Integrand size = 22, antiderivative size = 227 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {a+b x+c x^2}}{4 e^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} e^4}-\frac {3 (2 c d-b e) \sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 e^4} \]
-(c*x^2+b*x+a)^(3/2)/e/(e*x+d)+3/8*(8*c^2*d^2+b^2*e^2-4*c*e*(-a*e+2*b*d))* arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^4/c^(1/2)-3/2*(-b*e+2 *c*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^4-3/4*(-2*c*e*x-3*b*e+4*c*d) *(c*x^2+b*x+a)^(1/2)/e^3
Time = 10.30 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {6 e (-4 c d+3 b e+2 c e x) \sqrt {a+x (b+c x)}-\frac {8 e^3 (a+x (b+c x))^{3/2}}{d+e x}+\frac {3 \left (8 c^2 d^2+b^2 e^2+4 c e (-2 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+12 (2 c d-b e) \sqrt {c d^2+e (-b d+a e)} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 e^4} \]
(6*e*(-4*c*d + 3*b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)] - (8*e^3*(a + x*(b + c*x))^(3/2))/(d + e*x) + (3*(8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))* ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 12*(2*c* d - b*e)*Sqrt[c*d^2 + e*(-(b*d) + 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)])])/(8*e^4 )
Time = 0.54 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1161, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {3 \int \frac {(b+2 c x) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {c \left (3 d e b^2-4 \left (c d^2+a e^2\right ) b+4 a c d e-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right )}{(d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {3 d e b^2-4 \left (c d^2+a e^2\right ) b+4 a c d e-\left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{4 e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{4 e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{4 e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}}{4 e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3 \left (-\frac {-\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}}{4 e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {4 (2 c d-b e) \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}}{4 e^2}-\frac {\sqrt {a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2}\right )}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)}\) |
-((a + b*x + c*x^2)^(3/2)/(e*(d + e*x))) + (3*(-1/2*((4*c*d - 3*b*e - 2*c* e*x)*Sqrt[a + b*x + c*x^2])/e^2 - (-(((8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e )) + (4*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*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]*Sqrt[a + b*x + c*x^2])])/e) /(4*e^2)))/(2*e)
3.24.48.3.1 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)*(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. \(621\) vs. \(2(201)=402\).
Time = 0.40 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.74
| method | result | size |
| risch | \(\frac {\left (2 c e x +5 b e -8 c d \right ) \sqrt {c \,x^{2}+b x +a}}{4 e^{3}}+\frac {\frac {3 \left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {\left (16 a b \,e^{3}-32 a c d \,e^{2}-16 b^{2} d \,e^{2}+48 b c \,d^{2} e -32 c^{2} d^{3}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (8 a^{2} e^{4}-16 a b d \,e^{3}+16 a c \,d^{2} e^{2}+8 b^{2} d^{2} e^{2}-16 b c \,d^{3} e +8 c^{2} d^{4}\right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}}{8 e^{3}}\) | \(622\) |
| default | \(\text {Expression too large to display}\) | \(1096\) |
1/4*(2*c*e*x+5*b*e-8*c*d)*(c*x^2+b*x+a)^(1/2)/e^3+1/8/e^3*(3*(4*a*c*e^2+b^ 2*e^2-8*b*c*d*e+8*c^2*d^2)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c ^(1/2)-(16*a*b*e^3-32*a*c*d*e^2-16*b^2*d*e^2+48*b*c*d^2*e-32*c^2*d^3)/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))+1/e^3*(8*a^2*e^4-16*a*b*d*e^ 3+16*a*c*d^2*e^2+8*b^2*d^2*e^2-16*b*c*d^3*e+8*c^2*d^4)*(-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))))
Time = 9.26 (sec) , antiderivative size = 1583, normalized size of antiderivative = 6.97 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Too large to display} \]
[1/16*(3*(8*c^2*d^3 - 8*b*c*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*e - 8 *b*c*d*e^2 + (b^2 + 4*a*c)*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) - 12*(2*c^2*d^2 - b *c*d*e + (2*c^2*d*e - b*c*e^2)*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*(2*c^2*e^3*x^2 - 12*c^2*d^2*e + 9*b*c*d*e ^2 - 4*a*c*e^3 - (6*c^2*d*e^2 - 5*b*c*e^3)*x)*sqrt(c*x^2 + b*x + a))/(c*e^ 5*x + c*d*e^4), -1/16*(24*(2*c^2*d^2 - b*c*d*e + (2*c^2*d*e - b*c*e^2)*x)* sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*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)/(a*c*d^2 - a*b*d*e + a^2 *e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x )) - 3*(8*c^2*d^3 - 8*b*c*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*e - 8*b *c*d*e^2 + (b^2 + 4*a*c)*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) - 4*(2*c^2*e^3*x^2 - 12*c^2*d^2*e + 9*b*c*d*e^2 - 4*a*c*e^3 - (6*c^2*d*e^2 - 5*b*c*e^3)*x)*sqrt (c*x^2 + b*x + a))/(c*e^5*x + c*d*e^4), -1/8*(3*(8*c^2*d^3 - 8*b*c*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*e^3)*x)* sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2...
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \]