3.18.39 \(\int (A+B x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1739]

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

[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

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Rubi [A]
time = 0.51, 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 = {784, 78} \begin {gather*} \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} \end {gather*}

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 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)^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*}

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Mathematica [A]
time = 0.23, size = 740, normalized size = 1.93 \begin {gather*} \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 (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 )+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 )\right )\right )}{5544 (a+b x)} \end {gather*}

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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(292)=584\).
time = 0.82, size = 1068, normalized size = 2.79 Too large to display

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,method=_RETURNVERBOSE)

[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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1289 vs. \(2 (304) = 608\).
time = 0.30, size = 1289, normalized size = 3.37 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d^{5} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d^{5}}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x^{5} e^{5}}{12 \, b^{2}} - \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a x^{4} e^{5}}{132 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} x^{4}}{11 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} x^{3} e^{5}}{33 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} a x^{3}}{22 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d^{2} e^{3} + A d e^{4}\right )} x^{3}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{6} x e^{5}}{6 \, b^{6}} - \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{3} x^{2} e^{5}}{99 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} a^{5} x}{6 \, b^{5}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d^{2} e^{3} + A d e^{4}\right )} a^{4} x}{6 \, b^{4}} - \frac {5 \, {\left (B d^{3} e^{2} + A d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{3 \, b^{3}} + \frac {5 \, {\left (B d^{4} e + 2 \, A d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x}{6 \, b^{2}} - \frac {{\left (B d^{5} + 5 \, A d^{4} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{6 \, b} + \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} a^{2} x^{2}}{198 \, b^{4}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d^{2} e^{3} + A d e^{4}\right )} a x^{2}}{18 \, b^{3}} + \frac {10 \, {\left (B d^{3} e^{2} + A d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{7} e^{5}}{6 \, b^{7}} + \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{4} x e^{5}}{792 \, b^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} a^{6}}{6 \, b^{6}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, B d^{2} e^{3} + A d e^{4}\right )} a^{5}}{6 \, b^{5}} - \frac {5 \, {\left (B d^{3} e^{2} + A d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{3 \, b^{4}} + \frac {5 \, {\left (B d^{4} e + 2 \, A d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3}}{6 \, b^{3}} - \frac {{\left (B d^{5} + 5 \, A d^{4} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{6 \, b^{2}} - \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} a^{3} x}{396 \, b^{5}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d^{2} e^{3} + A d e^{4}\right )} a^{2} x}{36 \, b^{4}} - \frac {55 \, {\left (B d^{3} e^{2} + A d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x}{36 \, b^{3}} + \frac {5 \, {\left (B d^{4} e + 2 \, A d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x}{8 \, b^{2}} - \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{5} e^{5}}{5544 \, b^{7}} + \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (5 \, B d e^{4} + A e^{5}\right )} a^{4}}{2772 \, b^{6}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, B d^{2} e^{3} + A d e^{4}\right )} a^{3}}{252 \, b^{5}} + \frac {415 \, {\left (B d^{3} e^{2} + A d^{2} e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2}}{252 \, b^{4}} - \frac {45 \, {\left (B d^{4} e + 2 \, A d^{3} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} + \frac {{\left (B d^{5} + 5 \, A d^{4} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b^{2}} \end {gather*}

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]

