3.16.85 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=300 \[ -\frac {2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac {2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac {2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac {2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {2 \sqrt {d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8} \]

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Rubi [A]  time = 0.15, antiderivative size = 300, 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 = {27, 77} \begin {gather*} -\frac {2 b^5 (d+e x)^{11/2} (-6 a B e-A b e+7 b B d)}{11 e^8}+\frac {2 b^4 (d+e x)^{9/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{3 e^8}-\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{7 e^8}+\frac {2 b^2 (d+e x)^{5/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}-\frac {2 b (d+e x)^{3/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {2 \sqrt {d+e x} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8}+\frac {2 (b d-a e)^6 (B d-A e)}{e^8 \sqrt {d+e x}}+\frac {2 b^6 B (d+e x)^{13/2}}{13 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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]))))

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 259, normalized size = 0.86 \begin {gather*} \frac {2 \left (-273 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+1001 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-2145 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+3003 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-3003 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+3003 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+3003 (b d-a e)^6 (B d-A e)+231 b^6 B (d+e x)^7\right )}{3003 e^8 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3003*(b*d - a*e)^6*(B*d - A*e) + 3003*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x) - 3003*b*(b*d -
a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 3003*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e
*x)^3 - 2145*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 1001*b^4*(b*d - a*e)*(7*b*B*d - 2*A
*b*e - 5*a*B*e)*(d + e*x)^5 - 273*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 231*b^6*B*(d + e*x)^7))/(3003*
e^8*Sqrt[d + e*x])

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IntegrateAlgebraic [B]  time = 0.23, size = 1069, normalized size = 3.56 \begin {gather*} \frac {2 \left (3003 b^6 B d^7-3003 A b^6 e d^6-18018 a b^5 B e d^6+21021 b^6 B (d+e x) d^6+18018 a A b^5 e^2 d^5+45045 a^2 b^4 B e^2 d^5-21021 b^6 B (d+e x)^2 d^5-18018 A b^6 e (d+e x) d^5-108108 a b^5 B e (d+e x) d^5-45045 a^2 A b^4 e^3 d^4-60060 a^3 b^3 B e^3 d^4+21021 b^6 B (d+e x)^3 d^4+15015 A b^6 e (d+e x)^2 d^4+90090 a b^5 B e (d+e x)^2 d^4+90090 a A b^5 e^2 (d+e x) d^4+225225 a^2 b^4 B e^2 (d+e x) d^4+60060 a^3 A b^3 e^4 d^3+45045 