Integrand size = 22, antiderivative size = 331 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c} e^6}+\frac {5 \sqrt {c d^2-b d e+a e^2} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^6} \]
5/12*(2*c*e*x-3*b*e+8*c*d)*(c*x^2+b*x+a)^(3/2)/e^3/(e*x+d)-1/2*(c*x^2+b*x+ a)^(5/2)/e/(e*x+d)^2-5/16*(-b*e+2*c*d)*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4 *b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^6/c^(1/2)+5/8* (16*c^2*d^2+3*b^2*e^2-4*c*e*(-a*e+4*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))*(a*e^2-b*d*e+c*d^2)^ (1/2)/e^6+5/8*(16*c^2*d^2+5*b^2*e^2-4*c*e*(-a*e+5*b*d)-4*c*e*(-b*e+2*c*d)* x)*(c*x^2+b*x+a)^(1/2)/e^5
Time = 11.11 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {-\frac {2 (a+x (b+c x))^{5/2}}{(d+e x)^2}+\frac {5 (2 c d-b e) (a+x (b+c x))^{5/2}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {5 \left (-\frac {(a+x (b+c x))^{3/2} \left (3 b^2 e^2+2 c^2 d (4 d-3 e x)+c e (-11 b d+2 a e+3 b e x)\right )}{3 e^2}+\frac {-2 c^2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (5 b^2 e^2+8 c^2 d (2 d-e x)+4 c e (-5 b d+a e+b e x)\right )+c^{3/2} (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 c^2 \left (16 c^2 d^2+3 b^2 e^2+4 c e (-4 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 c^2 e^5}\right )}{-c d^2+e (b d-a e)}}{4 e} \]
((-2*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 + (5*(2*c*d - b*e)*(a + x*(b + c *x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (5*(-1/3*((a + x*(b + c*x))^(3/2)*(3*b^2*e^2 + 2*c^2*d*(4*d - 3*e*x) + c*e*(-11*b*d + 2*a*e + 3 *b*e*x)))/e^2 + (-2*c^2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)] *(5*b^2*e^2 + 8*c^2*d*(2*d - e*x) + 4*c*e*(-5*b*d + a*e + b*e*x)) + c^(3/2 )*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(16*c^2*d^2 + b^2*e^2 + 4*c*e*( -4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*c^2*(16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*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)])])/(4*c^2*e^5)))/(-(c*d^2) + e*(b*d - a*e)))/(4*e)
Time = 0.73 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1161, 1230, 1231, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^2}dx}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {\int \frac {\left (-3 e b^2+8 c d b-4 a c e+8 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\int \frac {2 c \left (e (b d-2 a e) \left (-3 e b^2+8 c d b-4 a c e\right )-2 d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{(d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\int \frac {e (b d-2 a e) \left (-3 e b^2+8 c d b-4 a c e\right )-2 d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {-\frac {4 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \sqrt {a e^2-b d e+c d^2} \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\) |
-1/2*(a + b*x + c*x^2)^(5/2)/(e*(d + e*x)^2) + (5*(((8*c*d - 3*b*e + 2*c*e *x)*(a + b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) - (-(((16*c^2*d^2 + 5*b^2*e ^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/e ^2) - (-(((2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*Arc Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e)) + (2*Sqr t[c*d^2 - b*d*e + a*e^2]*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2 + 4*a*c*e^2) *ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sq rt[a + b*x + c*x^2])])/e)/(2*e^2))/(2*e^2)))/(4*e)
3.24.61.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[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[(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. \(1434\) vs. \(2(299)=598\).
Time = 0.46 (sec) , antiderivative size = 1435, normalized size of antiderivative = 4.34
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1435\) |
| default | \(\text {Expression too large to display}\) | \(2822\) |
1/24*(8*c^2*e^2*x^2+26*b*c*e^2*x-36*c^2*d*e*x+56*a*c*e^2+33*b^2*e^2-162*b* c*d*e+144*c^2*d^2)*(c*x^2+b*x+a)^(1/2)/e^5+1/16/e^5*(5*(12*a*b*c*e^3-24*a* c^2*d*e^2+b^3*e^3-18*b^2*c*d*e^2+48*b*c^2*d^2*e-32*c^3*d^3)/e*ln((1/2*b+c* x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-(48*a^2*c*e^4+48*a*b^2*e^4-288*a*b *c*d*e^3+288*a*c^2*d^2*e^2-48*b^3*d*e^3+288*b^2*c*d^2*e^2-480*b*c^2*d^3*e+ 240*c^3*d^4)/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))+(48*a^2*b*e ^5-96*a^2*c*d*e^4-96*a*b^2*d*e^4+288*a*b*c*d^2*e^3-192*a*c^2*d^3*e^2+48*b^ 3*d^2*e^3-192*b^2*c*d^3*e^2+240*b*c^2*d^4*e-96*c^3*d^5)/e^3*(-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)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)))+1/e^4*(16*a^3*e^6-48*a^2*b*d*e^5+48*a^2*c*d^2*e^4 +48*a*b^2*d^2*e^4-96*a*b*c*d^3*e^3+48*a*c^2*d^4*e^2-16*b^3*d^3*e^3+48*b^2* c*d^4*e^2-48*b*c^2*d^5*e+16*c^3*d^6)*(-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)^(1/2)-3/4*( 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)^(1/2)+1/2*(b*e-2...
Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (299) = 598\).
Time = 73.71 (sec) , antiderivative size = 3019, normalized size of antiderivative = 9.12 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
[-1/96*(15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 6*(3*b^2*c + 4*a*c^2)*d^3*e^2 - (b^3 + 12*a*b*c)*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5)*x^2 + 2*(32*c^3*d^4*e - 48*b*c^2 *d^3*e^2 + 6*(3*b^2*c + 4*a*c^2)*d^2*e^3 - (b^3 + 12*a*b*c)*d*e^4)*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)*s qrt(c) - 4*a*c) - 30*(16*c^3*d^4 - 16*b*c^2*d^3*e + (3*b^2*c + 4*a*c^2)*d^ 2*e^2 + (16*c^3*d^2*e^2 - 16*b*c^2*d*e^3 + (3*b^2*c + 4*a*c^2)*e^4)*x^2 + 2*(16*c^3*d^3*e - 16*b*c^2*d^2*e^2 + (3*b^2*c + 4*a*c^2)*d*e^3)*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*(8*c^3*e ^5*x^4 + 240*c^3*d^4*e - 300*b*c^2*d^3*e^2 - 30*a*b*c*d*e^4 - 12*a^2*c*e^5 + 5*(15*b^2*c + 28*a*c^2)*d^2*e^3 - 2*(10*c^3*d*e^4 - 13*b*c^2*e^5)*x^3 + (80*c^3*d^2*e^3 - 110*b*c^2*d*e^4 + (33*b^2*c + 56*a*c^2)*e^5)*x^2 + 2*(1 80*c^3*d^3*e^2 - 230*b*c^2*d^2*e^3 - 27*a*b*c*e^5 + 10*(6*b^2*c + 11*a*c^2 )*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c*e^8*x^2 + 2*c*d*e^7*x + c*d^2*e^6), 1/48*(15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 6*(3*b^2*c + 4*a*c^2)*d^3*e^2 - (b ^3 + 12*a*b*c)*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5)*x^2 + 2*(32*c^3*d^4*e - 48*b*c^...
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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. 1500 vs. \(2 (299) = 598\).
Time = 0.51 (sec) , antiderivative size = 1500, normalized size of antiderivative = 4.53 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
1/24*sqrt(c*x^2 + b*x + a)*(2*x*(4*c^2*x/e^3 - (18*c^4*d*e^14 - 13*b*c^3*e ^15)/(c^2*e^18)) + (144*c^4*d^2*e^13 - 162*b*c^3*d*e^14 + 33*b^2*c^2*e^15 + 56*a*c^3*e^15)/(c^2*e^18)) + 5/4*(16*c^3*d^4 - 32*b*c^2*d^3*e + 19*b^2*c *d^2*e^2 + 20*a*c^2*d^2*e^2 - 3*b^3*d*e^3 - 20*a*b*c*d*e^3 + 3*a*b^2*e^4 + 4*a^2*c*e^4)*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))/(sqrt(-c*d^2 + b*d*e - a*e^2)*e^6) + 5/16*(3 2*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 24*a*c^2*d*e^2 - b^3*e^3 - 1 2*a*b*c*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/( sqrt(c)*e^6) + 1/4*(40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^3*d^4*e - 8 0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^2*d^3*e^2 + 49*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))^3*b^2*c*d^2*e^3 + 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^2*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*d*e^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c*d*e^4 + 9*(sqrt(c)*x - sqrt (c*x^2 + b*x + a))^3*a*b^2*e^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a ^2*c*e^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(7/2)*d^5 - 120*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(5/2)*d^4*e + 51*(sqrt(c)*x - sqrt(c *x^2 + b*x + a))^2*b^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(5/2)*d^3*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*sq rt(c)*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(3/2)*d^2*e ^3 - 21*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*sqrt(c)*d*e^4 - 36*...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]