1/6*(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)*A*a*d^5/b + 1/12*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*B*x^5*e^5/b^2 - 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*x^4*e^5/b^3 + 1/11*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*(5*B*d*e^4 + A*e^5)*x^4/b^2 + 5/33*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*x^3*e^5/b^4 - 3/
22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(5*B*d*e^4 + A*e^5)*a*x^3/b^3 + 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d^
2*e^3 + A*d*e^4)*x^3/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^6*x*e^5/b^6 - 16/99*(b^2*x^2 + 2*a*b*x + a^
2)^(7/2)*B*a^3*x^2*e^5/b^5 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(5*B*d*e^4 + A*e^5)*a^5*x/b^5 + 5/6*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*(2*B*d^2*e^3 + A*d*e^4)*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*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(5*B*d*e^4 + A*e^5)*a^2*
x^2/b^4 - 13/18*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d^2*e^3 + A*d*e^4)*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*(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*x*e^5/b^6 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(5*B*d*e^4 + A*e^5)*a^6/b^6 +
5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(2*B*d^2*e^3 + A*d*e^4)*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*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(5*B*d*e^4 + A
*e^5)*a^3*x/b^5 + 29/36*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d^2*e^3 + A*d*e^4)*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 - 923/5544*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^5*e^5/b^7 + 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*
(5*B*d*e^4 + A*e^5)*a^4/b^6 - 209/252*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*(2*B*d^2*e^3 + A*d*e^4)*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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (304) = 608\).
time = 3.62, size = 821, normalized size = 2.14 \begin {gather*} \frac {1}{7} \, B b^{5} d^{5} x^{7} + A a^{5} d^{5} x + \frac {1}{6} \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} x^{6} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} x^{5} + \frac {5}{2} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} x^{4} + \frac {5}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{5} x^{2} + \frac {1}{5544} \, {\left (462 \, B b^{5} x^{12} + 924 \, A a^{5} x^{6} + 504 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + 2772 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + 6160 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 3465 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + 792 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7}\right )} e^{5} + \frac {1}{2772} \, {\left (1260 \, B b^{5} d x^{11} + 2772 \, A a^{5} d x^{5} + 1386 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d x^{10} + 7700 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d x^{9} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d x^{8} + 9900 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d x^{7} + 2310 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d x^{6}\right )} e^{4} + \frac {1}{252} \, {\left (252 \, B b^{5} d^{2} x^{10} + 630 \, A a^{5} d^{2} x^{4} + 280 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} x^{9} + 1575 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} x^{8} + 3600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} x^{7} + 2100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} x^{6} + 504 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} x^{5}\right )} e^{3} + \frac {5}{252} \, {\left (56 \, B b^{5} d^{3} x^{9} + 168 \, A a^{5} d^{3} x^{3} + 63 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} x^{8} + 360 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} x^{7} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} x^{6} + 504 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} x^{5} + 126 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} x^{4}\right )} e^{2} + \frac {5}{168} \, {\left (21 \, B b^{5} d^{4} x^{8} + 84 \, A a^{5} d^{4} x^{2} + 24 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} x^{7} + 140 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} x^{6} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} x^{5} + 210 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} x^{4} + 56 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4} x^{3}\right )} e \end {gather*}

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]

1/7*B*b^5*d^5*x^7 + A*a^5*d^5*x + 1/6*(5*B*a*b^4 + A*b^5)*d^5*x^6 + (2*B*a^2*b^3 + A*a*b^4)*d^5*x^5 + 5/2*(B*a
^3*b^2 + A*a^2*b^3)*d^5*x^4 + 5/3*(B*a^4*b + 2*A*a^3*b^2)*d^5*x^3 + 1/2*(B*a^5 + 5*A*a^4*b)*d^5*x^2 + 1/5544*(
462*B*b^5*x^12 + 924*A*a^5*x^6 + 504*(5*B*a*b^4 + A*b^5)*x^11 + 2772*(2*B*a^2*b^3 + A*a*b^4)*x^10 + 6160*(B*a^
3*b^2 + A*a^2*b^3)*x^9 + 3465*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 792*(B*a^5 + 5*A*a^4*b)*x^7)*e^5 + 1/2772*(1260*B*
b^5*d*x^11 + 2772*A*a^5*d*x^5 + 1386*(5*B*a*b^4 + A*b^5)*d*x^10 + 7700*(2*B*a^2*b^3 + A*a*b^4)*d*x^9 + 17325*(
B*a^3*b^2 + A*a^2*b^3)*d*x^8 + 9900*(B*a^4*b + 2*A*a^3*b^2)*d*x^7 + 2310*(B*a^5 + 5*A*a^4*b)*d*x^6)*e^4 + 1/25
2*(252*B*b^5*d^2*x^10 + 630*A*a^5*d^2*x^4 + 280*(5*B*a*b^4 + A*b^5)*d^2*x^9 + 1575*(2*B*a^2*b^3 + A*a*b^4)*d^2
*x^8 + 3600*(B*a^3*b^2 + A*a^2*b^3)*d^2*x^7 + 2100*(B*a^4*b + 2*A*a^3*b^2)*d^2*x^6 + 504*(B*a^5 + 5*A*a^4*b)*d
^2*x^5)*e^3 + 5/252*(56*B*b^5*d^3*x^9 + 168*A*a^5*d^3*x^3 + 63*(5*B*a*b^4 + A*b^5)*d^3*x^8 + 360*(2*B*a^2*b^3
+ A*a*b^4)*d^3*x^7 + 840*(B*a^3*b^2 + A*a^2*b^3)*d^3*x^6 + 504*(B*a^4*b + 2*A*a^3*b^2)*d^3*x^5 + 126*(B*a^5 +
5*A*a^4*b)*d^3*x^4)*e^2 + 5/168*(21*B*b^5*d^4*x^8 + 84*A*a^5*d^4*x^2 + 24*(5*B*a*b^4 + A*b^5)*d^4*x^7 + 140*(2
*B*a^2*b^3 + A*a*b^4)*d^4*x^6 + 336*(B*a^3*b^2 + A*a^2*b^3)*d^4*x^5 + 210*(B*a^4*b + 2*A*a^3*b^2)*d^4*x^4 + 56
*(B*a^5 + 5*A*a^4*b)*d^4*x^3)*e

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

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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1446 vs. \(2 (304) = 608\).
time = 1.28, size = 1446, normalized size = 3.78 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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)

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