3.15.89 \(\int (A+B x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=158 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e) (B d-A e)}{4 e^3 (a+b x)}+\frac {b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^3 (a+b x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.14, 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 = {770, 77} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e) (B d-A e)}{4 e^3 (a+b x)}+\frac {b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)) + (b*B*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(6*e^3*(a + b*x))

Rule 77

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 770

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 163, normalized size = 1.03 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (3 a \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )\right )}{60 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x*Sqrt[(a + b*x)^2]*(3*a*(5*A*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + B*x*(10*d^3 + 20*d^2*e*x + 15*d*e
^2*x^2 + 4*e^3*x^3)) + b*x*(3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + B*x*(20*d^3 + 45*d^2*e*x +
36*d*e^2*x^2 + 10*e^3*x^3))))/(60*(a + b*x))

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 2.11, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

Defer[IntegrateAlgebraic][(A + B*x)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2], x]

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 134, normalized size = 0.85 \begin {gather*} \frac {1}{6} \, B b e^{3} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B b d e^{2} + {\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B b d^{2} e + A a e^{3} + 3 \, {\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{3} + 3 \, A a d e^{2} + 3 \, {\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a d^{2} e + {\left (B a + A b\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.18, size = 255, normalized size = 1.61 \begin {gather*} \frac {1}{6} \, B b x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, B b d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B b d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B b d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B a x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, A b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + B a d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + A b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A a x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + A a d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a d^{3} 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)^3*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 180, normalized size = 1.14 \begin {gather*} \frac {\left (10 b B \,e^{3} x^{5}+12 x^{4} A b \,e^{3}+12 x^{4} B a \,e^{3}+36 x^{4} b B d \,e^{2}+15 x^{3} a A \,e^{3}+45 x^{3} A b d \,e^{2}+45 x^{3} a B d \,e^{2}+45 x^{3} b B \,d^{2} e +60 x^{2} A a d \,e^{2}+60 x^{2} A b \,d^{2} e +60 x^{2} B a \,d^{2} e +20 x^{2} b B \,d^{3}+90 x A a \,d^{2} e +30 x A b \,d^{3}+30 x B a \,d^{3}+60 A a \,d^{3}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{60 b x +60 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3*((b*x+a)^2)^(1/2),x)

[Out]

1/60*x*(10*B*b*e^3*x^5+12*A*b*e^3*x^4+12*B*a*e^3*x^4+36*B*b*d*e^2*x^4+15*A*a*e^3*x^3+45*A*b*d*e^2*x^3+45*B*a*d
*e^2*x^3+45*B*b*d^2*e*x^3+60*A*a*d*e^2*x^2+60*A*b*d^2*e*x^2+60*B*a*d^2*e*x^2+20*B*b*d^3*x^2+90*A*a*d^2*e*x+30*
A*b*d^3*x+30*B*a*d^3*x+60*A*a*d^3)*((b*x+a)^2)^(1/2)/(b*x+a)

