\(\int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [1957]

Optimal result

Integrand size = 33, antiderivative size = 125 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^3}+\frac {e (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^3}+\frac {e^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)}{2 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)^2}{3 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3} \]

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^2 \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^2 \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(b d-a e)^2 (a+b x)^2}{b^2}+\frac {2 e (b d-a e) (a+b x)^3}{b^2}+\frac {e^2 (a+b x)^4}{b^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^3}+\frac {e (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^3}+\frac {e^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (10 a^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )}{30 (a+b x)} \]

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(10*a^2*(3*d^2 + 3*d*e*x + e^2*x^2) + 5*a*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + b^2*x^2*(10
*d^2 + 15*d*e*x + 6*e^2*x^2)))/(30*(a + b*x))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.37 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (6 b^{2} e^{2} x^{2}-3 a b \,e^{2} x +15 b^{2} d e x +e^{2} a^{2}-5 a b d e +10 b^{2} d^{2}\right )}{30 b^{3}}\)	\(68\)
gosper	\(\frac {x \left (6 b^{2} e^{2} x^{4}+15 x^{3} a b \,e^{2}+15 x^{3} b^{2} d e +10 x^{2} e^{2} a^{2}+40 x^{2} a b d e +10 x^{2} b^{2} d^{2}+30 a^{2} d e x +30 b \,d^{2} a x +30 a^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{30 b x +30 a}\)	\(107\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{2} x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a b \,e^{2}+2 b^{2} d e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{2} a^{2}+4 a b d e +b^{2} d^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a^{2} d e +2 b a \,d^{2}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d^{2} x}{b x +a}\)	\(167\)

[In]

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

[Out]

1/30*csgn(b*x+a)*(b*x+a)^3*(6*b^2*e^2*x^2-3*a*b*e^2*x+15*b^2*d*e*x+a^2*e^2-5*a*b*d*e+10*b^2*d^2)/b^3

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.65 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b^{2} d e + a b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \]

[In]

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

[Out]

1/5*b^2*e^2*x^5 + a^2*d^2*x + 1/2*(b^2*d*e + a*b*e^2)*x^4 + 1/3*(b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^3 + (a*b*d^2
+ a^2*d*e)*x^2

Sympy [A] (verification not implemented)

Time = 2.10 (sec) , antiderivative size = 704, normalized size of antiderivative = 5.63 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a d^{2} \left (\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases}\right ) + 2 a d e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + a e^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) + b d^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + 2 b d e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) + b e^{2} \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{4}}{20 b^{4}} + \frac {a^{3} x}{20 b^{3}} - \frac {a^{2} x^{2}}{20 b^{2}} + \frac {a x^{3}}{20 b} + \frac {x^{4}}{5}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{6} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {3 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9}}{8 a^{4} b^{4}} & \text {for}\: a b \neq 0 \\\frac {x^{4} \sqrt {a^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a*d**2*Piecewise(((a/(2*b) + x/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2), Ne(b**2, 0)), ((a**2 + 2*a*b*x)**(3/2)/(3*
a*b), Ne(a*b, 0)), (x*sqrt(a**2), True)) + 2*a*d*e*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**2/(6*b**2)
+ a*x/(6*b) + x**2/3), Ne(b**2, 0)), ((-a**2*(a**2 + 2*a*b*x)**(3/2)/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2*a**2*b
**2), Ne(a*b, 0)), (x**2*sqrt(a**2)/2, True)) + a*e**2*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a**3/(12*b
**3) - a**2*x/(12*b**2) + a*x**2/(12*b) + x**3/4), Ne(b**2, 0)), ((a**4*(a**2 + 2*a*b*x)**(3/2)/3 - 2*a**2*(a*
*2 + 2*a*b*x)**(5/2)/5 + (a**2 + 2*a*b*x)**(7/2)/7)/(4*a**3*b**3), Ne(a*b, 0)), (x**3*sqrt(a**2)/3, True)) + b
*d**2*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(-a**2/(6*b**2) + a*x/(6*b) + x**2/3), Ne(b**2, 0)), ((-a**2
*(a**2 + 2*a*b*x)**(3/2)/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2*a**2*b**2), Ne(a*b, 0)), (x**2*sqrt(a**2)/2, True))
+ 2*b*d*e*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(a**3/(12*b**3) - a**2*x/(12*b**2) + a*x**2/(12*b) + x*
*3/4), Ne(b**2, 0)), ((a**4*(a**2 + 2*a*b*x)**(3/2)/3 - 2*a**2*(a**2 + 2*a*b*x)**(5/2)/5 + (a**2 + 2*a*b*x)**(
7/2)/7)/(4*a**3*b**3), Ne(a*b, 0)), (x**3*sqrt(a**2)/3, True)) + b*e**2*Piecewise((sqrt(a**2 + 2*a*b*x + b**2*
x**2)*(-a**4/(20*b**4) + a**3*x/(20*b**3) - a**2*x**2/(20*b**2) + a*x**3/(20*b) + x**4/5), Ne(b**2, 0)), ((-a*
*6*(a**2 + 2*a*b*x)**(3/2)/3 + 3*a**4*(a**2 + 2*a*b*x)**(5/2)/5 - 3*a**2*(a**2 + 2*a*b*x)**(7/2)/7 + (a**2 + 2
*a*b*x)**(9/2)/9)/(8*a**4*b**4), Ne(a*b, 0)), (x**4*sqrt(a**2)/4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (86) = 172\).

