\(\int (d+e x) (a+b x+c x^2)^{3/2} \, dx\) [584]

Optimal result

Integrand size = 20, antiderivative size = 161 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}} \] Output:

-3/128*(-4*a*c+b^2)*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^3+1/16*(- 
b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^2+1/5*e*(c*x^2+b*x+a)^(5/2)/c+3 
/256*(-4*a*c+b^2)^2*(-b*e+2*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+ 
a)^(1/2))/c^(7/2)
 

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^4 e-10 b^3 c (3 d+e x)+8 b c^2 \left (25 a d+7 a e x+30 c d x^2+22 c e x^3\right )+4 b^2 c (-25 a e+c x (5 d+2 e x))+16 c^2 \left (8 a^2 e+2 c^2 x^3 (5 d+4 e x)+a c x (25 d+16 e x)\right )\right )-15 \left (b^2-4 a c\right )^2 (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{640 c^{7/2}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^4*e - 10*b^3*c*(3*d + e*x) + 8*b*c^2* 
(25*a*d + 7*a*e*x + 30*c*d*x^2 + 22*c*e*x^3) + 4*b^2*c*(-25*a*e + c*x*(5*d 
 + 2*e*x)) + 16*c^2*(8*a^2*e + 2*c^2*x^3*(5*d + 4*e*x) + a*c*x*(25*d + 16* 
e*x))) - 15*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + 
 Sqrt[a + x*(b + c*x)])])/(640*c^(7/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1160, 1087, 1087, 1092, 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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}\)

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}\)

Input:

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

Output:

(e*(a + b*x + c*x^2)^(5/2))/(5*c) + ((2*c*d - b*e)*(((b + 2*c*x)*(a + b*x 
+ c*x^2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a + b*x + c*x^ 
2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c 
*x^2])])/(8*c^(3/2))))/(16*c)))/(2*c)
 

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1160

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

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.45

method	result	size
risch	\(\frac {\left (128 c^{4} e \,x^{4}+176 b e \,c^{3} x^{3}+160 c^{4} d \,x^{3}+256 a \,c^{3} e \,x^{2}+8 b^{2} c^{2} e \,x^{2}+240 b \,c^{3} d \,x^{2}+56 a b \,c^{2} e x +400 a \,c^{3} d x -10 b^{3} c e x +20 b^{2} c^{2} d x +128 a^{2} c^{2} e -100 a \,b^{2} c e +200 a b \,c^{2} d +15 b^{4} e -30 b^{3} c d \right ) \sqrt {c \,x^{2}+b x +a}}{640 c^{3}}-\frac {3 \left (16 a^{2} b \,c^{2} e -32 a^{2} c^{3} d -8 a \,b^{3} c e +16 a \,b^{2} c^{2} d +b^{5} e -2 b^{4} c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}\)	\(233\)
default	\(d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\)	\(236\)

Input:

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

Output:

1/640/c^3*(128*c^4*e*x^4+176*b*c^3*e*x^3+160*c^4*d*x^3+256*a*c^3*e*x^2+8*b 
^2*c^2*e*x^2+240*b*c^3*d*x^2+56*a*b*c^2*e*x+400*a*c^3*d*x-10*b^3*c*e*x+20* 
b^2*c^2*d*x+128*a^2*c^2*e-100*a*b^2*c*e+200*a*b*c^2*d+15*b^4*e-30*b^3*c*d) 
*(c*x^2+b*x+a)^(1/2)-3/256*(16*a^2*b*c^2*e-32*a^2*c^3*d-8*a*b^3*c*e+16*a*b 
^2*c^2*d+b^5*e-2*b^4*c*d)/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/ 
2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.29 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (128 \, c^{5} e x^{4} + 16 \, {\left (10 \, c^{5} d + 11 \, b c^{4} e\right )} x^{3} + 8 \, {\left (30 \, b c^{4} d + {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} e\right )} x^{2} - 10 \, {\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} d + {\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} e + 2 \, {\left (10 \, {\left (b^{2} c^{3} + 20 \, a c^{4}\right )} d - {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2560 \, c^{4}}, -\frac {15 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (128 \, c^{5} e x^{4} + 16 \, {\left (10 \, c^{5} d + 11 \, b c^{4} e\right )} x^{3} + 8 \, {\left (30 \, b c^{4} d + {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} e\right )} x^{2} - 10 \, {\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} d + {\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} e + 2 \, {\left (10 \, {\left (b^{2} c^{3} + 20 \, a c^{4}\right )} d - {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1280 \, c^{4}}\right ] \] Input:

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

Output:

[-1/2560*(15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d - (b^5 - 8*a*b^3*c + 
16*a^2*b*c^2)*e)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b 
*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(128*c^5*e*x^4 + 16*(10*c^5*d + 1 
1*b*c^4*e)*x^3 + 8*(30*b*c^4*d + (b^2*c^3 + 32*a*c^4)*e)*x^2 - 10*(3*b^3*c 
^2 - 20*a*b*c^3)*d + (15*b^4*c - 100*a*b^2*c^2 + 128*a^2*c^3)*e + 2*(10*(b 
^2*c^3 + 20*a*c^4)*d - (5*b^3*c^2 - 28*a*b*c^3)*e)*x)*sqrt(c*x^2 + b*x + a 
))/c^4, -1/1280*(15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d - (b^5 - 8*a*b 
^3*c + 16*a^2*b*c^2)*e)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + 
 b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(128*c^5*e*x^4 + 16*(10*c^5*d + 
11*b*c^4*e)*x^3 + 8*(30*b*c^4*d + (b^2*c^3 + 32*a*c^4)*e)*x^2 - 10*(3*b^3* 
c^2 - 20*a*b*c^3)*d + (15*b^4*c - 100*a*b^2*c^2 + 128*a^2*c^3)*e + 2*(10*( 
b^2*c^3 + 20*a*c^4)*d - (5*b^3*c^2 - 28*a*b*c^3)*e)*x)*sqrt(c*x^2 + b*x + 
a))/c^4]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (151) = 302\).

Time = 0.76 (sec) , antiderivative size = 711, normalized size of antiderivative = 4.42 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c*e*x**4/5 + x**3*(11*b*c*e/10 + c**2*d 
)/(4*c) + x**2*(6*a*c*e/5 + b**2*e + 2*b*c*d - 7*b*(11*b*c*e/10 + c**2*d)/ 
(8*c))/(3*c) + x*(2*a*b*e + 2*a*c*d - 3*a*(11*b*c*e/10 + c**2*d)/(4*c) + b 
**2*d - 5*b*(6*a*c*e/5 + b**2*e + 2*b*c*d - 7*b*(11*b*c*e/10 + c**2*d)/(8* 
c))/(6*c))/(2*c) + (a**2*e + 2*a*b*d - 2*a*(6*a*c*e/5 + b**2*e + 2*b*c*d - 
 7*b*(11*b*c*e/10 + c**2*d)/(8*c))/(3*c) - 3*b*(2*a*b*e + 2*a*c*d - 3*a*(1 
1*b*c*e/10 + c**2*d)/(4*c) + b**2*d - 5*b*(6*a*c*e/5 + b**2*e + 2*b*c*d - 
7*b*(11*b*c*e/10 + c**2*d)/(8*c))/(6*c))/(4*c))/c) + (a**2*d - a*(2*a*b*e 
+ 2*a*c*d - 3*a*(11*b*c*e/10 + c**2*d)/(4*c) + b**2*d - 5*b*(6*a*c*e/5 + b 
**2*e + 2*b*c*d - 7*b*(11*b*c*e/10 + c**2*d)/(8*c))/(6*c))/(2*c) - b*(a**2 
*e + 2*a*b*d - 2*a*(6*a*c*e/5 + b**2*e + 2*b*c*d - 7*b*(11*b*c*e/10 + c**2 
*d)/(8*c))/(3*c) - 3*b*(2*a*b*e + 2*a*c*d - 3*a*(11*b*c*e/10 + c**2*d)/(4* 
c) + b**2*d - 5*b*(6*a*c*e/5 + b**2*e + 2*b*c*d - 7*b*(11*b*c*e/10 + c**2* 
d)/(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + 
 c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2 
*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(e*(a + b*x)**(7/ 
2)/(7*b) + (a + b*x)**(5/2)*(-a*e + b*d)/(5*b))/b, Ne(b, 0)), (a**(3/2)*(d 
*x + e*x**2/2), True))
 

Maxima [F(-2)]

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

integrate((e*x+d)*(c*x^2+b*x+a)^(3/2),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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.55 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{640} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c e x + \frac {10 \, c^{5} d + 11 \, b c^{4} e}{c^{4}}\right )} x + \frac {30 \, b c^{4} d + b^{2} c^{3} e + 32 \, a c^{4} e}{c^{4}}\right )} x + \frac {10 \, b^{2} c^{3} d + 200 \, a c^{4} d - 5 \, b^{3} c^{2} e + 28 \, a b c^{3} e}{c^{4}}\right )} x - \frac {30 \, b^{3} c^{2} d - 200 \, a b c^{3} d - 15 \, b^{4} c e + 100 \, a b^{2} c^{2} e - 128 \, a^{2} c^{3} e}{c^{4}}\right )} - \frac {3 \, {\left (2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {7}{2}}} \] Input:

