Integrand size = 25, antiderivative size = 628 \[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e+c \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) x\right )}{3 a^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 b^7 d e^2+b^5 c d \left (6 c d^2-37 a e^2\right )-a b^3 c^2 d \left (43 c d^2-26 a e^2\right )+24 a^3 c^3 e \left (c d^2+2 a e^2\right )+4 a^2 b c^3 d \left (17 c d^2+20 a e^2\right )+a b^4 c e \left (89 c d^2+70 a e^2\right )-2 a^2 b^2 c^2 e \left (79 c d^2+72 a e^2\right )-3 b^6 \left (4 c d^2 e+3 a e^3\right )+c \left (6 b^6 d e^2+2 b^4 c d \left (3 c d^2-16 a e^2\right )-2 a b^2 c^2 d \left (19 c d^2-2 a e^2\right )+8 a^2 c^3 d \left (5 c d^2+8 a e^2\right )-4 a^2 b c^2 e \left (25 c d^2+24 a e^2\right )+a b^3 c e \left (79 c d^2+62 a e^2\right )-3 b^5 \left (4 c d^2 e+3 a e^3\right )\right ) x\right )}{3 a^3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2}}{a^3 d x}+\frac {(5 b d+2 a e) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{7/2} d^2}+\frac {e^6 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:
1/3*(-2*b^3*c*d+6*a*b*c^2*d+2*b^4*e-8*a*b^2*c*e+4*a^2*c^2*e-2*c*(3*a*b*c*e -2*a*c^2*d-b^3*e+b^2*c*d)*x)/a^2/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b *x+a)^(3/2)-2/3*(6*b^7*d*e^2+b^5*c*d*(-37*a*e^2+6*c*d^2)-a*b^3*c^2*d*(-26* a*e^2+43*c*d^2)+24*a^3*c^3*e*(2*a*e^2+c*d^2)+4*a^2*b*c^3*d*(20*a*e^2+17*c* d^2)+a*b^4*c*e*(70*a*e^2+89*c*d^2)-2*a^2*b^2*c^2*e*(72*a*e^2+79*c*d^2)-3*b ^6*(3*a*e^3+4*c*d^2*e)+c*(6*b^6*d*e^2+2*b^4*c*d*(-16*a*e^2+3*c*d^2)-2*a*b^ 2*c^2*d*(-2*a*e^2+19*c*d^2)+8*a^2*c^3*d*(8*a*e^2+5*c*d^2)-4*a^2*b*c^2*e*(2 4*a*e^2+25*c*d^2)+a*b^3*c*e*(62*a*e^2+79*c*d^2)-3*b^5*(3*a*e^3+4*c*d^2*e)) *x)/a^3/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(1/2)-(c*x^2+b* x+a)^(1/2)/a^3/d/x+1/2*(2*a*e+5*b*d)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+ b*x+a)^(1/2))/a^(7/2)/d^2+e^6*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))/d^2/(a*e^2-b*d*e+c*d^2)^(5/2)
Time = 12.69 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-\frac {4 e \left (b^2-2 a c+b c x\right )}{a (a+x (b+c x))^{3/2}}+\frac {4 d \left (b^2-2 a c+b c x\right )}{a x (a+x (b+c x))^{3/2}}-\frac {4 e \left (\left (b^2-6 a c\right ) \left (3 b^2-4 a c\right )+b c \left (3 b^2-20 a c\right ) x\right )}{a^2 \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {4 e^2 \left (b^2 e-2 c (a e+c d x)+b c (-d+e x)\right )}{\left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{3/2}}+\frac {4 e^2 \left (3 b^4 e^3+b^3 c e^2 (d+3 e x)+8 c^2 \left (3 a^2 e^3+2 c^2 d^3 x+5 a c d e^2 x\right )+4 b c^2 \left (2 c d^2 (d-3 e x)+5 a e^2 (d-e x)\right )+2 b^2 c e \left (-11 a e^2+c d (-6 d+e x)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}}+\frac {6 \left (b^2-4 a c\right ) e \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{a^{5/2}}+\frac {d \left (\frac {2 \sqrt {a} \left (64 a^3 c^2+15 b^4 x (b+c x)+5 a b^2 \left (b^2-22 b c x-20 c^2 x^2\right )+4 a^2 c \left (-9 b^2+46 b c x+32 c^2 x^2\right )\right )}{x \sqrt {a+x (b+c x)}}-15 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )}{a^{7/2} \left (-b^2+4 a c\right )}-\frac {6 \left (b^2-4 a c\right ) e^6 \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}}}{6 \left (b^2-4 a c\right ) d^2} \] Input:
Integrate[1/(x^2*(d + e*x)*(a + b*x + c*x^2)^(5/2)),x]
Output:
((-4*e*(b^2 - 2*a*c + b*c*x))/(a*(a + x*(b + c*x))^(3/2)) + (4*d*(b^2 - 2* a*c + b*c*x))/(a*x*(a + x*(b + c*x))^(3/2)) - (4*e*((b^2 - 6*a*c)*(3*b^2 - 4*a*c) + b*c*(3*b^2 - 20*a*c)*x))/(a^2*(b^2 - 4*a*c)*Sqrt[a + x*(b + c*x) ]) + (4*e^2*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/((c*d^2 + e*(-(b *d) + a*e))*(a + x*(b + c*x))^(3/2)) + (4*e^2*(3*b^4*e^3 + b^3*c*e^2*(d + 3*e*x) + 8*c^2*(3*a^2*e^3 + 2*c^2*d^3*x + 5*a*c*d*e^2*x) + 4*b*c^2*(2*c*d^ 2*(d - 3*e*x) + 5*a*e^2*(d - e*x)) + 2*b^2*c*e*(-11*a*e^2 + c*d*(-6*d + e* x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) + (6*(b^2 - 4*a*c)*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] )/a^(5/2) + (d*((2*Sqrt[a]*(64*a^3*c^2 + 15*b^4*x*(b + c*x) + 5*a*b^2*(b^2 - 22*b*c*x - 20*c^2*x^2) + 4*a^2*c*(-9*b^2 + 46*b*c*x + 32*c^2*x^2)))/(x* Sqrt[a + x*(b + c*x)]) - 15*b*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x)/(2*Sqrt[ a]*Sqrt[a + x*(b + c*x)])]))/(a^(7/2)*(-b^2 + 4*a*c)) - (6*(b^2 - 4*a*c)*e ^6*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)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2))/(6*(b^2 - 4*a*c)*d^2)
Time = 1.10 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1289, 2009}
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.
