\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^4} \, dx\) [589]

Optimal result

Integrand size = 22, antiderivative size = 293 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {c \sqrt {a+b x+c x^2}}{e^3 (d+e x)}-\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{8 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^4 \left (c d^2-b d e+a e^2\right )^{3/2}} \] Output:

-c*(c*x^2+b*x+a)^(1/2)/e^3/(e*x+d)-1/8*(-b*e+2*c*d)*(b*d-2*a*e+(-b*e+2*c*d 
)*x)*(c*x^2+b*x+a)^(1/2)/e^2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^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))*arcta 
nh(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)
 

Mathematica [A] (verified)

Time = 11.13 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {-\frac {2 (a+x (b+c x))^{3/2}}{(d+e x)^3}+\frac {3 (2 c d-b e) (a+x (b+c x))^{3/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {3 \left (\frac {2 \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) (a+x (b+c x))^{3/2}}{d+e x}+\frac {2 \sqrt {a+x (b+c x)} \left (-b^3 e^3+4 c^3 d^2 (-2 d+e x)-b c e^2 (5 b d-10 a e+b e x)+2 c^2 e (b d (7 d-2 e x)+2 a e (-3 d+2 e x))\right )}{e^2}+\frac {16 c^{3/2} \left (c d^2+e (-b d+a e)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+(2 c d-b e) \sqrt {c d^2+e (-b d+a e)} \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{e^3}\right )}{8 \left (c d^2+e (-b d+a e)\right )^2}}{6 e} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x]
 

Output:

((-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)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.21, 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

\(\displaystyle \frac {-\frac {\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}-\frac {16 c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x]
 

Output:

-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)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 1092

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]
 

rule 1154

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]
 

rule 1161

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]
 

rule 1229

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]
 

rule 1269

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3084\) vs. \(2(265)=530\).

Time = 1.17 (sec) , antiderivative size = 3085, normalized size of antiderivative = 10.53

method	result	size
default	\(\text {Expression too large to display}\)	\(3085\)

Input:

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

1/e^4*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*(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)^(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*(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)^(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)*(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)^(5/2)+3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/3*(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*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*(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)+1/8*(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)+(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)))+(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)+1/2* 
(b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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))/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)*(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))))+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*(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)+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*(c*(x+d/e)^2+(b*e-2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1206 vs. \(2 (265) = 530\).

Time = 155.04 (sec) , antiderivative size = 4911, normalized size of antiderivative = 16.76 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**4,x)
 

Output:

Integral((a + b*x + c*x**2)**(3/2)/(d + e*x)**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x, algorithm="maxima")
 

Output:

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2059 vs. \(2 (265) = 530\).

Time = 3.86 (sec) , antiderivative size = 2059, normalized size of antiderivative = 7.03 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x, algorithm="giac")
 

Output:

-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^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x)
 

Output:

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 4565, normalized size of antiderivative = 15.58 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x)
 

Output:

( - 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*b*c*d**3*e**3 
- 108*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*b*c*d**2*e**4*x 
 - 108*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*b*c*d*e**5*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*b*c*e**6*x**3 
+ 72*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*c**2*d**4*e**2 + 
 216*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*c**2*d**3*e**3*x 
 + 216*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*c**2*d**2*e**4 
*x**2 + 72*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*s 
qrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*d*e** 
5*x**3 + 3*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*s 
qrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*b**3*d**3*e* 
*3 + 9*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)*b**3*d**2*e**...