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

Output:

1/640*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*c*e*x + (10*c^5*d + 11*b*c^4*e)/c^ 
4)*x + (30*b*c^4*d + b^2*c^3*e + 32*a*c^4*e)/c^4)*x + (10*b^2*c^3*d + 200* 
a*c^4*d - 5*b^3*c^2*e + 28*a*b*c^3*e)/c^4)*x - (30*b^3*c^2*d - 200*a*b*c^3 
*d - 15*b^4*c*e + 100*a*b^2*c^2*e - 128*a^2*c^3*e)/c^4) - 3/256*(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)*l 
og(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2)
 

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.89 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {e\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5\,c}+\frac {d\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )\,\left (3\,a\,c-\frac {3\,b^2}{4}\right )}{4\,c}-\frac {b\,e\,\left (\frac {3\,a\,\left (\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{4\,c}\right )}{4}+\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4}+\frac {b\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{8\,c}-\frac {3\,b^2\,\left (\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{4\,c}\right )}{16\,c}\right )}{2\,c}+\frac {d\,\left (\frac {b}{2}+c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \] Input:

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

Output:

(e*(a + b*x + c*x^2)^(5/2))/(5*c) + (d*((x/2 + b/(4*c))*(a + b*x + c*x^2)^ 
(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4)) 
/(2*c^(3/2)))*(3*a*c - (3*b^2)/4))/(4*c) - (b*e*((3*a*(log((b/2 + c*x)/c^( 
1/2) + (a + b*x + c*x^2)^(1/2))*(a/(2*c^(1/2)) - b^2/(8*c^(3/2))) + ((b + 
2*c*x)*(a + b*x + c*x^2)^(1/2))/(4*c)))/4 + (x*(a + b*x + c*x^2)^(3/2))/4 
+ (b*(a + b*x + c*x^2)^(3/2))/(8*c) - (3*b^2*(log((b/2 + c*x)/c^(1/2) + (a 
 + b*x + c*x^2)^(1/2))*(a/(2*c^(1/2)) - b^2/(8*c^(3/2))) + ((b + 2*c*x)*(a 
 + b*x + c*x^2)^(1/2))/(4*c)))/(16*c)))/(2*c) + (d*(b/2 + c*x)*(a + b*x + 
c*x^2)^(3/2))/(4*c)
 

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.66 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {256 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3} e -200 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2} e +400 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} d +112 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} e x +800 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} d x +512 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} e \,x^{2}+30 \sqrt {c \,x^{2}+b x +a}\, b^{4} c e -60 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} d -20 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} e x +40 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} d x +16 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} e \,x^{2}+480 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} d \,x^{2}+352 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} e \,x^{3}+320 \sqrt {c \,x^{2}+b x +a}\, c^{5} d \,x^{3}+256 \sqrt {c \,x^{2}+b x +a}\, c^{5} e \,x^{4}-240 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{2} e +480 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c^{3} d +120 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c e -240 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{2} d -15 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{5} e +30 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{4} c d}{1280 c^{4}} \] Input:

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

Output:

(256*sqrt(a + b*x + c*x**2)*a**2*c**3*e - 200*sqrt(a + b*x + c*x**2)*a*b** 
2*c**2*e + 400*sqrt(a + b*x + c*x**2)*a*b*c**3*d + 112*sqrt(a + b*x + c*x* 
*2)*a*b*c**3*e*x + 800*sqrt(a + b*x + c*x**2)*a*c**4*d*x + 512*sqrt(a + b* 
x + c*x**2)*a*c**4*e*x**2 + 30*sqrt(a + b*x + c*x**2)*b**4*c*e - 60*sqrt(a 
 + b*x + c*x**2)*b**3*c**2*d - 20*sqrt(a + b*x + c*x**2)*b**3*c**2*e*x + 4 
0*sqrt(a + b*x + c*x**2)*b**2*c**3*d*x + 16*sqrt(a + b*x + c*x**2)*b**2*c* 
*3*e*x**2 + 480*sqrt(a + b*x + c*x**2)*b*c**4*d*x**2 + 352*sqrt(a + b*x + 
c*x**2)*b*c**4*e*x**3 + 320*sqrt(a + b*x + c*x**2)*c**5*d*x**3 + 256*sqrt( 
a + b*x + c*x**2)*c**5*e*x**4 - 240*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + 
c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**2*e + 480*sqrt(c)*log(( 
2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c** 
3*d + 120*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt( 
4*a*c - b**2))*a*b**3*c*e - 240*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x* 
*2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*d - 15*sqrt(c)*log((2*sqr 
t(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*e + 30*s 
qrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b** 
2))*b**4*c*d)/(1280*c**4)