3.18.8 \(\int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [1708]

Optimal. Leaf size=158 \[ \frac {(b d-a e) (B d-A e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}+\frac {b B (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)} \]

[Out]

1/5*(-a*e+b*d)*(-A*e+B*d)*(e*x+d)^5*((b*x+a)^2)^(1/2)/e^3/(b*x+a)-1/6*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^6*((b*x+a
)^2)^(1/2)/e^3/(b*x+a)+1/7*b*B*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

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Rubi [A]
time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {784, 78} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e) (B d-A e)}{5 e^3 (a+b x)}+\frac {b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^3*(a + b*x)) + (b*B*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^3*(a + b*x))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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*} \int (A+B x) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (A+B x) (d+e x)^4 \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e) (d+e x)^4}{e^2}+\frac {b (-2 b B d+A b e+a B e) (d+e x)^5}{e^2}+\frac {b^2 B (d+e x)^6}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e) (B d-A e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}+\frac {b B (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 208, normalized size = 1.32 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (7 a \left (6 A \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+B x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )+b x \left (7 A \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{210 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x*Sqrt[(a + b*x)^2]*(7*a*(6*A*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + B*x*(15*d^4 + 4
0*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + b*x*(7*A*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*
d*e^3*x^3 + 5*e^4*x^4) + 2*B*x*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4))))/(210*(a
+ b*x))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 2.
time = 0.60, size = 475, normalized size = 3.01

method	result	size
gosper	\(\frac {x \left (30 x^{6} B \,e^{4} b +35 x^{5} A b \,e^{4}+35 x^{5} B \,e^{4} a +140 x^{5} B b d \,e^{3}+42 x^{4} A a \,e^{4}+168 x^{4} A b d \,e^{3}+168 x^{4} B a d \,e^{3}+252 x^{4} B b \,d^{2} e^{2}+210 x^{3} A a d \,e^{3}+315 x^{3} A b \,d^{2} e^{2}+315 x^{3} B a \,d^{2} e^{2}+210 x^{3} B b \,d^{3} e +420 x^{2} A a \,d^{2} e^{2}+280 x^{2} A b \,d^{3} e +280 x^{2} B a \,d^{3} e +70 x^{2} B b \,d^{4}+420 x A a \,d^{3} e +105 x A \,d^{4} b +105 x B a \,d^{4}+210 A a \,d^{4}\right ) \sqrt {\left (b x +a \right )^{2}}}{210 b x +210 a}\)	\(232\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, x^{7} B \,e^{4} b}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (A b +B a \right ) e^{4}+4 B b d \,e^{3}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A a \,e^{4}+4 \left (A b +B a \right ) d \,e^{3}+6 B b \,d^{2} e^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 A a d \,e^{3}+6 \left (A b +B a \right ) d^{2} e^{2}+4 B b \,d^{3} e \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 A a \,d^{2} e^{2}+4 \left (A b +B a \right ) d^{3} e +B b \,d^{4}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 A a \,d^{3} e +\left (A b +B a \right ) d^{4}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,d^{4} a x}{b x +a}\)	\(288\)
default	\(\frac {\mathrm {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (30 B \,b^{5} e^{4} x^{5}+35 A \,b^{5} e^{4} x^{4}-25 B a \,b^{4} e^{4} x^{4}+140 B \,b^{5} d \,e^{3} x^{4}-28 A a \,b^{4} e^{4} x^{3}+168 A \,b^{5} d \,e^{3} x^{3}+20 B \,a^{2} b^{3} e^{4} x^{3}-112 B a \,b^{4} d \,e^{3} x^{3}+252 B \,b^{5} d^{2} e^{2} x^{3}+21 A \,a^{2} b^{3} e^{4} x^{2}-126 A a \,b^{4} d \,e^{3} x^{2}+315 A \,b^{5} d^{2} e^{2} x^{2}-15 B \,a^{3} b^{2} e^{4} x^{2}+84 B \,a^{2} b^{3} d \,e^{3} x^{2}-189 B a \,b^{4} d^{2} e^{2} x^{2}+210 B \,b^{5} d^{3} e \,x^{2}-14 A \,a^{3} b^{2} e^{4} x +84 A \,a^{2} b^{3} d \,e^{3} x -210 A a \,b^{4} d^{2} e^{2} x +280 A \,b^{5} d^{3} e x +10 B \,a^{4} b \,e^{4} x -56 B \,a^{3} b^{2} d \,e^{3} x +126 B \,a^{2} b^{3} d^{2} e^{2} x -140 B a \,b^{4} d^{3} e x +70 B \,b^{5} d^{4} x +7 A \,a^{4} b \,e^{4}-42 A \,a^{3} b^{2} d \,e^{3}+105 A \,a^{2} b^{3} d^{2} e^{2}-140 b^{4} A \,d^{3} e a +105 A \,b^{5} d^{4}-5 B \,a^{5} e^{4}+28 B \,a^{4} b d \,e^{3}-63 B \,a^{3} b^{2} d^{2} e^{2}+70 B \,a^{2} b^{3} d^{3} e -35 b^{4} B \,d^{4} a \right )}{210 b^{6}}\)	\(475\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*((b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/210*csgn(b*x+a)*(b*x+a)^2*(30*B*b^5*e^4*x^5+35*A*b^5*e^4*x^4-25*B*a*b^4*e^4*x^4+140*B*b^5*d*e^3*x^4-28*A*a*b
^4*e^4*x^3+168*A*b^5*d*e^3*x^3+20*B*a^2*b^3*e^4*x^3-112*B*a*b^4*d*e^3*x^3+252*B*b^5*d^2*e^2*x^3+21*A*a^2*b^3*e
^4*x^2-126*A*a*b^4*d*e^3*x^2+315*A*b^5*d^2*e^2*x^2-15*B*a^3*b^2*e^4*x^2+84*B*a^2*b^3*d*e^3*x^2-189*B*a*b^4*d^2
*e^2*x^2+210*B*b^5*d^3*e*x^2-14*A*a^3*b^2*e^4*x+84*A*a^2*b^3*d*e^3*x-210*A*a*b^4*d^2*e^2*x+280*A*b^5*d^3*e*x+1
0*B*a^4*b*e^4*x-56*B*a^3*b^2*d*e^3*x+126*B*a^2*b^3*d^2*e^2*x-140*B*a*b^4*d^3*e*x+70*B*b^5*d^4*x+7*A*a^4*b*e^4-
42*A*a^3*b^2*d*e^3+105*A*a^2*b^3*d^2*e^2-140*A*a*b^4*d^3*e+105*A*b^5*d^4-5*B*a^5*e^4+28*B*a^4*b*d*e^3-63*B*a^3
*b^2*d^2*e^2+70*B*a^2*b^3*d^3*e-35*B*a*b^4*d^4)/b^6

