\(\int \frac {(d+e x)^{3/2}}{(a+b x+c x^2)^3} \, dx\) [547]

Optimal result

Integrand size = 22, antiderivative size = 441 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d-7 b^2 e+4 a c e+12 c (2 c d-b e) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d-\sqrt {b^2-4 a c} d-a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {c} \left (16 c^2 d^2+b \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^2-4 c e \left (4 b d+\sqrt {b^2-4 a c} d-a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{2 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-1/2*(e*x+d)^(1/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2 
+1/4*(e*x+d)^(1/2)*(12*b*c*d-7*b^2*e+4*a*c*e+12*c*(-b*e+2*c*d)*x)/(-4*a*c+ 
b^2)^2/(c*x^2+b*x+a)-3/4*c^(1/2)*(16*c^2*d^2+b*(3*b-2*(-4*a*c+b^2)^(1/2))* 
e^2-4*c*e*(4*b*d-(-4*a*c+b^2)^(1/2)*d-a*e))*arctanh(2^(1/2)*c^(1/2)*(e*x+d 
)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2))*2^(1/2)/(-4*a*c+b^2)^(5/2) 
/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)+3/4*c^(1/2)*(16*c^2*d^2+b*(3*b+2*( 
-4*a*c+b^2)^(1/2))*e^2-4*c*e*(4*b*d+(-4*a*c+b^2)^(1/2)*d-a*e))*arctanh(2^( 
1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2))*2^(1/2) 
/(-4*a*c+b^2)^(5/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4707\) vs. \(2(441)=882\).

Time = 16.35 (sec) , antiderivative size = 4707, normalized size of antiderivative = 10.67 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/2*((d + e*x)^(5/2)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 
 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-(((d + e*x)^(5/ 
2)*(-1/2*(a*c*e*(2*c*d - b*e)^2) + ((b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 
+ b^2*e^2 - c*e*(11*b*d - 6*a*e)))/2 + c*(-1/2*(c*e*(b*d - 2*a*e)*(2*c*d - 
 b*e)) + ((2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - c*e*(11*b*d - 6*a*e)))/2)* 
x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (-1/2*(e* 
(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*(d + e*x)^(3/ 
2)) + (2*((3*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b 
*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*( 
3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a 
*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^ 
2*e^2)))/4)*Sqrt[d + e*x])/c + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e 
]*(((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d 
- 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 
4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2) 
))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e 
*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4* 
c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b* 
d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 
 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b...
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1164, 27, 1235, 27, 1197, 27, 1480, 221}

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.

\(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {12 c d^2-7 b e d+2 a e^2+5 e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {12 c d^2-e (7 b d-2 a e)+5 e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {-\frac {\int \frac {3 \left (c d^2-b e d+a e^2\right ) \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right )}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \int \frac {16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {-\frac {3 \int \frac {e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)+4 c (2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 e \int \frac {8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)+4 c (2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{b^2-4 a c}-\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {-\frac {3 e \left (\frac {c \left (-4 c e \left (-d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt {b^2-4 a c}\right )+16 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{e \sqrt {b^2-4 a c}}-\frac {c \left (-4 c e \left (d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt {b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{e \sqrt {b^2-4 a c}}\right )}{b^2-4 a c}-\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {3 e \left (\frac {\sqrt {2} \sqrt {c} \left (-4 c e \left (d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt {b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {2} \sqrt {c} \left (-4 c e \left (-d \sqrt {b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt {b^2-4 a c}\right )+16 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{b^2-4 a c}-\frac {\sqrt {d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

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

Output:

-1/2*(Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b 
*x + c*x^2)^2) - (-((Sqrt[d + e*x]*(12*b*c*d - 7*b^2*e + 4*a*c*e + 12*c*(2 
*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) - (3*e*(-((Sqrt[2]*Sqrt 
[c]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d - Sqrt[ 
b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d 
- (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d - (b - Sqrt 
[b^2 - 4*a*c])*e])) + (Sqrt[2]*Sqrt[c]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 
 4*a*c])*e^2 - 4*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2] 
*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^ 
2 - 4*a*c]*e*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(b^2 - 4*a*c))/(4* 
(b^2 - 4*a*c))
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1164

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 + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.20