| \(\displaystyle \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (\frac {e^2}{d^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}}-\frac {e}{d^2 x \left (a+b x+c x^2\right )^{5/2}}+\frac {1}{d x^2 \left (a+b x+c x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{5/2} d^2}+\frac {5 b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{7/2} d}-\frac {2 e \left (b c x \left (3 b^2-20 a c\right )+\left (b^2-6 a c\right ) \left (3 b^2-4 a c\right )\right )}{3 a^2 d^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (32 a^2 c^2+b c x \left (5 b^2-28 a c\right )-32 a b^2 c+5 b^4\right )}{3 a^2 d x \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a+b x+c x^2}}{3 a^3 d x \left (b^2-4 a c\right )^2}+\frac {e^6 \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{d^2 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac {2 e^2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-4 c e (b d-3 a e)-3 b^2 e^2+8 c^2 d^2\right )+4 a c e (2 c d-b e)^2\right )}{3 d^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 e^2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 d^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 e \left (-2 a c+b^2+b c x\right )}{3 a d^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (-2 a c+b^2+b c x\right )}{3 a d x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[1/(x^2*(d + e*x)*(a + b*x + c*x^2)^(5/2)),x]
Output:
(-2*e*(b^2 - 2*a*c + b*c*x))/(3*a*(b^2 - 4*a*c)*d^2*(a + b*x + c*x^2)^(3/2 )) + (2*(b^2 - 2*a*c + b*c*x))/(3*a*(b^2 - 4*a*c)*d*x*(a + b*x + c*x^2)^(3 /2)) - (2*e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a *c)*d^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) + (2*(5*b^4 - 32* a*b^2*c + 32*a^2*c^2 + b*c*(5*b^2 - 28*a*c)*x))/(3*a^2*(b^2 - 4*a*c)^2*d*x *Sqrt[a + b*x + c*x^2]) - (2*e*((b^2 - 6*a*c)*(3*b^2 - 4*a*c) + b*c*(3*b^2 - 20*a*c)*x))/(3*a^2*(b^2 - 4*a*c)^2*d^2*Sqrt[a + b*x + c*x^2]) - (2*e^2* (4*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 3*b^2*e^ 2 - 4*c*e*(b*d - 3*a*e)) - c*(2*c*d - b*e)*(8*c^2*d^2 - 3*b^2*e^2 - 4*c*e* (2*b*d - 5*a*e))*x))/(3*(b^2 - 4*a*c)^2*d^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt [a + b*x + c*x^2]) - ((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*Sqrt[a + b*x + c*x^2])/(3*a^3*(b^2 - 4*a*c)^2*d*x) + (5*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]* Sqrt[a + b*x + c*x^2])])/(2*a^(7/2)*d) + (e*ArcTanh[(2*a + b*x)/(2*Sqrt[a] *Sqrt[a + b*x + c*x^2])])/(a^(5/2)*d^2) + (e^6*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])])/(d^2* (c*d^2 - b*d*e + a*e^2)^(5/2))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(1238\) vs. \(2(602)=1204\).
Time = 1.68 (sec) , antiderivative size = 1239, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1239\) |
| risch | \(\text {Expression too large to display}\) | \(12685\) |
Input:
int(1/x^2/(e*x+d)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a/x/(c*x^2+b*x+a)^(3/2)-5/2*b/a*(1/3/a/(c*x^2+b*x+a)^(3/2)-1/2*b/a *(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c* x+b)/(c*x^2+b*x+a)^(1/2))+1/a*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+b)/(4*a* c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^( 1/2))/x)))-4*c/a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4* a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+e/d^2*(1/3/(a*e^2-b*d*e+c*d^2)* e^2/(c*(x+d/e)^2+(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/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a *e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e )+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e- 2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+ d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b *d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2 )^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c *(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+ d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(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)*(c*(x+d/e)^2+(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))))-e/d^2*(1/3/a/(c*x^2+b*x+a)^(3/2)-1/2*b /a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*...
