\(\int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [1958]

Optimal result

Integrand size = 31, antiderivative size = 78 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^2}+\frac {e (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)}{3 b^2}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^2} \]

[In]

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

[Out]

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

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) \, 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) \, 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) (a+b x)^2}{b}+\frac {e (a+b x)^3}{b}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^2}+\frac {e (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}{12 \left (-a^2-a b x+\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]

[In]

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

[Out]

(x*(6*a^2*(2*d + e*x) + 4*a*b*x*(3*d + 2*e*x) + b^2*x^2*(4*d + 3*e*x))*(Sqrt[a^2]*b*x + a*(Sqrt[a^2] - Sqrt[(a
+ b*x)^2])))/(12*(-a^2 - a*b*x + Sqrt[a^2]*Sqrt[(a + b*x)^2]))

Maple [C] (warning: unable to verify)

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

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.41

method	result	size
default	\(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (b x +a \right )^{3} \left (-3 b e x +a e -4 b d \right )}{12 b^{2}}\)	\(32\)
gosper	\(\frac {x \left (3 b^{2} e \,x^{3}+8 x^{2} a b e +4 x^{2} b^{2} d +6 a^{2} e x +12 a b d x +12 a^{2} d \right ) \sqrt {\left (b x +a \right )^{2}}}{12 b x +12 a}\)	\(66\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} e \,x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 a b e +b^{2} d \right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e +2 b d a \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{2} d x}{b x +a}\)	\(113\)

[In]

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

[Out]

-1/12*csgn(b*x+a)*(b*x+a)^3*(-3*b*e*x+a*e-4*b*d)/b^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.56 (sec) , antiderivative size = 393, normalized size of antiderivative = 5.04 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=a d \left (\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases}\right ) + a e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (- \frac {a^{2}}{6 b^{2}} + \frac {a x}{6 b} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {- \frac {a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}}{2 a^{2} b^{2}} & \text {for}\: a b \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) + b e \left (\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {a^{3}}{12 b^{3}} - \frac {a^{2} x}{12 b^{2}} + \frac {a x^{2}}{12 b} + \frac {x^{3}}{4}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}}{4 a^{3} b^{3}} & \text {for}\: a b \neq 0 \\\frac {x^{3} \sqrt {a^{2}}}{3} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a*d*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)) + a*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), N
e(a*b, 0)), (x**2*sqrt(a**2)/2, True)) + b*d*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)) + b*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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (52) = 104\).

Time = 0.25 (sec) , antiderivative size = 251, normalized size of antiderivative = 3.22 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e x}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d + a e\right )} a x}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e x}{4 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d + a e\right )} a^{2}}{2 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e}{12 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d + a e\right )}}{3 \, b^{2}} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (52) = 104\).

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.41 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {1}{4} \, b^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a b e x^{3} \mathrm {sgn}\left (b x + a\right ) + a b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (4 \, a^{3} b d - a^{4} e\right )} \mathrm {sgn}\left (b x + a\right )}{12 \, b^{2}} \]

[In]

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

[Out]

1/4*b^2*e*x^4*sgn(b*x + a) + 1/3*b^2*d*x^3*sgn(b*x + a) + 2/3*a*b*e*x^3*sgn(b*x + a) + a*b*d*x^2*sgn(b*x + a)
+ 1/2*a^2*e*x^2*sgn(b*x + a) + a^2*d*x*sgn(b*x + a) + 1/12*(4*a^3*b*d - a^4*e)*sgn(b*x + a)/b^2

Mupad [B] (verification not implemented)

Time = 11.17 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.81 \[ \int (a+b x) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {d\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b}-\frac {a^2\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b}-\frac {5\,a\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{96\,b^4}+\frac {a\,\left (a+b\,x\right )\,\left (3\,b\,d-a\,e+2\,b\,e\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,b^2} \]

[In]

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

[Out]

(d*(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*x*(a^2 + b^
2*x^2 + 2*a*b*x)^(3/2))/(4*b) - (a^2*e*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*b) - (5*a*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))/(96*b^4) + (a*(a + b*x)*(3*b*d - a*
e + 2*b*e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*b^2)