + b^3*e ^3 - 12*a*b*c*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c )*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c*d^2*e^4 - b*d*e^5 + a*e^6)*sqrt(-c* d^2 + b*d*e - a*e^2)) - c^(3/2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/e^4 - 1/24*(144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c ^3*d^3*e^2 - 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d^2*e^3 + 78* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d*e^4 + 120*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))^5*a*c^2*d*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5* b^3*e^5 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^5 + 432*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^4*e - 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^3*e^2 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 4*b^2*c^(3/2)*d^2*e^3 + 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2 )*d^2*e^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*d*e^4 + 8 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(3/2)*d*e^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*sqrt(c)*e^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)*e^5 + 352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3* c^4*d^5 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e - 420*(sqrt (c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d^3*e^2 - 272*(sqrt(c)*x - sqrt(c *x^2 + b*x + a))^3*a*c^3*d^3*e^2 + 106*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^3*b^3*c*d^2*e^3 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d^...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \]