3.1738 \(\int (A+B x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=436 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{8 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (-5 a B e-A b e+6 b B d)}{12 e^7 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13}}{13 e^7 (a+b x)} \]

[Out]

1/7*(-a*e+b*d)^5*(-A*e+B*d)*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-1/8*(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*
(e*x+d)^8*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+5/9*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^9*((b*x+a)^2)^(1/2
)/e^7/(b*x+a)-b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^10*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+5/11*b^3*(-a*e+
b*d)*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^11*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-1/12*b^4*(-A*b*e-5*B*a*e+6*B*b*d)*(e*x+
d)^12*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/13*b^5*B*(e*x+d)^13*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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Rubi [A]  time = 1.01, antiderivative size = 436, 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} \[ -\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (-5 a B e-A b e+6 b B d)}{12 e^7 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{8 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13}}{13 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^5*(B*d - A*e)*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - ((b*d - a*e)^4*(6*b*
B*d - 5*A*b*e - a*B*e)*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)) + (5*b*(b*d - a*e)^3*(3*b*
B*d - 2*A*b*e - a*B*e)*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (b^2*(b*d - a*e)^2*(2*b*
B*d - A*b*e - a*B*e)*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d
- A*b*e - 2*a*B*e)*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5
*a*B*e)*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)) + (b^5*B*(d + e*x)^13*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(13*e^7*(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)^6 \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)^6 \, 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 {b^5 (b d-a e)^5 (-B d+A e) (d+e x)^6}{e^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^7}{e^6}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^8}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^9}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{10}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{11}}{e^6}+\frac {b^{10} B (d+e x)^{12}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^5 (B d-A e) (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x)}+\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}-\frac {b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^7 (a+b x)}+\frac {b^5 B (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}\\ \end {align*}

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Mathematica [B]  time = 0.37, size = 876, normalized size = 2.01 \[ \frac {x \sqrt {(a+b x)^2} \left (1287 \left (8 A \left (7 d^6+21 e x d^5+35 e^2 x^2 d^4+35 e^3 x^3 d^3+21 e^4 x^4 d^2+7 e^5 x^5 d+e^6 x^6\right )+B x \left (28 d^6+112 e x d^5+210 e^2 x^2 d^4+224 e^3 x^3 d^3+140 e^4 x^4 d^2+48 e^5 x^5 d+7 e^6 x^6\right )\right ) a^5+715 b x \left (9 A \left (28 d^6+112 e x d^5+210 e^2 x^2 d^4+224 e^3 x^3 d^3+140 e^4 x^4 d^2+48 e^5 x^5 d+7 e^6 x^6\right )+2 B x \left (84 d^6+378 e x d^5+756 e^2 x^2 d^4+840 e^3 x^3 d^3+540 e^4 x^4 d^2+189 e^5 x^5 d+28 e^6 x^6\right )\right ) a^4+286 b^2 x^2 \left (10 A \left (84 d^6+378 e x d^5+756 e^2 x^2 d^4+840 e^3 x^3 d^3+540 e^4 x^4 d^2+189 e^5 x^5 d+28 e^6 x^6\right )+3 B x \left (210 d^6+1008 e x d^5+2100 e^2 x^2 d^4+2400 e^3 x^3 d^3+1575 e^4 x^4 d^2+560 e^5 x^5 d+84 e^6 x^6\right )\right ) a^3+78 b^3 x^3 \left (11 A \left (210 d^6+1008 e x d^5+2100 e^2 x^2 d^4+2400 e^3 x^3 d^3+1575 e^4 x^4 d^2+560 e^5 x^5 d+84 e^6 x^6\right )+4 B x \left (462 d^6+2310 e x d^5+4950 e^2 x^2 d^4+5775 e^3 x^3 d^3+3850 e^4 x^4 d^2+1386 e^5 x^5 d+210 e^6 x^6\right )\right ) a^2+13 b^4 x^4 \left (12 A \left (462 d^6+2310 e x d^5+4950 e^2 x^2 d^4+5775 e^3 x^3 d^3+3850 e^4 x^4 d^2+1386 e^5 x^5 d+210 e^6 x^6\right )+5 B x \left (924 d^6+4752 e x d^5+10395 e^2 x^2 d^4+12320 e^3 x^3 d^3+8316 e^4 x^4 d^2+3024 e^5 x^5 d+462 e^6 x^6\right )\right ) a+b^5 x^5 \left (13 A \left (924 d^6+4752 e x d^5+10395 e^2 x^2 d^4+12320 e^3 x^3 d^3+8316 e^4 x^4 d^2+3024 e^5 x^5 d+462 e^6 x^6\right )+6 B x \left (1716 d^6+9009 e x d^5+20020 e^2 x^2 d^4+24024 e^3 x^3 d^3+16380 e^4 x^4 d^2+6006 e^5 x^5 d+924 e^6 x^6\right )\right )\right )}{72072 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(1287*a^5*(8*A*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*x^4 + 7
*d*e^5*x^5 + e^6*x^6) + B*x*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 + 48*d
*e^5*x^5 + 7*e^6*x^6)) + 715*a^4*b*x*(9*A*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*
e^4*x^4 + 48*d*e^5*x^5 + 7*e^6*x^6) + 2*B*x*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^
2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6)) + 286*a^3*b^2*x^2*(10*A*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840
*d^3*e^3*x^3 + 540*d^2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 3*B*x*(210*d^6 + 1008*d^5*e*x + 2100*d^4*e^2*x^
2 + 2400*d^3*e^3*x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6)) + 78*a^2*b^3*x^3*(11*A*(210*d^6 + 1008*
d^5*e*x + 2100*d^4*e^2*x^2 + 2400*d^3*e^3*x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6) + 4*B*x*(462*d^
6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6)) + 1
3*a*b^4*x^4*(12*A*(462*d^6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^
5*x^5 + 210*e^6*x^6) + 5*B*x*(924*d^6 + 4752*d^5*e*x + 10395*d^4*e^2*x^2 + 12320*d^3*e^3*x^3 + 8316*d^2*e^4*x^
4 + 3024*d*e^5*x^5 + 462*e^6*x^6)) + b^5*x^5*(13*A*(924*d^6 + 4752*d^5*e*x + 10395*d^4*e^2*x^2 + 12320*d^3*e^3
*x^3 + 8316*d^2*e^4*x^4 + 3024*d*e^5*x^5 + 462*e^6*x^6) + 6*B*x*(1716*d^6 + 9009*d^5*e*x + 20020*d^4*e^2*x^2 +
24024*d^3*e^3*x^3 + 16380*d^2*e^4*x^4 + 6006*d*e^5*x^5 + 924*e^6*x^6))))/(72072*(a + b*x))

