\(\int (d+e x)^2 \sqrt {b x+c x^2} \, dx\) [130]

Optimal result

Integrand size = 21, antiderivative size = 195 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {b \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \sqrt {b x+c x^2}}{64 c^3}+\frac {1}{32} \left (16 d^2-\frac {b e (16 c d-5 b e)}{c^2}\right ) x \sqrt {b x+c x^2}+\frac {e (16 c d-5 b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e^2 x \left (b x+c x^2\right )^{3/2}}{4 c}-\frac {b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{7/2}} \] Output:

1/64*b*(5*b^2*e^2-16*b*c*d*e+16*c^2*d^2)*(c*x^2+b*x)^(1/2)/c^3+1/32*(16*d^ 
2-b*e*(-5*b*e+16*c*d)/c^2)*x*(c*x^2+b*x)^(1/2)+1/24*e*(-5*b*e+16*c*d)*(c*x 
^2+b*x)^(3/2)/c^2+1/4*e^2*x*(c*x^2+b*x)^(3/2)/c-1/64*b^2*(5*b^2*e^2-16*b*c 
*d*e+16*c^2*d^2)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(7/2)
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.89 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (15 b^3 e^2-2 b^2 c e (24 d+5 e x)+8 b c^2 \left (6 d^2+4 d e x+e^2 x^2\right )+16 c^3 x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+\frac {6 b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{192 c^{7/2}} \] Input:

Integrate[(d + e*x)^2*Sqrt[b*x + c*x^2],x]
 

Output:

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3*e^2 - 2*b^2*c*e*(24*d + 5*e*x) + 8*b*c 
^2*(6*d^2 + 4*d*e*x + e^2*x^2) + 16*c^3*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) + 
 (6*b^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[x])/(S 
qrt[b] - Sqrt[b + c*x])])/(Sqrt[x]*Sqrt[b + c*x])))/(192*c^(7/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.79, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1166, 27, 1160, 1087, 1091, 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.

\(\displaystyle \int \sqrt {b x+c x^2} (d+e x)^2 \, dx\)

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

Input:

Int[(d + e*x)^2*Sqrt[b*x + c*x^2],x]
 

Output:

(e*(d + e*x)*(b*x + c*x^2)^(3/2))/(4*c) + ((5*e*(2*c*d - b*e)*(b*x + c*x^2 
)^(3/2))/(3*c) + ((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(((b + 2*c*x)*Sqrt 
[b*x + c*x^2])/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^( 
3/2))))/(2*c))/(8*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 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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(-\frac {5 \left (b^{2} \left (b^{2} e^{2}-\frac {16}{5} b c d e +\frac {16}{5} c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\sqrt {x \left (c x +b \right )}\, \left (\frac {16 \left (\frac {1}{6} e^{2} x^{2}+\frac {2}{3} d e x +d^{2}\right ) b \,c^{\frac {5}{2}}}{5}+\frac {32 x \left (\frac {1}{2} e^{2} x^{2}+\frac {4}{3} d e x +d^{2}\right ) c^{\frac {7}{2}}}{5}+e \left (\left (-\frac {2 e x}{3}-\frac {16 d}{5}\right ) c^{\frac {3}{2}}+b e \sqrt {c}\right ) b^{2}\right )\right )}{64 c^{\frac {7}{2}}}\)	\(132\)
risch	\(\frac {\left (48 c^{3} e^{2} x^{3}+8 b \,c^{2} e^{2} x^{2}+128 c^{3} d e \,x^{2}-10 b^{2} c \,e^{2} x +32 b \,c^{2} d e x +96 c^{3} d^{2} x +15 e^{2} b^{3}-48 b^{2} c d e +48 b \,c^{2} d^{2}\right ) x \left (c x +b \right )}{192 c^{3} \sqrt {x \left (c x +b \right )}}-\frac {b^{2} \left (5 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {7}{2}}}\)	\(164\)
default	\(d^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\)	\(249\)