Leaf count of result is larger than twice the leaf count of optimal. 5177 vs. \(2 (600) = 1200\).
Time = 38.37 (sec) , antiderivative size = 20807, normalized size of antiderivative = 33.13 \[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/x**2/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)
Output:
Integral(1/(x**2*(d + e*x)*(a + b*x + c*x**2)**(5/2)), x)
\[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 10475 vs. \(2 (600) = 1200\).
Time = 0.58 (sec) , antiderivative size = 10475, normalized size of antiderivative = 16.68 \[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
2*e^6*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^2*d^6 - 2*b*c*d^5*e + b^2*d^4*e^2 + 2*a*c*d^4*e^ 2 - 2*a*b*d^3*e^3 + a^2*d^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 2/3*((((6 *a^8*b^4*c^10*d^15 - 38*a^9*b^2*c^11*d^15 + 40*a^10*c^12*d^15 - 48*a^8*b^5 *c^9*d^14*e + 307*a^9*b^3*c^10*d^14*e - 340*a^10*b*c^11*d^14*e + 168*a^8*b ^6*c^8*d^13*e^2 - 1040*a^9*b^4*c^9*d^13*e^2 + 976*a^10*b^2*c^10*d^13*e^2 + 304*a^11*c^11*d^13*e^2 - 336*a^8*b^7*c^7*d^12*e^3 + 1876*a^9*b^5*c^8*d^12 *e^3 - 648*a^10*b^3*c^9*d^12*e^3 - 2280*a^11*b*c^10*d^12*e^3 + 420*a^8*b^8 *c^6*d^11*e^4 - 1820*a^9*b^6*c^7*d^11*e^4 - 2464*a^10*b^4*c^8*d^11*e^4 + 6 390*a^11*b^2*c^9*d^11*e^4 + 984*a^12*c^10*d^11*e^4 - 336*a^8*b^9*c^5*d^10* e^5 + 658*a^9*b^7*c^6*d^10*e^5 + 6496*a^10*b^5*c^7*d^10*e^5 - 7403*a^11*b^ 3*c^8*d^10*e^5 - 6396*a^12*b*c^9*d^10*e^5 + 168*a^8*b^10*c^4*d^9*e^6 + 448 *a^9*b^8*c^5*d^9*e^6 - 6720*a^10*b^6*c^6*d^9*e^6 - 60*a^11*b^4*c^7*d^9*e^6 + 15620*a^12*b^2*c^8*d^9*e^6 + 1760*a^13*c^9*d^9*e^6 - 48*a^8*b^11*c^3*d^ 8*e^7 - 620*a^9*b^9*c^4*d^8*e^7 + 2984*a^10*b^7*c^5*d^8*e^7 + 8991*a^11*b^ 5*c^6*d^8*e^7 - 16450*a^12*b^3*c^7*d^8*e^7 - 9680*a^13*b*c^8*d^8*e^7 + 6*a ^8*b^12*c^2*d^7*e^8 + 274*a^9*b^10*c^3*d^7*e^8 + 88*a^10*b^8*c^4*d^7*e^8 - 8810*a^11*b^6*c^5*d^7*e^8 + 3350*a^12*b^4*c^6*d^7*e^8 + 19430*a^13*b^2*c^ 7*d^7*e^8 + 1880*a^14*c^8*d^7*e^8 - 45*a^9*b^11*c^2*d^6*e^9 - 556*a^10*b^9 *c^3*d^6*e^9 + 2835*a^11*b^7*c^4*d^6*e^9 + 7456*a^12*b^5*c^5*d^6*e^9 - ...
Timed out. \[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:
int(1/(x^2*(d + e*x)*(a + b*x + c*x^2)^(5/2)),x)
Output:
int(1/(x^2*(d + e*x)*(a + b*x + c*x^2)^(5/2)), x)
Time = 0.38 (sec) , antiderivative size = 17328, normalized size of antiderivative = 27.59 \[ \int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(e*x+d)/(c*x^2+b*x+a)^(5/2),x)
Output:
(96*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**8*c**2*e**6*x - 48* sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b *d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**7*b**2*c*e**6*x + 192*s qrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b* d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**7*b*c**2*e**6*x**2 + 192 *sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**7*c**3*e**6*x**3 + 6*s qrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b* d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**6*b**4*e**6*x - 96*sqrt( a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**6*b**3*c*e**6*x**2 + 192*sqr t(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d* e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**6*b*c**3*e**6*x**4 + 96*sq rt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d *e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**6*c**4*e**6*x**5 + 12*sqr t(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d* e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**5*b**5*e**6*x**2 - 36*sqrt (a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**5*b**4*c*e**6*x**3 - 96*...