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fricas [B]  time = 0.74, size = 964, normalized size = 2.21 \[ \frac {1}{13} \, B b^{5} e^{6} x^{13} + A a^{5} d^{6} x + \frac {1}{12} \, {\left (6 \, B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{12} + \frac {1}{11} \, {\left (15 \, B b^{5} d^{2} e^{4} + 6 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{11} + \frac {1}{2} \, {\left (4 \, B b^{5} d^{3} e^{3} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{10} + \frac {5}{9} \, {\left (3 \, B b^{5} d^{4} e^{2} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 12 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, B b^{5} d^{5} e + 15 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 150 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{6} + A a^{5} e^{6} + 6 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 75 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 6 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, A a^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e + 150 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{2} + 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{3} + 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{4}\right )} x^{6} + {\left (3 \, A a^{5} d^{2} e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{6} + 12 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} e + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} e^{2} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e^{3}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, A a^{5} d^{3} e^{3} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{6} + 6 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{5} e + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, A a^{5} d^{4} e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{6} + 6 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{5} e\right )} x^{3} + \frac {1}{2} \, {\left (6 \, A a^{5} d^{5} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{6}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/13*B*b^5*e^6*x^13 + A*a^5*d^6*x + 1/12*(6*B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^12 + 1/11*(15*B*b^5*d^2*e
^4 + 6*(5*B*a*b^4 + A*b^5)*d*e^5 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^11 + 1/2*(4*B*b^5*d^3*e^3 + 3*(5*B*a*b^4 +
A*b^5)*d^2*e^4 + 6*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 2*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^10 + 5/9*(3*B*b^5*d^4*e^2
+ 4*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 15*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 12*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + (B*
a^4*b + 2*A*a^3*b^2)*e^6)*x^9 + 1/8*(6*B*b^5*d^5*e + 15*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 100*(2*B*a^2*b^3 + A*a*b
^4)*d^3*e^3 + 150*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 30*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + (B*a^5 + 5*A*a^4*b)*e^6
)*x^8 + 1/7*(B*b^5*d^6 + A*a^5*e^6 + 6*(5*B*a*b^4 + A*b^5)*d^5*e + 75*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 200*(B
*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 75*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 6*(B*a^5 + 5*A*a^4*b)*d*e^5)*x^7 + 1/6*(6
*A*a^5*d*e^5 + (5*B*a*b^4 + A*b^5)*d^6 + 30*(2*B*a^2*b^3 + A*a*b^4)*d^5*e + 150*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^
2 + 100*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^3 + 15*(B*a^5 + 5*A*a^4*b)*d^2*e^4)*x^6 + (3*A*a^5*d^2*e^4 + (2*B*a^2*b^
3 + A*a*b^4)*d^6 + 12*(B*a^3*b^2 + A*a^2*b^3)*d^5*e + 15*(B*a^4*b + 2*A*a^3*b^2)*d^4*e^2 + 4*(B*a^5 + 5*A*a^4*
b)*d^3*e^3)*x^5 + 5/4*(4*A*a^5*d^3*e^3 + 2*(B*a^3*b^2 + A*a^2*b^3)*d^6 + 6*(B*a^4*b + 2*A*a^3*b^2)*d^5*e + 3*(
B*a^5 + 5*A*a^4*b)*d^4*e^2)*x^4 + 1/3*(15*A*a^5*d^4*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^6 + 6*(B*a^5 + 5*A*a^4*b
)*d^5*e)*x^3 + 1/2*(6*A*a^5*d^5*e + (B*a^5 + 5*A*a^4*b)*d^6)*x^2

________________________________________________________________________________________

giac [B]  time = 0.27, size = 1702, normalized size = 3.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/13*B*b^5*x^13*e^6*sgn(b*x + a) + 1/2*B*b^5*d*x^12*e^5*sgn(b*x + a) + 15/11*B*b^5*d^2*x^11*e^4*sgn(b*x + a) +
2*B*b^5*d^3*x^10*e^3*sgn(b*x + a) + 5/3*B*b^5*d^4*x^9*e^2*sgn(b*x + a) + 3/4*B*b^5*d^5*x^8*e*sgn(b*x + a) + 1
/7*B*b^5*d^6*x^7*sgn(b*x + a) + 5/12*B*a*b^4*x^12*e^6*sgn(b*x + a) + 1/12*A*b^5*x^12*e^6*sgn(b*x + a) + 30/11*
B*a*b^4*d*x^11*e^5*sgn(b*x + a) + 6/11*A*b^5*d*x^11*e^5*sgn(b*x + a) + 15/2*B*a*b^4*d^2*x^10*e^4*sgn(b*x + a)
+ 3/2*A*b^5*d^2*x^10*e^4*sgn(b*x + a) + 100/9*B*a*b^4*d^3*x^9*e^3*sgn(b*x + a) + 20/9*A*b^5*d^3*x^9*e^3*sgn(b*
x + a) + 75/8*B*a*b^4*d^4*x^8*e^2*sgn(b*x + a) + 15/8*A*b^5*d^4*x^8*e^2*sgn(b*x + a) + 30/7*B*a*b^4*d^5*x^7*e*
sgn(b*x + a) + 6/7*A*b^5*d^5*x^7*e*sgn(b*x + a) + 5/6*B*a*b^4*d^6*x^6*sgn(b*x + a) + 1/6*A*b^5*d^6*x^6*sgn(b*x
+ a) + 10/11*B*a^2*b^3*x^11*e^6*sgn(b*x + a) + 5/11*A*a*b^4*x^11*e^6*sgn(b*x + a) + 6*B*a^2*b^3*d*x^10*e^5*sg
n(b*x + a) + 3*A*a*b^4*d*x^10*e^5*sgn(b*x + a) + 50/3*B*a^2*b^3*d^2*x^9*e^4*sgn(b*x + a) + 25/3*A*a*b^4*d^2*x^
9*e^4*sgn(b*x + a) + 25*B*a^2*b^3*d^3*x^8*e^3*sgn(b*x + a) + 25/2*A*a*b^4*d^3*x^8*e^3*sgn(b*x + a) + 150/7*B*a
^2*b^3*d^4*x^7*e^2*sgn(b*x + a) + 75/7*A*a*b^4*d^4*x^7*e^2*sgn(b*x + a) + 10*B*a^2*b^3*d^5*x^6*e*sgn(b*x + a)
+ 5*A*a*b^4*d^5*x^6*e*sgn(b*x + a) + 2*B*a^2*b^3*d^6*x^5*sgn(b*x + a) + A*a*b^4*d^6*x^5*sgn(b*x + a) + B*a^3*b
^2*x^10*e^6*sgn(b*x + a) + A*a^2*b^3*x^10*e^6*sgn(b*x + a) + 20/3*B*a^3*b^2*d*x^9*e^5*sgn(b*x + a) + 20/3*A*a^
2*b^3*d*x^9*e^5*sgn(b*x + a) + 75/4*B*a^3*b^2*d^2*x^8*e^4*sgn(b*x + a) + 75/4*A*a^2*b^3*d^2*x^8*e^4*sgn(b*x +
a) + 200/7*B*a^3*b^2*d^3*x^7*e^3*sgn(b*x + a) + 200/7*A*a^2*b^3*d^3*x^7*e^3*sgn(b*x + a) + 25*B*a^3*b^2*d^4*x^
6*e^2*sgn(b*x + a) + 25*A*a^2*b^3*d^4*x^6*e^2*sgn(b*x + a) + 12*B*a^3*b^2*d^5*x^5*e*sgn(b*x + a) + 12*A*a^2*b^
3*d^5*x^5*e*sgn(b*x + a) + 5/2*B*a^3*b^2*d^6*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^6*x^4*sgn(b*x + a) + 5/9*B*a^4
*b*x^9*e^6*sgn(b*x + a) + 10/9*A*a^3*b^2*x^9*e^6*sgn(b*x + a) + 15/4*B*a^4*b*d*x^8*e^5*sgn(b*x + a) + 15/2*A*a
^3*b^2*d*x^8*e^5*sgn(b*x + a) + 75/7*B*a^4*b*d^2*x^7*e^4*sgn(b*x + a) + 150/7*A*a^3*b^2*d^2*x^7*e^4*sgn(b*x +
a) + 50/3*B*a^4*b*d^3*x^6*e^3*sgn(b*x + a) + 100/3*A*a^3*b^2*d^3*x^6*e^3*sgn(b*x + a) + 15*B*a^4*b*d^4*x^5*e^2
*sgn(b*x + a) + 30*A*a^3*b^2*d^4*x^5*e^2*sgn(b*x + a) + 15/2*B*a^4*b*d^5*x^4*e*sgn(b*x + a) + 15*A*a^3*b^2*d^5
*x^4*e*sgn(b*x + a) + 5/3*B*a^4*b*d^6*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^6*x^3*sgn(b*x + a) + 1/8*B*a^5*x^8*e
^6*sgn(b*x + a) + 5/8*A*a^4*b*x^8*e^6*sgn(b*x + a) + 6/7*B*a^5*d*x^7*e^5*sgn(b*x + a) + 30/7*A*a^4*b*d*x^7*e^5
*sgn(b*x + a) + 5/2*B*a^5*d^2*x^6*e^4*sgn(b*x + a) + 25/2*A*a^4*b*d^2*x^6*e^4*sgn(b*x + a) + 4*B*a^5*d^3*x^5*e
^3*sgn(b*x + a) + 20*A*a^4*b*d^3*x^5*e^3*sgn(b*x + a) + 15/4*B*a^5*d^4*x^4*e^2*sgn(b*x + a) + 75/4*A*a^4*b*d^4
*x^4*e^2*sgn(b*x + a) + 2*B*a^5*d^5*x^3*e*sgn(b*x + a) + 10*A*a^4*b*d^5*x^3*e*sgn(b*x + a) + 1/2*B*a^5*d^6*x^2