Time = 0.25 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.62 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{2} x - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{2} x}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{2} x^{2}}{5 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{2}}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{2}}{2 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{2} x}{20 \, b^{2}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{2}}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (2 \, b d e + a e^{2}\right )} a^{2} x}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d^{2} + 2 \, a d e\right )} a x}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (2 \, b d e + a e^{2}\right )} a^{3}}{2 \, b^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d^{2} + 2 \, a d e\right )} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d e + a e^{2}\right )} x}{4 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d e + a e^{2}\right )} a}{12 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d^{2} + 2 \, a d e\right )}}{3 \, b^{2}} \]

[In]

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

[Out]

1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*d^2*x - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3*e^2*x/b^2 + 1/5*(b^2*x^2 + 2
*a*b*x + a^2)^(3/2)*e^2*x^2/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*d^2/b - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2
)*a^4*e^2/b^3 - 7/20*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*e^2*x/b^2 + 9/20*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*e^
2/b^3 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*(2*b*d*e + a*e^2)*a^2*x/b^2 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*(b*d
^2 + 2*a*d*e)*a*x/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*(2*b*d*e + a*e^2)*a^3/b^3 - 1/2*sqrt(b^2*x^2 + 2*a*b*x
+ a^2)*(b*d^2 + 2*a*d*e)*a^2/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(2*b*d*e + a*e^2)*x/b^2 - 5/12*(b^2*x^
2 + 2*a*b*x + a^2)^(3/2)*(2*b*d*e + a*e^2)*a/b^3 + 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*d^2 + 2*a*d*e)/b^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (86) = 172\).

Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.45 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, a b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{2} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{2} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (10 \, a^{3} b^{2} d^{2} - 5 \, a^{4} b d e + a^{5} e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{30 \, b^{3}} \]

[In]

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

[Out]

1/5*b^2*e^2*x^5*sgn(b*x + a) + 1/2*b^2*d*e*x^4*sgn(b*x + a) + 1/2*a*b*e^2*x^4*sgn(b*x + a) + 1/3*b^2*d^2*x^3*s
gn(b*x + a) + 4/3*a*b*d*e*x^3*sgn(b*x + a) + 1/3*a^2*e^2*x^3*sgn(b*x + a) + a*b*d^2*x^2*sgn(b*x + a) + a^2*d*e
*x^2*sgn(b*x + a) + a^2*d^2*x*sgn(b*x + a) + 1/30*(10*a^3*b^2*d^2 - 5*a^4*b*d*e + a^5*e^2)*sgn(b*x + a)/b^3

Mupad [B] (verification not implemented)

Time = 11.21 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.50 \[ \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a\,d^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}-\frac {11\,a^2\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{160\,b^5}-\frac {a^3\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2}+\frac {d\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b}-\frac {7\,a\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{60\,b^3}+\frac {a\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {a^2\,d\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b}-\frac {a\,d\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{48\,b^4} \]

[In]

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

[Out]

a*d^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (d^2*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*
(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(24*b^3) + (e^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b) - (11*a^2*e^2*(8*b
^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(160*b^5) - (a^3*e^2*(x/2 + a/(2
*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*b^2) + (d*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(2*b) - (7*a*e^2*(a^2
+ b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*a*b*x) - 4*a^2*b*x))/(60*b^3) + (a*e^
2*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(4*b^2) - (a^2*d*e*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*b)
- (a*d*e*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(48*b^4)