a^4 b^2 B e^4 d^3-15015 b^6 B (d+e x)^4 d^3-12012 A b^6 e (d+e x)^3 d^3-72072 a b^5 B e (d+e x)^3 d^3-60060 a A b^5 e^2 (d+e x)^2 d^3-150150 a^2 b^4 B e^2 (d+e x)^2 d^3-180180 a^2 A b^4 e^3 (d+e x) d^3-240240 a^3 b^3 B e^3 (d+e x) d^3-45045 a^4 A b^2 e^5 d^2-18018 a^5 b B e^5 d^2+7007 b^6 B (d+e x)^5 d^2+6435 A b^6 e (d+e x)^4 d^2+38610 a b^5 B e (d+e x)^4 d^2+36036 a A b^5 e^2 (d+e x)^3 d^2+90090 a^2 b^4 B e^2 (d+e x)^3 d^2+90090 a^2 A b^4 e^3 (d+e x)^2 d^2+120120 a^3 b^3 B e^3 (d+e x)^2 d^2+180180 a^3 A b^3 e^4 (d+e x) d^2+135135 a^4 b^2 B e^4 (d+e x) d^2+18018 a^5 A b e^6 d+3003 a^6 B e^6 d-1911 b^6 B (d+e x)^6 d-2002 A b^6 e (d+e x)^5 d-12012 a b^5 B e (d+e x)^5 d-12870 a A b^5 e^2 (d+e x)^4 d-32175 a^2 b^4 B e^2 (d+e x)^4 d-36036 a^2 A b^4 e^3 (d+e x)^3 d-48048 a^3 b^3 B e^3 (d+e x)^3 d-60060 a^3 A b^3 e^4 (d+e x)^2 d-45045 a^4 b^2 B e^4 (d+e x)^2 d-90090 a^4 A b^2 e^5 (d+e x) d-36036 a^5 b B e^5 (d+e x) d-3003 a^6 A e^7+231 b^6 B (d+e x)^7+273 A b^6 e (d+e x)^6+1638 a b^5 B e (d+e x)^6+2002 a A b^5 e^2 (d+e x)^5+5005 a^2 b^4 B e^2 (d+e x)^5+6435 a^2 A b^4 e^3 (d+e x)^4+8580 a^3 b^3 B e^3 (d+e x)^4+12012 a^3 A b^3 e^4 (d+e x)^3+9009 a^4 b^2 B e^4 (d+e x)^3+15015 a^4 A b^2 e^5 (d+e x)^2+6006 a^5 b B e^5 (d+e x)^2+18018 a^5 A b e^6 (d+e x)+3003 a^6 B e^6 (d+e x)\right )}{3003 e^8 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3003*b^6*B*d^7 - 3003*A*b^6*d^6*e - 18018*a*b^5*B*d^6*e + 18018*a*A*b^5*d^5*e^2 + 45045*a^2*b^4*B*d^5*e^2
- 45045*a^2*A*b^4*d^4*e^3 - 60060*a^3*b^3*B*d^4*e^3 + 60060*a^3*A*b^3*d^3*e^4 + 45045*a^4*b^2*B*d^3*e^4 - 4504
5*a^4*A*b^2*d^2*e^5 - 18018*a^5*b*B*d^2*e^5 + 18018*a^5*A*b*d*e^6 + 3003*a^6*B*d*e^6 - 3003*a^6*A*e^7 + 21021*
b^6*B*d^6*(d + e*x) - 18018*A*b^6*d^5*e*(d + e*x) - 108108*a*b^5*B*d^5*e*(d + e*x) + 90090*a*A*b^5*d^4*e^2*(d
+ e*x) + 225225*a^2*b^4*B*d^4*e^2*(d + e*x) - 180180*a^2*A*b^4*d^3*e^3*(d + e*x) - 240240*a^3*b^3*B*d^3*e^3*(d
+ e*x) + 180180*a^3*A*b^3*d^2*e^4*(d + e*x) + 135135*a^4*b^2*B*d^2*e^4*(d + e*x) - 90090*a^4*A*b^2*d*e^5*(d +
e*x) - 36036*a^5*b*B*d*e^5*(d + e*x) + 18018*a^5*A*b*e^6*(d + e*x) + 3003*a^6*B*e^6*(d + e*x) - 21021*b^6*B*d
^5*(d + e*x)^2 + 15015*A*b^6*d^4*e*(d + e*x)^2 + 90090*a*b^5*B*d^4*e*(d + e*x)^2 - 60060*a*A*b^5*d^3*e^2*(d +
e*x)^2 - 150150*a^2*b^4*B*d^3*e^2*(d + e*x)^2 + 90090*a^2*A*b^4*d^2*e^3*(d + e*x)^2 + 120120*a^3*b^3*B*d^2*e^3
*(d + e*x)^2 - 60060*a^3*A*b^3*d*e^4*(d + e*x)^2 - 45045*a^4*b^2*B*d*e^4*(d + e*x)^2 + 15015*a^4*A*b^2*e^5*(d
+ e*x)^2 + 6006*a^5*b*B*e^5*(d + e*x)^2 + 21021*b^6*B*d^4*(d + e*x)^3 - 12012*A*b^6*d^3*e*(d + e*x)^3 - 72072*
a*b^5*B*d^3*e*(d + e*x)^3 + 36036*a*A*b^5*d^2*e^2*(d + e*x)^3 + 90090*a^2*b^4*B*d^2*e^2*(d + e*x)^3 - 36036*a^
2*A*b^4*d*e^3*(d + e*x)^3 - 48048*a^3*b^3*B*d*e^3*(d + e*x)^3 + 12012*a^3*A*b^3*e^4*(d + e*x)^3 + 9009*a^4*b^2
*B*e^4*(d + e*x)^3 - 15015*b^6*B*d^3*(d + e*x)^4 + 6435*A*b^6*d^2*e*(d + e*x)^4 + 38610*a*b^5*B*d^2*e*(d + e*x
)^4 - 12870*a*A*b^5*d*e^2*(d + e*x)^4 - 32175*a^2*b^4*B*d*e^2*(d + e*x)^4 + 6435*a^2*A*b^4*e^3*(d + e*x)^4 + 8
580*a^3*b^3*B*e^3*(d + e*x)^4 + 7007*b^6*B*d^2*(d + e*x)^5 - 2002*A*b^6*d*e*(d + e*x)^5 - 12012*a*b^5*B*d*e*(d
+ e*x)^5 + 2002*a*A*b^5*e^2*(d + e*x)^5 + 5005*a^2*b^4*B*e^2*(d + e*x)^5 - 1911*b^6*B*d*(d + e*x)^6 + 273*A*b
^6*e*(d + e*x)^6 + 1638*a*b^5*B*e*(d + e*x)^6 + 231*b^6*B*(d + e*x)^7))/(3003*e^8*Sqrt[d + e*x])

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fricas [B]  time = 0.49, size = 778, normalized size = 2.59 \begin {gather*} \frac {2 \, {\left (231 \, B b^{6} e^{7} x^{7} + 14336 \, B b^{6} d^{7} - 3003 \, A a^{6} e^{7} - 13312 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 36608 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 54912 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 48048 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 24024 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 6006 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 21 \, {\left (14 \, B b^{6} d e^{6} - 13 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 7 \, {\left (56 \, B b^{6} d^{2} e^{5} - 52 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 143 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} - 5 \, {\left (112 \, B b^{6} d^{3} e^{4} - 104 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 286 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 429 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + {\left (896 \, B b^{6} d^{4} e^{3} - 832 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 2288 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 3432 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 3003 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} - {\left (1792 \, B b^{6} d^{5} e^{2} - 1664 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 4576 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 6864 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 6006 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 3003 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + {\left (7168 \, B b^{6} d^{6} e - 6656 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 18304 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 27456 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 24024 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 12012 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 3003 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{3003 \, {\left (e^{9} x + d e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*B*b^6*e^7*x^7 + 14336*B*b^6*d^7 - 3003*A*a^6*e^7 - 13312*(6*B*a*b^5 + A*b^6)*d^6*e + 36608*(5*B*a^
2*b^4 + 2*A*a*b^5)*d^5*e^2 - 54912*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 48048*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3