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (\left (c d -\frac {b e}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+4 c^{2} d^{2}+\left (a \,e^{2}-4 b d e \right ) c +\frac {3 b^{2} e^{2}}{4}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} c \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (e \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} c \left (\left (\frac {b e}{2}-c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+4 c^{2} d^{2}+\left (a \,e^{2}-4 b d e \right ) c +\frac {3 b^{2} e^{2}}{4}\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (-2 c^{3} d \,x^{3}+\left (\left (b \,x^{3}-\frac {1}{3} a \,x^{2}\right ) e -\frac {10 d x \left (\frac {9 b x}{10}+a \right )}{3}\right ) c^{2}+\left (\left (\frac {4}{3} a b x +\frac {19}{12} b^{2} x^{2}+a^{2}\right ) e -\frac {5 \left (\frac {2 b x}{5}+a \right ) d b}{3}\right ) c +\frac {b^{2} \left (\left (\frac {5 b x}{3}+a \right ) e +\frac {2 b d}{3}\right )}{4}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{16 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (c \,x^{2}+b x +a \right )^{2} \left (a c -\frac {b^{2}}{4}\right )^{2}}\)	\(530\)
derivativedivides	\(2 e^{5} \left (\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {c \left (4 a c \,e^{2}-19 b^{2} e^{2}+72 b c d e -72 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {\left (b e -2 c d \right ) \left (16 a c \,e^{2}+5 b^{2} e^{2}-36 b c d e +36 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {3 \left (4 e^{4} a^{2} c +a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} a \,c^{2}-b^{3} d \,e^{3}+9 b^{2} c \,d^{2} e^{2}-16 b \,c^{2} d^{3} e +8 d^{4} c^{3}\right ) \sqrt {e x +d}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (-4 a c \,e^{2}-3 b^{2} e^{2}+16 b c d e -16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\right )\)	\(703\)
default	\(2 e^{5} \left (\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}+\frac {c \left (4 a c \,e^{2}-19 b^{2} e^{2}+72 b c d e -72 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {\left (b e -2 c d \right ) \left (16 a c \,e^{2}+5 b^{2} e^{2}-36 b c d e +36 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {3 \left (4 e^{4} a^{2} c +a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} a \,c^{2}-b^{3} d \,e^{3}+9 b^{2} c \,d^{2} e^{2}-16 b \,c^{2} d^{3} e +8 d^{4} c^{3}\right ) \sqrt {e x +d}}{8 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {3 c \left (\frac {\left (-4 a c \,e^{2}-3 b^{2} e^{2}+16 b c d e -16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}+3 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\right )\)	\(703\)

Input:

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

Output:

-3/16/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/((-b*e+2*c*d+(-4* 
e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(((c*d-1/2*b*e)*(-4*e^2*(a*c-1/4*b^2))^ 
(1/2)+4*c^2*d^2+(a*e^2-4*b*d*e)*c+3/4*b^2*e^2)*((b*e-2*c*d+(-4*e^2*(a*c-1/ 
4*b^2))^(1/2))*c)^(1/2)*e*2^(1/2)*(c*x^2+b*x+a)^2*c*arctanh((e*x+d)^(1/2)* 
c*2^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+(e*2^(1/2)* 
(c*x^2+b*x+a)^2*c*((1/2*b*e-c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+4*c^2*d^2+(a 
*e^2-4*b*d*e)*c+3/4*b^2*e^2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(- 
4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+(-2*c^3*d*x^3+((b*x^3-1/3*a*x^2)*e-1 
0/3*d*x*(9/10*b*x+a))*c^2+((4/3*a*b*x+19/12*b^2*x^2+a^2)*e-5/3*(2/5*b*x+a) 
*d*b)*c+1/4*b^2*((5/3*b*x+a)*e+2/3*b*d))*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2) 
)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)*(-4*e^2*(a*c-1/4*b^2))^(1/2))*((-b*e+2*c*d 
+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))/(-4*e^2*(a*c-1/4*b^2))^(1/2)/(c*x 
^2+b*x+a)^2/(a*c-1/4*b^2)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11128 vs. \(2 (381) = 762\).

Time = 0.47 (sec) , antiderivative size = 11128, normalized size of antiderivative = 25.23 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (381) = 762\).

Time = 1.02 (sec) , antiderivative size = 2116, normalized size of antiderivative = 4.80 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3/16*(2*(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)^2*sqrt(-4*c^2*d + 2*(b*c - sq 
rt(b^2 - 4*a*c)*c)*e)*(2*c*d*e - b*e^2) + (8*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 
- 4*a*c)*d^2*e - 8*(b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c)*d*e^2 + (b^4 - 16 
*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c 
)*c)*e)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e) - (32*(b^6*c^3 - 12*a*b^4* 
c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3*e - 48*(b^7*c^2 - 12*a*b^5*c^3 + 48 
*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e^2 + 2*(11*b^8*c - 128*a*b^6*c^2 + 480*a 
^2*b^4*c^3 - 512*a^3*b^2*c^4 - 256*a^4*c^5)*d*e^3 - (3*b^9 - 32*a*b^7*c + 
96*a^2*b^5*c^2 - 256*a^4*b*c^4)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4 
*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(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 - 16*a^2*b*c^2*e + sqrt((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 - 16*a^2*b*c^2*e 
)^2 - 4*(b^4*c*d^2 - 8*a*b^2*c^2*d^2 + 16*a^2*c^3*d^2 - b^5*d*e + 8*a*b^3* 
c*d*e - 16*a^2*b*c^2*d*e + a*b^4*e^2 - 8*a^2*b^2*c*e^2 + 16*a^3*c^2*e^2)*( 
b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)))/( 
((b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(b^2 - 4*a*c)*d^ 
2 - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*sqrt(b^2 - 4*a*c)*d 
*e + (a*b^6 - 12*a^2*b^4*c + 48*a^3*b^2*c^2 - 64*a^4*c^3)*sqrt(b^2 - 4*a*c 
)*e^2)*abs(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)*abs(c)) - 3/16*(2*(b^4*e - 
8*a*b^2*c*e + 16*a^2*c^2*e)^2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c...
 

Mupad [B] (verification not implemented)

Time = 18.77 (sec) , antiderivative size = 13764, normalized size of antiderivative = 31.21 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

log((27*c^3*e^3*(b*e - 2*c*d)*(5*b^4*e^4 + 256*c^4*d^4 + 16*a^2*c^2*e^4 + 
192*a*c^3*d^2*e^2 + 336*b^2*c^2*d^2*e^2 + 40*a*b^2*c*e^4 - 512*b*c^3*d^3*e 
 - 80*b^3*c*d*e^3 - 192*a*b*c^2*d*e^3))/(16*(4*a*c - b^2)^6) - (3*2^(1/2)* 
((3*2^(1/2)*((3*c^2*e^3*(b^2*e^2 + 8*c^2*d^2 + 4*a*c*e^2 - 8*b*c*d*e))/(4* 
a*c - b^2) - (3*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2 
)*(-(b^15*e^5 + e^5*(-(4*a*c - b^2)^15)^(1/2) + 524288*a^5*c^10*d^5 - 512* 
b^10*c^5*d^5 + 10240*a*b^8*c^6*d^5 - 81920*a^7*b*c^7*e^5 + 163840*a^7*c^8* 
d*e^4 + 1280*b^11*c^4*d^4*e - 81920*a^2*b^6*c^7*d^5 + 327680*a^3*b^4*c^8*d 
^5 - 655360*a^4*b^2*c^9*d^5 - 560*a^2*b^11*c^2*e^5 + 4160*a^3*b^9*c^3*e^5 
- 11520*a^4*b^7*c^4*e^5 - 1024*a^5*b^5*c^5*e^5 + 61440*a^6*b^3*c^6*e^5 + 6 
55360*a^6*c^9*d^3*e^2 - 1120*b^12*c^3*d^3*e^2 + 400*b^13*c^2*d^2*e^3 + 20* 
a*b^13*c*e^5 - 50*b^14*c*d*e^4 - 166400*a^2*b^8*c^5*d^3*e^2 + 44800*a^2*b^ 
9*c^4*d^2*e^3 + 614400*a^3*b^6*c^6*d^3*e^2 - 102400*a^3*b^7*c^5*d^2*e^3 - 
1024000*a^4*b^4*c^7*d^3*e^2 - 102400*a^4*b^5*c^6*d^2*e^3 + 327680*a^5*b^2* 
c^8*d^3*e^2 + 819200*a^5*b^3*c^7*d^2*e^3 - 25600*a*b^9*c^5*d^4*e + 600*a*b 
^12*c^2*d*e^4 - 1310720*a^5*b*c^9*d^4*e + 21760*a*b^10*c^4*d^3*e^2 - 7040* 
a*b^11*c^3*d^2*e^3 + 204800*a^2*b^7*c^6*d^4*e - 160*a^2*b^10*c^3*d*e^4 - 8 
19200*a^3*b^5*c^7*d^4*e - 28800*a^3*b^8*c^4*d*e^4 + 1638400*a^4*b^3*c^8*d^ 
4*e + 166400*a^4*b^6*c^5*d*e^4 - 358400*a^5*b^4*c^6*d*e^4 - 983040*a^6*b*c 
^8*d^2*e^3 + 204800*a^6*b^2*c^7*d*e^4)/((4*a*c - b^2)^10*(a*e^2 + c*d^2...
 

Reduce [F]

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

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

Output:

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