________________________________________________________________________________________

maxima [B]  time = 0.62, size = 698, normalized size = 4.42 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B e^{3} x^{3}}{6 \, b^{2}} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A d^{3} x + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{4} e^{3} x}{2 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a e^{3} x^{2}}{10 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a d^{3}}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{5} e^{3}}{2 \, b^{5}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{2} e^{3} x}{5 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{3} e^{3}}{15 \, b^{5}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} x}{2 \, b^{3}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x}{2 \, b} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}}{2 \, b^{4}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{2 \, b^{2}} - \frac {7 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x}{4 \, b^{2}} + \frac {9 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{20 \, b^{4}} - \frac {5 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{4 \, b^{3}} + \frac {{\left (B d^{3} + 3 \, A d^{2} 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)^3*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*e^3*x^3/b^2 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*d^3*x + 1/2*sqrt(b^2*x
^2 + 2*a*b*x + a^2)*B*a^4*e^3*x/b^4 - 3/10*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a*e^3*x^2/b^3 + 1/2*sqrt(b^2*x^2
+ 2*a*b*x + a^2)*A*a*d^3/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^5*e^3/b^5 + 2/5*(b^2*x^2 + 2*a*b*x + a^2)^(
3/2)*B*a^2*e^3*x/b^4 - 7/15*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^3*e^3/b^5 - 1/2*(3*B*d*e^2 + A*e^3)*sqrt(b^2*x
^2 + 2*a*b*x + a^2)*a^3*x/b^3 + 3/2*(B*d^2*e + A*d*e^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*x/b^2 - 1/2*(B*d^3 +
3*A*d^2*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x/b + 1/5*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2/
b^2 - 1/2*(3*B*d*e^2 + A*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4/b^4 + 3/2*(B*d^2*e + A*d*e^2)*sqrt(b^2*x^2 + 2
*a*b*x + a^2)*a^3/b^3 - 1/2*(B*d^3 + 3*A*d^2*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^2 - 7/20*(3*B*d*e^2 + A*e^
3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b^3 + 3/4*(B*d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x/b^2 + 9
/20*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^4 - 5/4*(B*d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*a/b^3 + 1/3*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)/b^2

________________________________________________________________________________________

mupad [B]  time = 3.32, size = 935, normalized size = 5.92 \begin {gather*} A\,d^3\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {A\,e^3\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}+\frac {B\,e^3\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b^2}+\frac {B\,d^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}}{24\,b^4}+\frac {A\,d^2\,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}}{8\,b^4}+\frac {3\,A\,d\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}+\frac {3\,B\,d^2\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {B\,a^2\,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 )}{24\,b^5}-\frac {A\,a^2\,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}}{60\,b^6}+\frac {3\,B\,d\,e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}-\frac {3\,B\,a\,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 )}{40\,b^5}-\frac {7\,A\,a\,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 )}{60\,b^4}-\frac {3\,A\,a^2\,d\,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 {3\,B\,a^2\,d^2\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2}-\frac {7\,B\,a\,d\,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 )}{20\,b^4}-\frac {5\,A\,a\,d\,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}}{32\,b^5}-\frac {5\,B\,a\,d^2\,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}}{32\,b^5}-\frac {B\,a^2\,d\,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}}{20\,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)^3,x)

[Out]

A*d^3*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (A*e^3*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) +
(B*e^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b^2) + (B*d^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))/(24*b^4) + (A*d^2*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))/(8*b^4) + (3*A*d*e^2*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(4*b^2) + (3*B*d^2*e*x*(a^2 +
b^2*x^2 + 2*a*b*x)^(3/2))/(4*b^2) - (B*a^2*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))/(24*b^5) - (A*a^2*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))/(60*b^6) + (3*B*d*e^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(5*b^2) - (3*B*a*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)))/(40*b^5) - (7*A*a*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))/(60*b^4) - (3*A*a^2*d*e^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)
^(1/2))/(4*b^2) - (3*B*a^2*d^2*e*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*b^2) - (7*B*a*d*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))/(20*b^4) - (5*A*
a*d*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))/(32*b^5) - (5*B*a*d^
2*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))/(32*b^5) - (B*a^2*d*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))/(20*b^6)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 168, normalized size = 1.06 \begin {gather*} A a d^{3} x + \frac {B b e^{3} x^{6}}{6} + x^{5} \left (\frac {A b e^{3}}{5} + \frac {B a e^{3}}{5} + \frac {3 B b d e^{2}}{5}\right ) + x^{4} \left (\frac {A a e^{3}}{4} + \frac {3 A b d e^{2}}{4} + \frac {3 B a d e^{2}}{4} + \frac {3 B b d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + A b d^{2} e + B a d^{2} e + \frac {B b d^{3}}{3}\right ) + x^{2} \left (\frac {3 A a d^{2} e}{2} + \frac {A b d^{3}}{2} + \frac {B a d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*((b*x+a)**2)**(1/2),x)

[Out]

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

________________________________________________________________________________________