*e^4 - 24024*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 6006*(B*a^6 + 6*A*a^5*b)*d*e^6 - 21*(14*B*b^6*d*e^6 - 13*(6*B
*a*b^5 + A*b^6)*e^7)*x^6 + 7*(56*B*b^6*d^2*e^5 - 52*(6*B*a*b^5 + A*b^6)*d*e^6 + 143*(5*B*a^2*b^4 + 2*A*a*b^5)*
e^7)*x^5 - 5*(112*B*b^6*d^3*e^4 - 104*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 286*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 429*
(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + (896*B*b^6*d^4*e^3 - 832*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 2288*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^2*e^5 - 3432*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 3003*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 -
(1792*B*b^6*d^5*e^2 - 1664*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 4576*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 6864*(4*B*a^
3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 6006*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 3003*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^
2 + (7168*B*b^6*d^6*e - 6656*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 18304*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 27456*(4*
B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 24024*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 12012*(2*B*a^5*b + 5*A*a^4*b^2)
*d*e^6 + 3003*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/(e^9*x + d*e^8)

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giac [B]  time = 0.32, size = 1131, normalized size = 3.77

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*(x*e + d)^(13/2)*B*b^6*e^96 - 1911*(x*e + d)^(11/2)*B*b^6*d*e^96 + 7007*(x*e + d)^(9/2)*B*b^6*d^2*
e^96 - 15015*(x*e + d)^(7/2)*B*b^6*d^3*e^96 + 21021*(x*e + d)^(5/2)*B*b^6*d^4*e^96 - 21021*(x*e + d)^(3/2)*B*b
^6*d^5*e^96 + 21021*sqrt(x*e + d)*B*b^6*d^6*e^96 + 1638*(x*e + d)^(11/2)*B*a*b^5*e^97 + 273*(x*e + d)^(11/2)*A
*b^6*e^97 - 12012*(x*e + d)^(9/2)*B*a*b^5*d*e^97 - 2002*(x*e + d)^(9/2)*A*b^6*d*e^97 + 38610*(x*e + d)^(7/2)*B
*a*b^5*d^2*e^97 + 6435*(x*e + d)^(7/2)*A*b^6*d^2*e^97 - 72072*(x*e + d)^(5/2)*B*a*b^5*d^3*e^97 - 12012*(x*e +
d)^(5/2)*A*b^6*d^3*e^97 + 90090*(x*e + d)^(3/2)*B*a*b^5*d^4*e^97 + 15015*(x*e + d)^(3/2)*A*b^6*d^4*e^97 - 1081
08*sqrt(x*e + d)*B*a*b^5*d^5*e^97 - 18018*sqrt(x*e + d)*A*b^6*d^5*e^97 + 5005*(x*e + d)^(9/2)*B*a^2*b^4*e^98 +
2002*(x*e + d)^(9/2)*A*a*b^5*e^98 - 32175*(x*e + d)^(7/2)*B*a^2*b^4*d*e^98 - 12870*(x*e + d)^(7/2)*A*a*b^5*d*
e^98 + 90090*(x*e + d)^(5/2)*B*a^2*b^4*d^2*e^98 + 36036*(x*e + d)^(5/2)*A*a*b^5*d^2*e^98 - 150150*(x*e + d)^(3
/2)*B*a^2*b^4*d^3*e^98 - 60060*(x*e + d)^(3/2)*A*a*b^5*d^3*e^98 + 225225*sqrt(x*e + d)*B*a^2*b^4*d^4*e^98 + 90
090*sqrt(x*e + d)*A*a*b^5*d^4*e^98 + 8580*(x*e + d)^(7/2)*B*a^3*b^3*e^99 + 6435*(x*e + d)^(7/2)*A*a^2*b^4*e^99