Input:

int((e*x+d)^2*(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-5/64/c^(7/2)*(b^2*(b^2*e^2-16/5*b*c*d*e+16/5*c^2*d^2)*arctanh((x*(c*x+b)) 
^(1/2)/x/c^(1/2))-(x*(c*x+b))^(1/2)*(16/5*(1/6*e^2*x^2+2/3*d*e*x+d^2)*b*c^ 
(5/2)+32/5*x*(1/2*e^2*x^2+4/3*d*e*x+d^2)*c^(7/2)+e*((-2/3*e*x-16/5*d)*c^(3 
/2)+b*e*c^(1/2))*b^2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.73 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\left [\frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 48 \, b^{2} c^{2} d e + 15 \, b^{3} c e^{2} + 8 \, {\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - 5 \, b^{2} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{4}}, \frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) + {\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 48 \, b^{2} c^{2} d e + 15 \, b^{3} c e^{2} + 8 \, {\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - 5 \, b^{2} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{4}}\right ] \] Input:

integrate((e*x+d)^2*(c*x^2+b*x)^(1/2),x, algorithm="fricas")
 

Output:

[1/384*(3*(16*b^2*c^2*d^2 - 16*b^3*c*d*e + 5*b^4*e^2)*sqrt(c)*log(2*c*x + 
b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(48*c^4*e^2*x^3 + 48*b*c^3*d^2 - 48*b 
^2*c^2*d*e + 15*b^3*c*e^2 + 8*(16*c^4*d*e + b*c^3*e^2)*x^2 + 2*(48*c^4*d^2 
 + 16*b*c^3*d*e - 5*b^2*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^4, 1/192*(3*(16*b 
^2*c^2*d^2 - 16*b^3*c*d*e + 5*b^4*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*s 
qrt(-c)/(c*x + b)) + (48*c^4*e^2*x^3 + 48*b*c^3*d^2 - 48*b^2*c^2*d*e + 15* 
b^3*c*e^2 + 8*(16*c^4*d*e + b*c^3*e^2)*x^2 + 2*(48*c^4*d^2 + 16*b*c^3*d*e 
- 5*b^2*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^4]
 

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.46 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\begin {cases} - \frac {b \left (b d^{2} - \frac {3 b \left (2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{4 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 c} + \sqrt {b x + c x^{2}} \left (\frac {e^{2} x^{3}}{4} + \frac {x^{2} \left (\frac {b e^{2}}{8} + 2 c d e\right )}{3 c} + \frac {x \left (2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{2 c} + \frac {b d^{2} - \frac {3 b \left (2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d^{2} \left (b x\right )^{\frac {3}{2}}}{3} + \frac {2 d e \left (b x\right )^{\frac {5}{2}}}{5 b} + \frac {e^{2} \left (b x\right )^{\frac {7}{2}}}{7 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**2*(c*x**2+b*x)**(1/2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.45 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {1}{2} \, \sqrt {c x^{2} + b x} d^{2} x - \frac {\sqrt {c x^{2} + b x} b d e x}{2 \, c} + \frac {5 \, \sqrt {c x^{2} + b x} b^{2} e^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} e^{2} x}{4 \, c} - \frac {b^{2} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {b^{3} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {5 \, b^{4} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {\sqrt {c x^{2} + b x} b d^{2}}{4 \, c} - \frac {\sqrt {c x^{2} + b x} b^{2} d e}{4 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d e}{3 \, c} + \frac {5 \, \sqrt {c x^{2} + b x} b^{3} e^{2}}{64 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b e^{2}}{24 \, c^{2}} \] Input:

integrate((e*x+d)^2*(c*x^2+b*x)^(1/2),x, algorithm="maxima")
 

Output:

