3.1739 \(\int (A+B x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=383 \[ \frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (-6 a B e+A b e+5 b B d)}{11 b^7}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2 (-2 a B e+A b e+b B d)}{9 b^7}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{8 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{7 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^5}{6 b^7}+\frac {B e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^7} \]

[Out]

1/6*(A*b-B*a)*(-a*e+b*d)^5*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^7+1/7*(-a*e+b*d)^4*(5*A*b*e-6*B*a*e+B*b*d)*(b*x+a)^6*
((b*x+a)^2)^(1/2)/b^7+5/8*e*(-a*e+b*d)^3*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^7+10/9*e^2*(-a*
e+b*d)^2*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^7+1/2*e^3*(-a*e+b*d)*(A*b*e-3*B*a*e+2*B*b*d)*(b*x
+a)^9*((b*x+a)^2)^(1/2)/b^7+1/11*e^4*(A*b*e-6*B*a*e+5*B*b*d)*(b*x+a)^10*((b*x+a)^2)^(1/2)/b^7+1/12*B*e^5*(b*x+
a)^11*((b*x+a)^2)^(1/2)/b^7

________________________________________________________________________________________

Rubi [A]  time = 0.79, antiderivative size = 383, 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} \[ \frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (-6 a B e+A b e+5 b B d)}{11 b^7}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2 (-2 a B e+A b e+b B d)}{9 b^7}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{8 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{7 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^5}{6 b^7}+\frac {B e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^7} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((A*b - a*B)*(b*d - a*e)^5*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^7) + ((b*d - a*e)^4*(b*B*d + 5*A*b*
e - 6*a*B*e)*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^7) + (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*
e)*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*b^7) + (10*e^2*(b*d - a*e)^2*(b*B*d + A*b*e - 2*a*B*e)*(a + b
*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^7) + (e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^9*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(2*b^7) + (e^4*(5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(11*b^7) + (B*e^5*(a + b*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*b^7)

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)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) (d+e x)^5 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(A b-a B) (b d-a e)^5 \left (a b+b^2 x\right )^5}{b^6}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) \left (a b+b^2 x\right )^6}{b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \left (a b+b^2 x\right )^7}{b^8}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) \left (a b+b^2 x\right )^8}{b^9}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) \left (a b+b^2 x\right )^9}{b^{10}}+\frac {e^4 (5 b B d+A b e-6 a B e) \left (a b+b^2 x\right )^{10}}{b^{11}}+\frac {B e^5 \left (a b+b^2 x\right )^{11}}{b^{12}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^5 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^7}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^7}+\frac {e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^7}+\frac {B e^5 (a+b x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{12 b^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 740, normalized size = 1.93 \[ \frac {x \sqrt {(a+b x)^2} \left (132 a^5 \left (7 A \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+B x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )\right )+165 a^4 b x \left (4 A \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+B x \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )\right )+110 a^3 b^2 x^2 \left (3 A \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+B x \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right )+22 a^2 b^3 x^3 \left (5 A \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+2 B x \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )+2 a b^4 x^4 \left (11 A \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+5 B x \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )\right )+b^5 x^5 \left (A \left (924 d^5+3960 d^4 e x+6930 d^3 e^2 x^2+6160 d^2 e^3 x^3+2772 d e^4 x^4+504 e^5 x^5\right )+B x \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )\right )\right )}{5544 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(132*a^5*(7*A*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x
^5) + B*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)) + 165*a^4*b*x*(
4*A*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + B*x*(56*d^5 + 210*d^
4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5)) + 110*a^3*b^2*x^2*(3*A*(56*d^5 + 210*
d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + B*x*(126*d^5 + 504*d^4*e*x + 840*d
^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)) + 22*a^2*b^3*x^3*(5*A*(126*d^5 + 504*d^4*e*x + 840
*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 2*B*x*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^
2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5)) + 2*a*b^4*x^4*(11*A*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^
2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 5*B*x*(462*d^5 + 1980*d^4*e*x + 3465*d^3*e^2*x^2 + 3
080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5)) + b^5*x^5*(B*x*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5
544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + A*(924*d^5 + 3960*d^4*e*x + 6930*d^3*e^2*x^2 + 6160*d^2*e^3*
x^3 + 2772*d*e^4*x^4 + 504*e^5*x^5))))/(5544*(a + b*x))

________________________________________________________________________________________

fricas [B]  time = 0.88, size = 813, normalized size = 2.12 \[ \frac {1}{12} \, B b^{5} e^{5} x^{12} + A a^{5} d^{5} x + \frac {1}{11} \, {\left (5 \, B b^{5} d e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B b^{5} d^{2} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B b^{5} d^{3} e^{2} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{4} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (B b^{5} d^{4} e + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{4} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{5} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e + 50 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{2} + 100 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{3} + 25 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (A a^{5} e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} + 25 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e + 100 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{2} + 50 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{4}\right )} x^{6} + {\left (A a^{5} d e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{2} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{3}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, A a^{5} d^{2} e^{3} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} e + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e^{2}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, A a^{5} d^{3} e^{2} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{5} d^{4} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{5}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*B*b^5*e^5*x^12 + A*a^5*d^5*x + 1/11*(5*B*b^5*d*e^4 + (5*B*a*b^4 + A*b^5)*e^5)*x^11 + 1/2*(2*B*b^5*d^2*e^3
+ (5*B*a*b^4 + A*b^5)*d*e^4 + (2*B*a^2*b^3 + A*a*b^4)*e^5)*x^10 + 5/9*(2*B*b^5*d^3*e^2 + 2*(5*B*a*b^4 + A*b^5
)*d^2*e^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*d*e^4 + 2*(B*a^3*b^2 + A*a^2*b^3)*e^5)*x^9 + 5/8*(B*b^5*d^4*e + 2*(5*B*a
*b^4 + A*b^5)*d^3*e^2 + 10*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*d*e^4 + (B*a^4*b + 2*A
*a^3*b^2)*e^5)*x^8 + 1/7*(B*b^5*d^5 + 5*(5*B*a*b^4 + A*b^5)*d^4*e + 50*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^2 + 100*(
B*a^3*b^2 + A*a^2*b^3)*d^2*e^3 + 25*(B*a^4*b + 2*A*a^3*b^2)*d*e^4 + (B*a^5 + 5*A*a^4*b)*e^5)*x^7 + 1/6*(A*a^5*
e^5 + (5*B*a*b^4 + A*b^5)*d^5 + 25*(2*B*a^2*b^3 + A*a*b^4)*d^4*e + 100*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^2 + 50*(B
*a^4*b + 2*A*a^3*b^2)*d^2*e^3 + 5*(B*a^5 + 5*A*a^4*b)*d*e^4)*x^6 + (A*a^5*d*e^4 + (2*B*a^2*b^3 + A*a*b^4)*d^5
+ 10*(B*a^3*b^2 + A*a^2*b^3)*d^4*e + 10*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^2 + 2*(B*a^5 + 5*A*a^4*b)*d^2*e^3)*x^5 +
5/4*(2*A*a^5*d^2*e^3 + 2*(B*a^3*b^2 + A*a^2*b^3)*d^5 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^4*e + 2*(B*a^5 + 5*A*a^4*b
)*d^3*e^2)*x^4 + 5/3*(2*A*a^5*d^3*e^2 + (B*a^4*b + 2*A*a^3*b^2)*d^5 + (B*a^5 + 5*A*a^4*b)*d^4*e)*x^3 + 1/2*(5*
A*a^5*d^4*e + (B*a^5 + 5*A*a^4*b)*d^5)*x^2

________________________________________________________________________________________

giac [B]  time = 0.28, size = 1446, normalized size = 3.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/12*B*b^5*x^12*e^5*sgn(b*x + a) + 5/11*B*b^5*d*x^11*e^4*sgn(b*x + a) + B*b^5*d^2*x^10*e^3*sgn(b*x + a) + 10/9
*B*b^5*d^3*x^9*e^2*sgn(b*x + a) + 5/8*B*b^5*d^4*x^8*e*sgn(b*x + a) + 1/7*B*b^5*d^5*x^7*sgn(b*x + a) + 5/11*B*a
*b^4*x^11*e^5*sgn(b*x + a) + 1/11*A*b^5*x^11*e^5*sgn(b*x + a) + 5/2*B*a*b^4*d*x^10*e^4*sgn(b*x + a) + 1/2*A*b^
5*d*x^10*e^4*sgn(b*x + a) + 50/9*B*a*b^4*d^2*x^9*e^3*sgn(b*x + a) + 10/9*A*b^5*d^2*x^9*e^3*sgn(b*x + a) + 25/4
*B*a*b^4*d^3*x^8*e^2*sgn(b*x + a) + 5/4*A*b^5*d^3*x^8*e^2*sgn(b*x + a) + 25/7*B*a*b^4*d^4*x^7*e*sgn(b*x + a) +
5/7*A*b^5*d^4*x^7*e*sgn(b*x + a) + 5/6*B*a*b^4*d^5*x^6*sgn(b*x + a) + 1/6*A*b^5*d^5*x^6*sgn(b*x + a) + B*a^2*
b^3*x^10*e^5*sgn(b*x + a) + 1/2*A*a*b^4*x^10*e^5*sgn(b*x + a) + 50/9*B*a^2*b^3*d*x^9*e^4*sgn(b*x + a) + 25/9*A
*a*b^4*d*x^9*e^4*sgn(b*x + a) + 25/2*B*a^2*b^3*d^2*x^8*e^3*sgn(b*x + a) + 25/4*A*a*b^4*d^2*x^8*e^3*sgn(b*x + a
) + 100/7*B*a^2*b^3*d^3*x^7*e^2*sgn(b*x + a) + 50/7*A*a*b^4*d^3*x^7*e^2*sgn(b*x + a) + 25/3*B*a^2*b^3*d^4*x^6*
e*sgn(b*x + a) + 25/6*A*a*b^4*d^4*x^6*e*sgn(b*x + a) + 2*B*a^2*b^3*d^5*x^5*sgn(b*x + a) + A*a*b^4*d^5*x^5*sgn(
b*x + a) + 10/9*B*a^3*b^2*x^9*e^5*sgn(b*x + a) + 10/9*A*a^2*b^3*x^9*e^5*sgn(b*x + a) + 25/4*B*a^3*b^2*d*x^8*e^
4*sgn(b*x + a) + 25/4*A*a^2*b^3*d*x^8*e^4*sgn(b*x + a) + 100/7*B*a^3*b^2*d^2*x^7*e^3*sgn(b*x + a) + 100/7*A*a^
2*b^3*d^2*x^7*e^3*sgn(b*x + a) + 50/3*B*a^3*b^2*d^3*x^6*e^2*sgn(b*x + a) + 50/3*A*a^2*b^3*d^3*x^6*e^2*sgn(b*x
+ a) + 10*B*a^3*b^2*d^4*x^5*e*sgn(b*x + a) + 10*A*a^2*b^3*d^4*x^5*e*sgn(b*x + a) + 5/2*B*a^3*b^2*d^5*x^4*sgn(b
*x + a) + 5/2*A*a^2*b^3*d^5*x^4*sgn(b*x + a) + 5/8*B*a^4*b*x^8*e^5*sgn(b*x + a) + 5/4*A*a^3*b^2*x^8*e^5*sgn(b*
x + a) + 25/7*B*a^4*b*d*x^7*e^4*sgn(b*x + a) + 50/7*A*a^3*b^2*d*x^7*e^4*sgn(b*x + a) + 25/3*B*a^4*b*d^2*x^6*e^
3*sgn(b*x + a) + 50/3*A*a^3*b^2*d^2*x^6*e^3*sgn(b*x + a) + 10*B*a^4*b*d^3*x^5*e^2*sgn(b*x + a) + 20*A*a^3*b^2*
d^3*x^5*e^2*sgn(b*x + a) + 25/4*B*a^4*b*d^4*x^4*e*sgn(b*x + a) + 25/2*A*a^3*b^2*d^4*x^4*e*sgn(b*x + a) + 5/3*B
*a^4*b*d^5*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^5*x^3*sgn(b*x + a) + 1/7*B*a^5*x^7*e^5*sgn(b*x + a) + 5/7*A*a^4
*b*x^7*e^5*sgn(b*x + a) + 5/6*B*a^5*d*x^6*e^4*sgn(b*x + a) + 25/6*A*a^4*b*d*x^6*e^4*sgn(b*x + a) + 2*B*a^5*d^2
*x^5*e^3*sgn(b*x + a) + 10*A*a^4*b*d^2*x^5*e^3*sgn(b*x + a) + 5/2*B*a^5*d^3*x^4*e^2*sgn(b*x + a) + 25/2*A*a^4*
b*d^3*x^4*e^2*sgn(b*x + a) + 5/3*B*a^5*d^4*x^3*e*sgn(b*x + a) + 25/3*A*a^4*b*d^4*x^3*e*sgn(b*x + a) + 1/2*B*a^
5*d^5*x^2*sgn(b*x + a) + 5/2*A*a^4*b*d^5*x^2*sgn(b*x + a) + 1/6*A*a^5*x^6*e^5*sgn(b*x + a) + A*a^5*d*x^5*e^4*s
gn(b*x + a) + 5/2*A*a^5*d^2*x^4*e^3*sgn(b*x + a) + 10/3*A*a^5*d^3*x^3*e^2*sgn(b*x + a) + 5/2*A*a^5*d^4*x^2*e*s
gn(b*x + a) + A*a^5*d^5*x*sgn(b*x + a)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 1068, normalized size = 2.79 \[ \frac {\left (462 B \,e^{5} b^{5} x^{11}+504 x^{10} A \,b^{5} e^{5}+2520 x^{10} B \,e^{5} a \,b^{4}+2520 x^{10} B \,b^{5} d \,e^{4}+2772 x^{9} A a \,b^{4} e^{5}+2772 x^{9} A \,b^{5} d \,e^{4}+5544 x^{9} B \,e^{5} a^{2} b^{3}+13860 x^{9} B a \,b^{4} d \,e^{4}+5544 x^{9} B \,b^{5} d^{2} e^{3}+6160 x^{8} A \,a^{2} b^{3} e^{5}+15400 x^{8} A a \,b^{4} d \,e^{4}+6160 x^{8} A \,b^{5} d^{2} e^{3}+6160 x^{8} B \,e^{5} a^{3} b^{2}+30800 x^{8} B \,a^{2} b^{3} d \,e^{4}+30800 x^{8} B a \,b^{4} d^{2} e^{3}+6160 x^{8} B \,b^{5} d^{3} e^{2}+6930 x^{7} A \,a^{3} b^{2} e^{5}+34650 x^{7} A \,a^{2} b^{3} d \,e^{4}+34650 x^{7} A a \,b^{4} d^{2} e^{3}+6930 x^{7} A \,b^{5} d^{3} e^{2}+3465 x^{7} B \,e^{5} a^{4} b +34650 x^{7} B \,a^{3} b^{2} d \,e^{4}+69300 x^{7} B \,a^{2} b^{3} d^{2} e^{3}+34650 x^{7} B a \,b^{4} d^{3} e^{2}+3465 x^{7} B \,b^{5} d^{4} e +3960 x^{6} A \,a^{4} b \,e^{5}+39600 x^{6} A \,a^{3} b^{2} d \,e^{4}+79200 x^{6} A \,a^{2} b^{3} d^{2} e^{3}+39600 x^{6} A a \,b^{4} d^{3} e^{2}+3960 x^{6} A \,b^{5} d^{4} e +792 x^{6} B \,e^{5} a^{5}+19800 x^{6} B \,a^{4} b d \,e^{4}+79200 x^{6} B \,a^{3} b^{2} d^{2} e^{3}+79200 x^{6} B \,a^{2} b^{3} d^{3} e^{2}+19800 x^{6} B a \,b^{4} d^{4} e +792 x^{6} B \,b^{5} d^{5}+924 x^{5} A \,a^{5} e^{5}+23100 x^{5} A \,a^{4} b d \,e^{4}+92400 x^{5} A \,a^{3} b^{2} d^{2} e^{3}+92400 x^{5} A \,a^{2} b^{3} d^{3} e^{2}+23100 x^{5} A a \,b^{4} d^{4} e +924 x^{5} A \,d^{5} b^{5}+4620 x^{5} B \,a^{5} d \,e^{4}+46200 x^{5} B \,a^{4} b \,d^{2} e^{3}+92400 x^{5} B \,a^{3} b^{2} d^{3} e^{2}+46200 x^{5} B \,a^{2} b^{3} d^{4} e +4620 x^{5} B a \,b^{4} d^{5}+5544 A \,a^{5} d \,e^{4} x^{4}+55440 A \,a^{4} b \,d^{2} e^{3} x^{4}+110880 A \,a^{3} b^{2} d^{3} e^{2} x^{4}+55440 A \,a^{2} b^{3} d^{4} e \,x^{4}+5544 A a \,b^{4} d^{5} x^{4}+11088 B \,a^{5} d^{2} e^{3} x^{4}+55440 B \,a^{4} b \,d^{3} e^{2} x^{4}+55440 B \,a^{3} b^{2} d^{4} e \,x^{4}+11088 B \,a^{2} b^{3} d^{5} x^{4}+13860 x^{3} A \,a^{5} d^{2} e^{3}+69300 x^{3} A \,a^{4} b \,d^{3} e^{2}+69300 x^{3} A \,a^{3} b^{2} d^{4} e +13860 x^{3} A \,d^{5} a^{2} b^{3}+13860 x^{3} B \,a^{5} d^{3} e^{2}+34650 x^{3} B \,a^{4} b \,d^{4} e +13860 x^{3} B \,a^{3} b^{2} d^{5}+18480 x^{2} A \,a^{5} d^{3} e^{2}+46200 x^{2} A \,a^{4} b \,d^{4} e +18480 x^{2} A \,d^{5} a^{3} b^{2}+9240 x^{2} B \,a^{5} d^{4} e +9240 x^{2} B \,a^{4} b \,d^{5}+13860 x A \,a^{5} d^{4} e +13860 x A \,d^{5} a^{4} b +2772 x B \,a^{5} d^{5}+5544 A \,d^{5} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{5544 \left (b x +a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/5544*x*(462*B*b^5*e^5*x^11+504*A*b^5*e^5*x^10+2520*B*a*b^4*e^5*x^10+2520*B*b^5*d*e^4*x^10+2772*A*a*b^4*e^5*x
^9+2772*A*b^5*d*e^4*x^9+5544*B*a^2*b^3*e^5*x^9+13860*B*a*b^4*d*e^4*x^9+5544*B*b^5*d^2*e^3*x^9+6160*A*a^2*b^3*e
^5*x^8+15400*A*a*b^4*d*e^4*x^8+6160*A*b^5*d^2*e^3*x^8+6160*B*a^3*b^2*e^5*x^8+30800*B*a^2*b^3*d*e^4*x^8+30800*B
*a*b^4*d^2*e^3*x^8+6160*B*b^5*d^3*e^2*x^8+6930*A*a^3*b^2*e^5*x^7+34650*A*a^2*b^3*d*e^4*x^7+34650*A*a*b^4*d^2*e
^3*x^7+6930*A*b^5*d^3*e^2*x^7+3465*B*a^4*b*e^5*x^7+34650*B*a^3*b^2*d*e^4*x^7+69300*B*a^2*b^3*d^2*e^3*x^7+34650
*B*a*b^4*d^3*e^2*x^7+3465*B*b^5*d^4*e*x^7+3960*A*a^4*b*e^5*x^6+39600*A*a^3*b^2*d*e^4*x^6+79200*A*a^2*b^3*d^2*e
^3*x^6+39600*A*a*b^4*d^3*e^2*x^6+3960*A*b^5*d^4*e*x^6+792*B*a^5*e^5*x^6+19800*B*a^4*b*d*e^4*x^6+79200*B*a^3*b^
2*d^2*e^3*x^6+79200*B*a^2*b^3*d^3*e^2*x^6+19800*B*a*b^4*d^4*e*x^6+792*B*b^5*d^5*x^6+924*A*a^5*e^5*x^5+23100*A*
a^4*b*d*e^4*x^5+92400*A*a^3*b^2*d^2*e^3*x^5+92400*A*a^2*b^3*d^3*e^2*x^5+23100*A*a*b^4*d^4*e*x^5+924*A*b^5*d^5*
x^5+4620*B*a^5*d*e^4*x^5+46200*B*a^4*b*d^2*e^3*x^5+92400*B*a^3*b^2*d^3*e^2*x^5+46200*B*a^2*b^3*d^4*e*x^5+4620*
B*a*b^4*d^5*x^5+5544*A*a^5*d*e^4*x^4+55440*A*a^4*b*d^2*e^3*x^4+110880*A*a^3*b^2*d^3*e^2*x^4+55440*A*a^2*b^3*d^
4*e*x^4+5544*A*a*b^4*d^5*x^4+11088*B*a^5*d^2*e^3*x^4+55440*B*a^4*b*d^3*e^2*x^4+55440*B*a^3*b^2*d^4*e*x^4+11088
*B*a^2*b^3*d^5*x^4+13860*A*a^5*d^2*e^3*x^3+69300*A*a^4*b*d^3*e^2*x^3+69300*A*a^3*b^2*d^4*e*x^3+13860*A*a^2*b^3
*d^5*x^3+13860*B*a^5*d^3*e^2*x^3+34650*B*a^4*b*d^4*e*x^3+13860*B*a^3*b^2*d^5*x^3+18480*A*a^5*d^3*e^2*x^2+46200
*A*a^4*b*d^4*e*x^2+18480*A*a^3*b^2*d^5*x^2+9240*B*a^5*d^4*e*x^2+9240*B*a^4*b*d^5*x^2+13860*A*a^5*d^4*e*x+13860
*A*a^4*b*d^5*x+2772*B*a^5*d^5*x+5544*A*a^5*d^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