*sgn(b*x + a) + 5/2*A*a^4*b*d^6*x^2*sgn(b*x + a) + 1/7*A*a^5*x^7*e^6*sgn(b*x + a) + A*a^5*d*x^6*e^5*sgn(b*x +
a) + 3*A*a^5*d^2*x^5*e^4*sgn(b*x + a) + 5*A*a^5*d^3*x^4*e^3*sgn(b*x + a) + 5*A*a^5*d^4*x^3*e^2*sgn(b*x + a) +
3*A*a^5*d^5*x^2*e*sgn(b*x + a) + A*a^5*d^6*x*sgn(b*x + a)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 1264, normalized size = 2.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/72072*x*(5544*B*b^5*e^6*x^12+6006*A*b^5*e^6*x^11+30030*B*a*b^4*e^6*x^11+36036*B*b^5*d*e^5*x^11+32760*A*a*b^4
*e^6*x^10+39312*A*b^5*d*e^5*x^10+65520*B*a^2*b^3*e^6*x^10+196560*B*a*b^4*d*e^5*x^10+98280*B*b^5*d^2*e^4*x^10+7
2072*A*a^2*b^3*e^6*x^9+216216*A*a*b^4*d*e^5*x^9+108108*A*b^5*d^2*e^4*x^9+72072*B*a^3*b^2*e^6*x^9+432432*B*a^2*
b^3*d*e^5*x^9+540540*B*a*b^4*d^2*e^4*x^9+144144*B*b^5*d^3*e^3*x^9+80080*A*a^3*b^2*e^6*x^8+480480*A*a^2*b^3*d*e
^5*x^8+600600*A*a*b^4*d^2*e^4*x^8+160160*A*b^5*d^3*e^3*x^8+40040*B*a^4*b*e^6*x^8+480480*B*a^3*b^2*d*e^5*x^8+12
01200*B*a^2*b^3*d^2*e^4*x^8+800800*B*a*b^4*d^3*e^3*x^8+120120*B*b^5*d^4*e^2*x^8+45045*A*a^4*b*e^6*x^7+540540*A
*a^3*b^2*d*e^5*x^7+1351350*A*a^2*b^3*d^2*e^4*x^7+900900*A*a*b^4*d^3*e^3*x^7+135135*A*b^5*d^4*e^2*x^7+9009*B*a^
5*e^6*x^7+270270*B*a^4*b*d*e^5*x^7+1351350*B*a^3*b^2*d^2*e^4*x^7+1801800*B*a^2*b^3*d^3*e^3*x^7+675675*B*a*b^4*
d^4*e^2*x^7+54054*B*b^5*d^5*e*x^7+10296*A*a^5*e^6*x^6+308880*A*a^4*b*d*e^5*x^6+1544400*A*a^3*b^2*d^2*e^4*x^6+2
059200*A*a^2*b^3*d^3*e^3*x^6+772200*A*a*b^4*d^4*e^2*x^6+61776*A*b^5*d^5*e*x^6+61776*B*a^5*d*e^5*x^6+772200*B*a
^4*b*d^2*e^4*x^6+2059200*B*a^3*b^2*d^3*e^3*x^6+1544400*B*a^2*b^3*d^4*e^2*x^6+308880*B*a*b^4*d^5*e*x^6+10296*B*
b^5*d^6*x^6+72072*A*a^5*d*e^5*x^5+900900*A*a^4*b*d^2*e^4*x^5+2402400*A*a^3*b^2*d^3*e^3*x^5+1801800*A*a^2*b^3*d
^4*e^2*x^5+360360*A*a*b^4*d^5*e*x^5+12012*A*b^5*d^6*x^5+180180*B*a^5*d^2*e^4*x^5+1201200*B*a^4*b*d^3*e^3*x^5+1
801800*B*a^3*b^2*d^4*e^2*x^5+720720*B*a^2*b^3*d^5*e*x^5+60060*B*a*b^4*d^6*x^5+216216*A*a^5*d^2*e^4*x^4+1441440
*A*a^4*b*d^3*e^3*x^4+2162160*A*a^3*b^2*d^4*e^2*x^4+864864*A*a^2*b^3*d^5*e*x^4+72072*A*a*b^4*d^6*x^4+288288*B*a
^5*d^3*e^3*x^4+1081080*B*a^4*b*d^4*e^2*x^4+864864*B*a^3*b^2*d^5*e*x^4+144144*B*a^2*b^3*d^6*x^4+360360*A*a^5*d^
3*e^3*x^3+1351350*A*a^4*b*d^4*e^2*x^3+1081080*A*a^3*b^2*d^5*e*x^3+180180*A*a^2*b^3*d^6*x^3+270270*B*a^5*d^4*e^
2*x^3+540540*B*a^4*b*d^5*e*x^3+180180*B*a^3*b^2*d^6*x^3+360360*A*a^5*d^4*e^2*x^2+720720*A*a^4*b*d^5*e*x^2+2402
40*A*a^3*b^2*d^6*x^2+144144*B*a^5*d^5*e*x^2+120120*B*a^4*b*d^6*x^2+216216*A*a^5*d^5*e*x+180180*A*a^4*b*d^6*x+3
6036*B*a^5*d^6*x+72072*A*a^5*d^6)*((b*x+a)^2)^(5/2)/(b*x+a)^5

________________________________________________________________________________________

maxima [B]  time = 0.70, size = 1744, normalized size = 4.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/13*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e^6*x^6/b^2 - 19/156*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*e^6*x^5/b^3 +
251/1716*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*e^6*x^4/b^4 - 68/429*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*e^6*
x^3/b^5 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d^6*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^7*e^6*x/b^7 +