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 976 vs. \(2 (123) = 246\).
time = 0.30, size = 976, normalized size = 6.18 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A d^{4} x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a d^{4}}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B x^{4} e^{4}}{7 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a x^{3} e^{4}}{42 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (4 \, B d e^{3} + A e^{4}\right )} x^{3}}{6 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{5} x e^{4}}{2 \, b^{5}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{2} x^{2} e^{4}}{14 \, b^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (4 \, B d e^{3} + A e^{4}\right )} a^{4} x}{2 \, b^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{3} x}{b^{3}} + \frac {{\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x}{b^{2}} - \frac {{\left (B d^{4} + 4 \, A d^{3} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x}{2 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (4 \, B d e^{3} + A e^{4}\right )} a x^{2}}{10 \, b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x^{2}}{5 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{6} e^{4}}{2 \, b^{6}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{3} x e^{4}}{7 \, b^{5}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (4 \, B d e^{3} + A e^{4}\right )} a^{5}}{2 \, b^{5}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{4}}{b^{4}} + \frac {{\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{b^{3}} - \frac {{\left (B d^{4} + 4 \, A d^{3} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{2 \, b^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (4 \, B d e^{3} + A e^{4}\right )} a^{2} x}{5 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a x}{10 \, b^{3}} + \frac {{\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x}{2 \, b^{2}} + \frac {10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{4} e^{4}}{21 \, b^{6}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (4 \, B d e^{3} + A e^{4}\right )} a^{3}}{15 \, b^{5}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{2}}{10 \, b^{4}} - \frac {5 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{6 \, b^{3}} + \frac {{\left (B d^{4} + 4 \, A d^{3} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*((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^4*x + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*a*d^4/b + 1/7*(b^2*x^2 + 2*a*b
*x + a^2)^(3/2)*B*x^4*e^4/b^2 - 11/42*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a*x^3*e^4/b^3 + 1/6*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*(4*B*d*e^3 + A*e^4)*x^3/b^2 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^5*x*e^4/b^5 + 5/14*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*B*a^2*x^2*e^4/b^4 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*(4*B*d*e^3 + A*e^4)*a^4*x/b^4 - s
qrt(b^2*x^2 + 2*a*b*x + a^2)*(3*B*d^2*e^2 + 2*A*d*e^3)*a^3*x/b^3 + (2*B*d^3*e + 3*A*d^2*e^2)*sqrt(b^2*x^2 + 2*
a*b*x + a^2)*a^2*x/b^2 - 1/2*(B*d^4 + 4*A*d^3*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x/b - 3/10*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*(4*B*d*e^3 + A*e^4)*a*x^2/b^3 + 2/5*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(3*B*d^2*e^2 + 2*A*d*e^3)*x^
2/b^2 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^6*e^4/b^6 - 3/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^3*x*e^4/b^5
+ 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*(4*B*d*e^3 + A*e^4)*a^5/b^5 - sqrt(b^2*x^2 + 2*a*b*x + a^2)*(3*B*d^2*e^2 +
2*A*d*e^3)*a^4/b^4 + (2*B*d^3*e + 3*A*d^2*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3/b^3 - 1/2*(B*d^4 + 4*A*d^3*e
)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^2 + 2/5*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(4*B*d*e^3 + A*e^4)*a^2*x/b^4 -
7/10*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(3*B*d^2*e^2 + 2*A*d*e^3)*a*x/b^3 + 1/2*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^
2 + 2*a*b*x + a^2)^(3/2)*x/b^2 + 10/21*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^4*e^4/b^6 - 7/15*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*(4*B*d*e^3 + A*e^4)*a^3/b^5 + 9/10*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(3*B*d^2*e^2 + 2*A*d*e^3)*a^2
/b^4 - 5/6*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a/b^3 + 1/3*(B*d^4 + 4*A*d^3*e)*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)/b^2