________________________________________________________________________________________

maxima [B]  time = 0.81, size = 1330, normalized size = 3.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e^5*x^5/b^2 - 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*e^5*x^4/b^3 +
5/33*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*e^5*x^3/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d^5*x + 1/6*(b^
2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^6*e^5*x/b^6 - 16/99*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*e^5*x^2/b^5 + 1/6*(
b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*d^5/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^7*e^5/b^7 + 131/792*(b^2*x^
2 + 2*a*b*x + a^2)^(7/2)*B*a^4*e^5*x/b^6 - 923/5544*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^5*e^5/b^7 + 1/11*(5*B*
d*e^4 + A*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^4/b^2 - 3/22*(5*B*d*e^4 + A*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7
/2)*a*x^3/b^3 + 1/2*(2*B*d^2*e^3 + A*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 - 1/6*(5*B*d*e^4 + A*e^5)*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*x/b^5 + 5/6*(2*B*d^2*e^3 + A*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/
b^4 - 5/3*(B*d^3*e^2 + A*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 5/6*(B*d^4*e + 2*A*d^3*e^2)*(b^2
*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(B*d^5 + 5*A*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b + 31/198
*(5*B*d*e^4 + A*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^2/b^4 - 13/18*(2*B*d^2*e^3 + A*d*e^4)*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*a*x^2/b^3 + 10/9*(B*d^3*e^2 + A*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^2/b^2 - 1/6*(5*B
*d*e^4 + A*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6/b^6 + 5/6*(2*B*d^2*e^3 + A*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2
)^(5/2)*a^5/b^5 - 5/3*(B*d^3*e^2 + A*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 5/6*(B*d^4*e + 2*A*d^3
*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(B*d^5 + 5*A*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^
2 - 65/396*(5*B*d*e^4 + A*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x/b^5 + 29/36*(2*B*d^2*e^3 + A*d*e^4)*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x/b^4 - 55/36*(B*d^3*e^2 + A*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 +
5/8*(B*d^4*e + 2*A*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 + 461/2772*(5*B*d*e^4 + A*e^5)*(b^2*x^2 + 2
*a*b*x + a^2)^(7/2)*a^4/b^6 - 209/252*(2*B*d^2*e^3 + A*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 415/25
2*(B*d^3*e^2 + A*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 45/56*(B*d^4*e + 2*A*d^3*e^2)*(b^2*x^2 + 2
*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(B*d^5 + 5*A*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b^2

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^5\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((A + B*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B x\right ) \left (d + e x\right )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**5*((a + b*x)**2)**(5/2), x)

________________________________________________________________________________________