\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^5} \, dx\) [590]

Optimal result

Integrand size = 22, antiderivative size = 225 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {3 \left (b^2-4 a c\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{8 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac {3 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-3/64*(-4*a*c+b^2)*(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)^2/(e*x+d)^2+1/8*(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)/(e*x+d)^4+3/128*(-4*a*c+b^2)^2*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)^(5/2)
 

Mathematica [A] (verified)

Time = 10.56 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{16 \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]
 

Output:

-1/16*((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)*ArcT 
anh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*S 
qrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(c*d^2 + e* 
(-(b*d) + a*e))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{16 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

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

Input:

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]
 

Output:

((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))
 

Defintions of rubi rules used

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 1152

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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5073\) vs. \(2(207)=414\).

Time = 1.32 (sec) , antiderivative size = 5074, normalized size of antiderivative = 22.55

method	result	size
default	\(\text {Expression too large to display}\)	\(5074\)

Input:

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1264 vs. \(2 (207) = 414\).

Time = 9.59 (sec) , antiderivative size = 2570, normalized size of antiderivative = 11.42 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x, algorithm="fricas")
 

Output:

[1/256*(3*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4*x^4 + 4*(b^4 - 8*a*b^2*c + 1 
6*a^2*c^2)*d*e^3*x^3 + 6*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2*e^2*x^2 + 4*(b 
^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3*e*x + (b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^4) 
*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* 
(40*a^3*b*d*e^4 - 16*a^4*e^5 - (3*b^3*c - 20*a*b*c^2)*d^5 + (3*b^4 - 22*a* 
b^2*c - 40*a^2*c^2)*d^4*e - (a*b^3 - 84*a^2*b*c)*d^3*e^2 - 2*(13*a^2*b^2 + 
 28*a^3*c)*d^2*e^3 + (16*c^4*d^5 - 40*b*c^3*d^4*e + 2*(13*b^2*c^2 + 28*a*c 
^3)*d^3*e^2 + (b^3*c - 84*a*b*c^2)*d^2*e^3 - (3*b^4 - 22*a*b^2*c - 40*a^2* 
c^2)*d*e^4 + (3*a*b^3 - 20*a^2*b*c)*e^5)*x^3 + (24*b*c^3*d^5 - 4*(17*b^2*c 
^2 - 8*a*c^3)*d^4*e + 5*(11*b^3*c + 4*a*b*c^2)*d^3*e^2 - (11*b^4 + 74*a*b^ 
2*c + 8*a^2*c^2)*d^2*e^3 + (13*a*b^3 + 68*a^2*b*c)*d*e^4 - 2*(a^2*b^2 + 20 
*a^3*c)*e^5)*x^2 - (24*a^3*b*e^5 - 2*(b^2*c^2 + 20*a*c^3)*d^5 + (13*b^3*c 
+ 68*a*b*c^2)*d^4*e - (11*b^4 + 74*a*b^2*c + 8*a^2*c^2)*d^3*e^2 + 5*(11*a* 
b^3 + 4*a^2*b*c)*d^2*e^3 - 4*(17*a^2*b^2 - 8*a^3*c)*d*e^4)*x)*sqrt(c*x^2 + 
 b*x + a))/(c^3*d^10 - 3*b*c^2*d^9*e - 3*a^2*b*d^5*e^5 + a^3*d^4*e^6 + 3*( 
b^2*c + a*c^2)*d^8*e^2 - (b^3 + 6*a*b*c)*d^7*e^3 + 3*(a*b^2 + a^2*c)*d^6*e 
^4 + (c^3*d^6*e^4 - 3*b*c^2*d^5*e^5 - 3*a^2*b*d*e^9 + a^3*e^10 + 3*(b^2...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{5}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**5,x)
 

Output:

Integral((a + b*x + c*x**2)**(3/2)/(d + e*x)**5, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,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 [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^5} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x)
 

Output:

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^5, x)
 

Reduce [B] (verification not implemented)

Time = 11.34 (sec) , antiderivative size = 3483, normalized size of antiderivative = 15.48 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,x)
 

Output:

(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**2*c**2*d**4 + 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**2*c**2*d**3*e*x + 288*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**2*c**2*d**2*e**2*x**2 + 1 
92*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**2*c**2*d*e**3*x**3 + 
 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**2*c**2*e**4*x**4 - 
24*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**2*c*d**4 - 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*b**2*c*d**3*e*x - 144*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**2*c*d**2*e**2*x**2 - 96*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*b**2*c*d*e**3*x**3 - 24*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*b**2*c*e**4*x**4 + 3*sqr...