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Fricas [A]
time = 1.76, size = 178, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, B b d^{4} x^{3} + A a d^{4} x + \frac {1}{2} \, {\left (B a + A b\right )} d^{4} x^{2} + \frac {1}{210} \, {\left (30 \, B b x^{7} + 42 \, A a x^{5} + 35 \, {\left (B a + A b\right )} x^{6}\right )} e^{4} + \frac {1}{15} \, {\left (10 \, B b d x^{6} + 15 \, A a d x^{4} + 12 \, {\left (B a + A b\right )} d x^{5}\right )} e^{3} + \frac {1}{10} \, {\left (12 \, B b d^{2} x^{5} + 20 \, A a d^{2} x^{3} + 15 \, {\left (B a + A b\right )} d^{2} x^{4}\right )} e^{2} + \frac {1}{3} \, {\left (3 \, B b d^{3} x^{4} + 6 \, A a d^{3} x^{2} + 4 \, {\left (B a + A b\right )} d^{3} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*B*b*d^4*x^3 + A*a*d^4*x + 1/2*(B*a + A*b)*d^4*x^2 + 1/210*(30*B*b*x^7 + 42*A*a*x^5 + 35*(B*a + A*b)*x^6)*e
^4 + 1/15*(10*B*b*d*x^6 + 15*A*a*d*x^4 + 12*(B*a + A*b)*d*x^5)*e^3 + 1/10*(12*B*b*d^2*x^5 + 20*A*a*d^2*x^3 + 1
5*(B*a + A*b)*d^2*x^4)*e^2 + 1/3*(3*B*b*d^3*x^4 + 6*A*a*d^3*x^2 + 4*(B*a + A*b)*d^3*x^3)*e

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Sympy [A]
time = 0.05, size = 226, normalized size = 1.43 \begin {gather*} A a d^{4} x + \frac {B b e^{4} x^{7}}{7} + x^{6} \left (\frac {A b e^{4}}{6} + \frac {B a e^{4}}{6} + \frac {2 B b d e^{3}}{3}\right ) + x^{5} \left (\frac {A a e^{4}}{5} + \frac {4 A b d e^{3}}{5} + \frac {4 B a d e^{3}}{5} + \frac {6 B b d^{2} e^{2}}{5}\right ) + x^{4} \left (A a d e^{3} + \frac {3 A b d^{2} e^{2}}{2} + \frac {3 B a d^{2} e^{2}}{2} + B b d^{3} e\right ) + x^{3} \cdot \left (2 A a d^{2} e^{2} + \frac {4 A b d^{3} e}{3} + \frac {4 B a d^{3} e}{3} + \frac {B b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a d^{3} e + \frac {A b d^{4}}{2} + \frac {B a d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*((b*x+a)**2)**(1/2),x)

[Out]

A*a*d**4*x + B*b*e**4*x**7/7 + x**6*(A*b*e**4/6 + B*a*e**4/6 + 2*B*b*d*e**3/3) + x**5*(A*a*e**4/5 + 4*A*b*d*e*
*3/5 + 4*B*a*d*e**3/5 + 6*B*b*d**2*e**2/5) + x**4*(A*a*d*e**3 + 3*A*b*d**2*e**2/2 + 3*B*a*d**2*e**2/2 + B*b*d*
*3*e) + x**3*(2*A*a*d**2*e**2 + 4*A*b*d**3*e/3 + 4*B*a*d**3*e/3 + B*b*d**4/3) + x**2*(2*A*a*d**3*e + A*b*d**4/
2 + B*a*d**4/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (123) = 246\).
time = 0.87, size = 328, normalized size = 2.08 \begin {gather*} \frac {1}{7} \, B b x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B b d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, B b d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + B b d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B b d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, B a x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, B a d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, A b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, B a d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, A b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A a x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + A a d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a d^{4} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/7*B*b*x^7*e^4*sgn(b*x + a) + 2/3*B*b*d*x^6*e^3*sgn(b*x + a) + 6/5*B*b*d^2*x^5*e^2*sgn(b*x + a) + B*b*d^3*x^4
*e*sgn(b*x + a) + 1/3*B*b*d^4*x^3*sgn(b*x + a) + 1/6*B*a*x^6*e^4*sgn(b*x + a) + 1/6*A*b*x^6*e^4*sgn(b*x + a) +
4/5*B*a*d*x^5*e^3*sgn(b*x + a) + 4/5*A*b*d*x^5*e^3*sgn(b*x + a) + 3/2*B*a*d^2*x^4*e^2*sgn(b*x + a) + 3/2*A*b*
d^2*x^4*e^2*sgn(b*x + a) + 4/3*B*a*d^3*x^3*e*sgn(b*x + a) + 4/3*A*b*d^3*x^3*e*sgn(b*x + a) + 1/2*B*a*d^4*x^2*s
gn(b*x + a) + 1/2*A*b*d^4*x^2*sgn(b*x + a) + 1/5*A*a*x^5*e^4*sgn(b*x + a) + A*a*d*x^4*e^3*sgn(b*x + a) + 2*A*a
*d^2*x^3*e^2*sgn(b*x + a) + 2*A*a*d^3*x^2*e*sgn(b*x + a) + A*a*d^4*x*sgn(b*x + a)

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Mupad [B]
time = 4.25, size = 1400, normalized size = 8.86 \begin {gather*} A\,d^4\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {A\,e^4\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b^2}+\frac {B\,e^4\,x^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{7\,b^2}+\frac {B\,d^4\,\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^4}+\frac {A\,d^3\,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}}{6\,b^4}+\frac {B\,d^3\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{b^2}-\frac {B\,a^2\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{35\,b^6}-\frac {A\,a^2\,e^4\,\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 )}{24\,b^5}+\frac {3\,A\,d^2\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b^2}+\frac {4\,A\,d\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}+\frac {2\,B\,d\,e^3\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{3\,b^2}-\frac {3\,A\,a\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{40\,b^5}-\frac {11\,B\,a\,e^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^5+5\,b^3\,x^3\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-14\,a^3\,b^2\,x^2-13\,a^4\,b\,x-9\,a\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )+12\,a^2\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{210\,b^6}+\frac {6\,B\,d^2\,e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}-\frac {B\,a^2\,d^3\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b^2}-\frac {3\,B\,a\,d\,e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{10\,b^5}-\frac {B\,a^2\,d^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}}{10\,b^6}-\frac {7\,A\,a\,d\,e^3\,\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 )}{15\,b^4}-\frac {5\,B\,a\,d^3\,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}}{24\,b^5}-\frac {3\,A\,a^2\,d^2\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b^2}-\frac {7\,B\,a\,d^2\,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 )}{10\,b^4}-\frac {B\,a^2\,d\,e^3\,\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 )}{6\,b^5}-\frac {5\,A\,a\,d^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}}{16\,b^5}-\frac {A\,a^2\,d\,e^3\,\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}}{15\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2)*(A + B*x)*(d + e*x)^4,x)

[Out]

A*d^4*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (A*e^4*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b^2) +
(B*e^4*x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(7*b^2) + (B*d^4*(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^4) + (A*d^3*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))/(6*b^4) + (B*d^3*e*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/b^2 - (B*a^2*e^4*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*
x^2 + 2*a*b*x)))/(35*b^6) - (A*a^2*e^4*(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))/(24*b^5) + (3*A*d^2*e^2*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(2*b^2) + (4*A*d*e^3*x^
2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) + (2*B*d*e^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(3*b^2) - (3*A*a*
e^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7
*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(40*b^5) - (11*B*a*e^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^5 + 5*b^3*x^3*(a^
2 + b^2*x^2 + 2*a*b*x) - 14*a^3*b^2*x^2 - 13*a^4*b*x - 9*a*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) + 12*a^2*b*x*(a^2
+ b^2*x^2 + 2*a*b*x)))/(210*b^6) + (6*B*d^2*e^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) - (B*a^2*d^3*e*(
x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/b^2 - (3*B*a*d*e^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(4*b^2*x^2*
(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(10*b^5) - (
B*a^2*d^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))/(10*b^6) - (7*
A*a*d*e^3*(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))/(
15*b^4) - (5*B*a*d^3*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))/(24*b
^5) - (3*A*a^2*d^2*e^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*b^2) - (7*B*a*d^2*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))/(10*b^4) - (B*a^2*d*e^3*
(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))/(6*b^5) - (
5*A*a*d^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))/(16*b^5) - (A*
a^2*d*e^3*(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))/(15*b^6)

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