1/2*sqrt(c*x^2 + b*x)*d^2*x - 1/2*sqrt(c*x^2 + b*x)*b*d*e*x/c + 5/32*sqrt( 
c*x^2 + b*x)*b^2*e^2*x/c^2 + 1/4*(c*x^2 + b*x)^(3/2)*e^2*x/c - 1/8*b^2*d^2 
*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(3/2) + 1/8*b^3*d*e*log(2* 
c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 5/128*b^4*e^2*log(2*c*x + 
 b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 1/4*sqrt(c*x^2 + b*x)*b*d^2/c 
- 1/4*sqrt(c*x^2 + b*x)*b^2*d*e/c^2 + 2/3*(c*x^2 + b*x)^(3/2)*d*e/c + 5/64 
*sqrt(c*x^2 + b*x)*b^3*e^2/c^3 - 5/24*(c*x^2 + b*x)^(3/2)*b*e^2/c^2
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.88 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, e^{2} x + \frac {16 \, c^{3} d e + b c^{2} e^{2}}{c^{3}}\right )} x + \frac {48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2}}{c^{3}}\right )} x + \frac {3 \, {\left (16 \, b c^{2} d^{2} - 16 \, b^{2} c d e + 5 \, b^{3} e^{2}\right )}}{c^{3}}\right )} + \frac {{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {7}{2}}} \] Input:

integrate((e*x+d)^2*(c*x^2+b*x)^(1/2),x, algorithm="giac")
 

Output:

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*e^2*x + (16*c^3*d*e + b*c^2*e^2)/c^3)*x + 
 (48*c^3*d^2 + 16*b*c^2*d*e - 5*b^2*c*e^2)/c^3)*x + 3*(16*b*c^2*d^2 - 16*b 
^2*c*d*e + 5*b^3*e^2)/c^3) + 1/128*(16*b^2*c^2*d^2 - 16*b^3*c*d*e + 5*b^4* 
e^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(7/2)
 

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.18 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=d^2\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {5\,b\,e^2\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}-\frac {b^2\,d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {e^2\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}+\frac {b^3\,d\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{8\,c^{5/2}}+\frac {d\,e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{12\,c^2} \] Input:

int((b*x + c*x^2)^(1/2)*(d + e*x)^2,x)
 

Output:

d^2*(b*x + c*x^2)^(1/2)*(x/2 + b/(4*c)) - (5*b*e^2*((b^3*log((b + 2*c*x)/c 
^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^ 
2*x^2 - 3*b^2 + 2*b*c*x))/(24*c^2)))/(8*c) - (b^2*d^2*log((b/2 + c*x)/c^(1 
/2) + (b*x + c*x^2)^(1/2)))/(8*c^(3/2)) + (e^2*x*(b*x + c*x^2)^(3/2))/(4*c 
) + (b^3*d*e*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(8*c^(5/2)) 
 + (d*e*(b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2 + 2*b*c*x))/(12*c^2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.31 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {15 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c \,e^{2}-48 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{2} d e -10 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{2} e^{2} x +48 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{3} d^{2}+32 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{3} d e x +8 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{3} e^{2} x^{2}+96 \sqrt {x}\, \sqrt {c x +b}\, c^{4} d^{2} x +128 \sqrt {x}\, \sqrt {c x +b}\, c^{4} d e \,x^{2}+48 \sqrt {x}\, \sqrt {c x +b}\, c^{4} e^{2} x^{3}-15 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{4} e^{2}+48 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{3} c d e -48 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{2} c^{2} d^{2}}{192 c^{4}} \] Input:

int((e*x+d)^2*(c*x^2+b*x)^(1/2),x)
 

Output:

(15*sqrt(x)*sqrt(b + c*x)*b**3*c*e**2 - 48*sqrt(x)*sqrt(b + c*x)*b**2*c**2 
*d*e - 10*sqrt(x)*sqrt(b + c*x)*b**2*c**2*e**2*x + 48*sqrt(x)*sqrt(b + c*x 
)*b*c**3*d**2 + 32*sqrt(x)*sqrt(b + c*x)*b*c**3*d*e*x + 8*sqrt(x)*sqrt(b + 
 c*x)*b*c**3*e**2*x**2 + 96*sqrt(x)*sqrt(b + c*x)*c**4*d**2*x + 128*sqrt(x 
)*sqrt(b + c*x)*c**4*d*e*x**2 + 48*sqrt(x)*sqrt(b + c*x)*c**4*e**2*x**3 - 
15*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**4*e**2 + 48*s 
qrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**3*c*d*e - 48*sqrt 
(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**2*c**2*d**2)/(192*c* 
*4)