211/1287*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^4*e^6*x^2/b^6 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*d^6/b - 1
/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^8*e^6/b^8 - 1709/10296*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^5*e^6*x/b^7
+ 1715/10296*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^6*e^6/b^8 + 1/12*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)*x^5/b^2 - 17/132*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^4/b^3 + 3/11*(5*B*d^2*e^4 + 2
*A*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^4/b^2 + 5/33*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a
^2*x^3/b^4 - 9/22*(5*B*d^2*e^4 + 2*A*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 1/2*(4*B*d^3*e^3 + 3*A
*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 + 1/6*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^
6*x/b^6 - 1/2*(5*B*d^2*e^4 + 2*A*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*x/b^5 + 5/6*(4*B*d^3*e^3 + 3*A*d^2
*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/b^4 - 5/6*(3*B*d^4*e^2 + 4*A*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5
/2)*a^3*x/b^3 + 1/2*(2*B*d^5*e + 5*A*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(B*d^6 + 6*A*d^5
*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b - 16/99*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^2/
b^5 + 31/66*(5*B*d^2*e^4 + 2*A*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^2/b^4 - 13/18*(4*B*d^3*e^3 + 3*A*d
^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^2/b^3 + 5/9*(3*B*d^4*e^2 + 4*A*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^
(7/2)*x^2/b^2 + 1/6*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7/b^7 - 1/2*(5*B*d^2*e^4 + 2*A*d*e^5
)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6/b^6 + 5/6*(4*B*d^3*e^3 + 3*A*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^
5/b^5 - 5/6*(3*B*d^4*e^2 + 4*A*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 1/2*(2*B*d^5*e + 5*A*d^4*e^2
)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(B*d^6 + 6*A*d^5*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 +
131/792*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x/b^6 - 65/132*(5*B*d^2*e^4 + 2*A*d*e^5)*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x/b^5 + 29/36*(4*B*d^3*e^3 + 3*A*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x
/b^4 - 55/72*(3*B*d^4*e^2 + 4*A*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 3/8*(2*B*d^5*e + 5*A*d^4*e^
2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 - 923/5544*(6*B*d*e^5 + A*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5/b^
7 + 461/924*(5*B*d^2*e^4 + 2*A*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4/b^6 - 209/252*(4*B*d^3*e^3 + 3*A*d^2
*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 415/504*(3*B*d^4*e^2 + 4*A*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^
(7/2)*a^2/b^4 - 27/56*(2*B*d^5*e + 5*A*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(B*d^6 + 6*A*d^5*e
)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^6\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((A + B*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B x\right ) \left (d + e x\right )^{6} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**6*((a + b*x)**2)**(5/2), x)

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