Integrand size = 22, antiderivative size = 296 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{7/2}} \]
1/16*(-b*e+2*c*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b* d*e+c*d^2)^2/(e*x+d)^4-1/5*e*(c*x^2+b*x+a)^(5/2)/(a*e^2-b*d*e+c*d^2)/(e*x+ d)^5+3/256*(-4*a*c+b^2)^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)^(7/ 2)-3/128*(-4*a*c+b^2)*(-b*e+2*c*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a )^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2
Time = 10.94 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {e (a+x (b+c x))^{5/2}}{5 \left (c d^2+e (-b d+a e)\right ) (d+e x)^5}-\frac {(2 c d-b e) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{32 \left (c d^2+e (-b d+a e)\right )^2} \]
-1/5*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) - ((2*c*d - b*e)*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^ (3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c *d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*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*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))) /(32*(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 0.45 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1157, 1152, 1152, 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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1157 |
\(\displaystyle \frac {(2 c d-b e) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(d+e x)^5}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {(2 c d-b e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {(2 c d-b e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {(2 c d-b e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(2 c d-b e) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right )}{16 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\) |
-1/5*(e*(a + b*x + c*x^2)^(5/2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) + ( (2*c*d - b*e)*(((b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/( 8*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (3*(b^2 - 4*a*c)*(((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e *x)^2) - ((b^2 - 4*a*c)*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])])/(8*(c*d^2 - b*d*e + a*e^2)^( 3/2))))/(16*(c*d^2 - b*d*e + a*e^2))))/(2*(c*d^2 - b*d*e + a*e^2))
3.24.52.3.1 Defintions of rubi rules used
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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
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[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[m + 2*p + 3, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(5188\) vs. \(2(274)=548\).
Time = 0.70 (sec) , antiderivative size = 5189, normalized size of antiderivative = 17.53
Leaf count of result is larger than twice the leaf count of optimal. 2394 vs. \(2 (274) = 548\).
Time = 44.81 (sec) , antiderivative size = 4830, normalized size of antiderivative = 16.32 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
[-1/2560*(15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2 *c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^ 2*e^4)*x^3 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 - 8*a *b^3*c + 16*a^2*b*c^2)*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c ^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*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*(464*a^4*b*d*e^6 - 128*a^5*e^7 - 10*(3*b^3*c^2 - 20*a*b*c^3)*d^7 + (45*b^4*c - 320*a*b^2*c^2 - 528*a^2*c^3)*d^6*e - (15*b^5 - 100*a*b^3*c - 1424*a^2*b*c^2)*d^5*e^2 + (5*a*b^4 - 1064*a^2*b^2*c - 944*a^3*c^2)*d^4*e^3 + 2*(129*a^2*b^3 + 724*a^ 3*b*c)*d^3*e^4 - 8*(73*a^3*b^2 + 68*a^4*c)*d^2*e^5 + (32*c^5*d^6*e - 96*b* c^4*d^5*e^2 + 4*(19*b^2*c^3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3) *d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16*a^2*c^3)*d^2*e^5 + (15*b^5 - 80* a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e ^7)*x^4 + 2*(80*c^5*d^7 - 248*b*c^4*d^6*e + 2*(107*b^2*c^3 + 220*a*c^4)...
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
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. 8409 vs. \(2 (274) = 548\).
Time = 0.98 (sec) , antiderivative size = 8409, normalized size of antiderivative = 28.41 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
3/128*(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 1 6*a^2*b*c^2*e)*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^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a ^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/ 640*(30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c*d*e^8 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^2*d*e^8 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*c^3*d*e^8 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5 *e^9 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*e^9 - 240*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*e^9 - 1280*(sqrt(c)*x - sqrt(c*x^ 2 + b*x + a))^8*c^(11/2)*d^6*e^3 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) )^8*b*c^(9/2)*d^5*e^4 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^( 7/2)*d^4*e^5 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^4*e^ 5 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^3*c^(5/2)*d^3*e^6 + 7680* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(7/2)*d^3*e^6 + 270*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(3/2)*d^2*e^7 - 6000*(sqrt(c)*x - sqrt(c *x^2 + b*x + a))^8*a*b^2*c^(5/2)*d^2*e^7 + 480*(sqrt(c)*x - sqrt(c*x^2 + b *x + a))^8*a^2*c^(7/2)*d^2*e^7 - 135*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8 *b^5*sqrt(c)*d*e^8 + 1080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^3*c^(3 /2)*d*e^8 + 1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b*c^(5/2)*d*...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \]