\(\int \frac {x (a+b x+c x^2)^{3/2}}{(d+e x)^4} \, dx\) [81]

Optimal result

Integrand size = 23, antiderivative size = 367 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {(8 c d-3 b e) \sqrt {a+b x+c x^2}}{2 e^4 (d+e x)}+\frac {\left (8 c d^2-e (7 b d-6 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {(4 d+3 e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac {\sqrt {c} (8 c d-3 b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^5}+\frac {\left (64 c^3 d^4-b^2 e^3 (5 b d-6 a e)-24 c^2 d^2 e (5 b d-4 a e)+12 c e^2 \left (5 b^2 d^2-7 a b d e+2 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^5 \left (c d^2-b d e+a e^2\right )^{3/2}} \] Output:

1/2*(-3*b*e+8*c*d)*(c*x^2+b*x+a)^(1/2)/e^4/(e*x+d)+1/8*(8*c*d^2-e*(-6*a*e+ 
7*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c* 
d^2)/(e*x+d)^2+1/3*(3*e*x+4*d)*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)^3-1/2*c^(1/ 
2)*(-3*b*e+8*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^5+1 
/16*(64*c^3*d^4-b^2*e^3*(-6*a*e+5*b*d)-24*c^2*d^2*e*(-4*a*e+5*b*d)+12*c*e^ 
2*(2*a^2*e^2-7*a*b*d*e+5*b^2*d^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))/e^5/(a*e^2-b*d*e+c*d^2)^(3/ 
2)
 

Mathematica [A] (verified)

Time = 13.44 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.87 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {\frac {2 d \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{5/2}}{(d+e x)^3}-\frac {\left (4 c d^2+e (-5 b d+6 a e)\right ) (a+x (b+c x))^{5/2}}{2 (d+e x)^2}+\frac {\left (24 c^2 d^3+b e^2 (5 b d-6 a e)+2 c d e (-15 b d+14 a e)\right ) (a+x (b+c x))^{5/2}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {\frac {(a+x (b+c x))^{3/2} \left (b^2 e^3 (5 b d-6 a e)-8 c^3 d^3 (4 d-3 e x)+c e^2 \left (-12 a^2 e^2+2 a b e (28 d-3 e x)+5 b^2 d (-9 d+e x)\right )+2 c^2 d e (3 b d (12 d-5 e x)+2 a e (-12 d+7 e x))\right )}{4 e^2}+\frac {3 \left (-2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (b^2 e^3 (-5 b d+6 a e)+16 c^3 d^3 (2 d-e x)+c e^2 \left (12 a^2 e^2+b^2 d (41 d-5 e x)+2 a b e (-26 d+3 e x)\right )+2 c^2 d e (2 a e (12 d-5 e x)+b d (-34 d+11 e x))\right )+8 \sqrt {c} (8 c d-3 b e) \left (c d^2+e (-b d+a e)\right )^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+\left (c d^2+e (-b d+a e)\right )^{3/2} \left (64 c^3 d^4-24 c^2 d^2 e (5 b d-4 a e)+b^2 e^3 (-5 b d+6 a e)+12 c e^2 \left (5 b^2 d^2-7 a b d e+2 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{8 e^5}}{-c d^2+e (b d-a e)}}{6 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

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

Output:

((2*d*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(5/2))/(d + e*x)^3 - (( 
4*c*d^2 + e*(-5*b*d + 6*a*e))*(a + x*(b + c*x))^(5/2))/(2*(d + e*x)^2) + ( 
(24*c^2*d^3 + b*e^2*(5*b*d - 6*a*e) + 2*c*d*e*(-15*b*d + 14*a*e))*(a + x*( 
b + c*x))^(5/2))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (((a + x*(b + 
c*x))^(3/2)*(b^2*e^3*(5*b*d - 6*a*e) - 8*c^3*d^3*(4*d - 3*e*x) + c*e^2*(-1 
2*a^2*e^2 + 2*a*b*e*(28*d - 3*e*x) + 5*b^2*d*(-9*d + e*x)) + 2*c^2*d*e*(3* 
b*d*(12*d - 5*e*x) + 2*a*e*(-12*d + 7*e*x))))/(4*e^2) + (3*(-2*e*(c*d^2 + 
e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]*(b^2*e^3*(-5*b*d + 6*a*e) + 16*c^3 
*d^3*(2*d - e*x) + c*e^2*(12*a^2*e^2 + b^2*d*(41*d - 5*e*x) + 2*a*b*e*(-26 
*d + 3*e*x)) + 2*c^2*d*e*(2*a*e*(12*d - 5*e*x) + b*d*(-34*d + 11*e*x))) + 
8*Sqrt[c]*(8*c*d - 3*b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*ArcTanh[(b + 2*c*x) 
/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (c*d^2 + e*(-(b*d) + a*e))^(3/2)*(64 
*c^3*d^4 - 24*c^2*d^2*e*(5*b*d - 4*a*e) + b^2*e^3*(-5*b*d + 6*a*e) + 12*c* 
e^2*(5*b^2*d^2 - 7*a*b*d*e + 2*a^2*e^2))*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*e 
^5))/(-(c*d^2) + e*(b*d - a*e)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1229, 27, 1230, 25, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int -\frac {\left (-5 d e b^2+8 c d^2 b+6 a e^2 b-8 a c d e+2 c \left (8 c d^2-e (7 b d-6 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^2}dx}{4 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (-5 d e b^2+8 c d^2 b+6 a e^2 b-8 a c d e+2 c \left (8 c d^2-e (7 b d-6 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}-\frac {\int -\frac {b e \left (-5 d e b^2+8 c d^2 b+6 a e^2 b-8 a c d e\right )-4 c (b d-a e) \left (8 c d^2-e (7 b d-6 a e)\right )-8 c (8 c d-3 b e) \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b e \left (-5 d e b^2+8 c d^2 b+6 a e^2 b-8 a c d e\right )-4 c (b d-a e) \left (8 c d^2-e (7 b d-6 a e)\right )-8 c (8 c d-3 b e) \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {\left (12 c e^2 \left (2 a^2 e^2-7 a b d e+5 b^2 d^2\right )-b^2 e^3 (5 b d-6 a e)-24 c^2 d^2 e (5 b d-4 a e)+64 c^3 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {8 c (8 c d-3 b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\left (12 c e^2 \left (2 a^2 e^2-7 a b d e+5 b^2 d^2\right )-b^2 e^3 (5 b d-6 a e)-24 c^2 d^2 e (5 b d-4 a e)+64 c^3 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {16 c (8 c d-3 b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (12 c e^2 \left (2 a^2 e^2-7 a b d e+5 b^2 d^2\right )-b^2 e^3 (5 b d-6 a e)-24 c^2 d^2 e (5 b d-4 a e)+64 c^3 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {8 \sqrt {c} (8 c d-3 b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {-\frac {2 \left (12 c e^2 \left (2 a^2 e^2-7 a b d e+5 b^2 d^2\right )-b^2 e^3 (5 b d-6 a e)-24 c^2 d^2 e (5 b d-4 a e)+64 c^3 d^4\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {8 \sqrt {c} (8 c d-3 b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (12 c e^2 \left (2 a^2 e^2-7 a b d e+5 b^2 d^2\right )-b^2 e^3 (5 b d-6 a e)-24 c^2 d^2 e (5 b d-4 a e)+64 c^3 d^4\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}-\frac {8 \sqrt {c} (8 c d-3 b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (8 c d^2-e (7 b d-6 a e)\right )-4 c d e (9 b d-8 a e)+b e^2 (5 b d-6 a e)+32 c^2 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (4 c d^2-e (3 b d-2 a e)\right )+d \left (8 c d^2-e (5 b d-2 a e)\right )\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-1/12*((d*(8*c*d^2 - e*(5*b*d - 2*a*e)) + 3*e*(4*c*d^2 - e*(3*b*d - 2*a*e) 
)*x)*(a + b*x + c*x^2)^(3/2))/(e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + 
(((32*c^2*d^3 - 4*c*d*e*(9*b*d - 8*a*e) + b*e^2*(5*b*d - 6*a*e) + 2*c*e*(8 
*c*d^2 - e*(7*b*d - 6*a*e))*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x)) + (( 
-8*Sqrt[c]*(8*c*d - 3*b*e)*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(b + 2*c*x)/(2* 
Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e + ((64*c^3*d^4 - b^2*e^3*(5*b*d - 6*a*e 
) - 24*c^2*d^2*e*(5*b*d - 4*a*e) + 12*c*e^2*(5*b^2*d^2 - 7*a*b*d*e + 2*a^2 
*e^2))*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]))/(2*e^2))/(8* 
e^2*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(2194\) vs. \(2(335)=670\).

Time = 1.59 (sec) , antiderivative size = 2195, normalized size of antiderivative = 5.98

method	result	size
risch	\(\text {Expression too large to display}\)	\(2195\)
default	\(\text {Expression too large to display}\)	\(4942\)

Input:

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

Output:

1/e^4*c*(c*x^2+b*x+a)^(1/2)+1/2/e^4*(c^(1/2)*(3*b*e-8*c*d)/e*ln((1/2*b+c*x 
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/e^2*(2*a*c*e^2+b^2*e^2-8*b*c*d*e+10*c^2*d 
^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))+2/e^3*(2*a*b*e^3-6*a*c*d 
*e^2-3*b^2*d*e^2+12*b*c*d^2*e-10*c^2*d^3)*(-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)^(1/2)+1/2* 
(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)))+2/e^4*(a^2*e^4-4*a*b*d*e^3+6*a*c*d^2*e^2+3*b^2*d^2*e^2-8*b*c*d^3*e+ 
5*c^2*d^4)*(-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)^(1/2)-3/4*(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)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)))+1/2*c/(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)/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(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. 2470 vs. \(2 (335) = 670\).

Time = 4.51 (sec) , antiderivative size = 2470, normalized size of antiderivative = 6.73 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(64*c^3*d^4 - 120*b*c^2*d^3*e + 60*b^2*c*d^2*e^2 + 96*a*c^2*d^2*e^2 - 
5*b^3*d*e^3 - 84*a*b*c*d*e^3 + 6*a*b^2*e^4 + 24*a^2*c*e^4)*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^5 - b*d*e^6 + a*e^7)*sqrt(-c*d^2 + b*d*e - a*e^2)) + sqrt(c*x^2 
 + b*x + a)*c/e^4 + 1/24*(288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d^ 
4*e^2 - 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d^3*e^3 + 252*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d^2*e^4 + 288*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^5*a*c^2*d^2*e^4 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5 
*b^3*d*e^5 - 228*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*d*e^5 + 30*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*e^6 + 24*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^5*a^2*c*e^6 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/ 
2)*d^5*e - 1464*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^4*e^2 + 
540*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*d^3*e^3 + 672*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)*d^3*e^3 - 21*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^4*b^3*sqrt(c)*d^2*e^4 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^4*a*b*c^(3/2)*d^2*e^4 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a* 
b^2*sqrt(c)*d*e^5 - 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)* 
d*e^5 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*sqrt(c)*e^6 + 832*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^6 - 400*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^3*b*c^3*d^5*e - 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.80 (sec) , antiderivative size = 6174, normalized size of antiderivative = 16.82 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(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**2*c*d**3*e**4 + 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**2*c*d**2*e**5*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**2*c*d*e**6*x**2 + 72*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*e**7*x**3 + 18*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*d**3*e**4 + 54*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*d**2*e**5*x + 54*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*d*e**6*x**2 + 18*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*e**7*x**3 - 252*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**4*e**3 - 756*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**4*x - 756*sqrt(a*e**2 - b*d*e ...