- 48048*(x*e + d)^(5/2)*B*a^3*b^3*d*e^99 - 36036*(x*e + d)^(5/2)*A*a^2*b^4*d*e^99 + 120120*(x*e + d)^(3/2)*B*
a^3*b^3*d^2*e^99 + 90090*(x*e + d)^(3/2)*A*a^2*b^4*d^2*e^99 - 240240*sqrt(x*e + d)*B*a^3*b^3*d^3*e^99 - 180180
*sqrt(x*e + d)*A*a^2*b^4*d^3*e^99 + 9009*(x*e + d)^(5/2)*B*a^4*b^2*e^100 + 12012*(x*e + d)^(5/2)*A*a^3*b^3*e^1
00 - 45045*(x*e + d)^(3/2)*B*a^4*b^2*d*e^100 - 60060*(x*e + d)^(3/2)*A*a^3*b^3*d*e^100 + 135135*sqrt(x*e + d)*
B*a^4*b^2*d^2*e^100 + 180180*sqrt(x*e + d)*A*a^3*b^3*d^2*e^100 + 6006*(x*e + d)^(3/2)*B*a^5*b*e^101 + 15015*(x
*e + d)^(3/2)*A*a^4*b^2*e^101 - 36036*sqrt(x*e + d)*B*a^5*b*d*e^101 - 90090*sqrt(x*e + d)*A*a^4*b^2*d*e^101 +
3003*sqrt(x*e + d)*B*a^6*e^102 + 18018*sqrt(x*e + d)*A*a^5*b*e^102)*e^(-104) + 2*(B*b^6*d^7 - 6*B*a*b^5*d^6*e
- A*b^6*d^6*e + 15*B*a^2*b^4*d^5*e^2 + 6*A*a*b^5*d^5*e^2 - 20*B*a^3*b^3*d^4*e^3 - 15*A*a^2*b^4*d^4*e^3 + 15*B*
a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 - 6*B*a^5*b*d^2*e^5 - 15*A*a^4*b^2*d^2*e^5 + B*a^6*d*e^6 + 6*A*a^5*b*d*
e^6 - A*a^6*e^7)*e^(-8)/sqrt(x*e + d)

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maple [B]  time = 0.06, size = 913, normalized size = 3.04 \begin {gather*} -\frac {2 \left (-231 B \,b^{6} x^{7} e^{7}-273 A \,b^{6} e^{7} x^{6}-1638 B a \,b^{5} e^{7} x^{6}+294 B \,b^{6} d \,e^{6} x^{6}-2002 A a \,b^{5} e^{7} x^{5}+364 A \,b^{6} d \,e^{6} x^{5}-5005 B \,a^{2} b^{4} e^{7} x^{5}+2184 B a \,b^{5} d \,e^{6} x^{5}-392 B \,b^{6} d^{2} e^{5} x^{5}-6435 A \,a^{2} b^{4} e^{7} x^{4}+2860 A a \,b^{5} d \,e^{6} x^{4}-520 A \,b^{6} d^{2} e^{5} x^{4}-8580 B \,a^{3} b^{3} e^{7} x^{4}+7150 B \,a^{2} b^{4} d \,e^{6} x^{4}-3120 B a \,b^{5} d^{2} e^{5} x^{4}+560 B \,b^{6} d^{3} e^{4} x^{4}-12012 A \,a^{3} b^{3} e^{7} x^{3}+10296 A \,a^{2} b^{4} d \,e^{6} x^{3}-4576 A a \,b^{5} d^{2} e^{5} x^{3}+832 A \,b^{6} d^{3} e^{4} x^{3}-9009 B \,a^{4} b^{2} e^{7} x^{3}+13728 B \,a^{3} b^{3} d \,e^{6} x^{3}-11440 B \,a^{2} b^{4} d^{2} e^{5} x^{3}+4992 B a \,b^{5} d^{3} e^{4} x^{3}-896 B \,b^{6} d^{4} e^{3} x^{3}-15015 A \,a^{4} b^{2} e^{7} x^{2}+24024 A \,a^{3} b^{3} d \,e^{6} x^{2}-20592 A \,a^{2} b^{4} d^{2} e^{5} x^{2}+9152 A a \,b^{5} d^{3} e^{4} x^{2}-1664 A \,b^{6} d^{4} e^{3} x^{2}-6006 B \,a^{5} b \,e^{7} x^{2}+18018 B \,a^{4} b^{2} d \,e^{6} x^{2}-27456 B \,a^{3} b^{3} d^{2} e^{5} x^{2}+22880 B \,a^{2} b^{4} d^{3} e^{4} x^{2}-9984 B a \,b^{5} d^{4} e^{3} x^{2}+1792 B \,b^{6} d^{5} e^{2} x^{2}-18018 A \,a^{5} b \,e^{7} x +60060 A \,a^{4} b^{2} d \,e^{6} x -96096 A \,a^{3} b^{3} d^{2} e^{5} x +82368 A \,a^{2} b^{4} d^{3} e^{4} x -36608 A a \,b^{5} d^{4} e^{3} x +6656 A \,b^{6} d^{5} e^{2} x -3003 B \,a^{6} e^{7} x +24024 B \,a^{5} b d \,e^{6} x -72072 B \,a^{4} b^{2} d^{2} e^{5} x +109824 B \,a^{3} b^{3} d^{3} e^{4} x -91520 B \,a^{2} b^{4} d^{4} e^{3} x +39936 B a \,b^{5} d^{5} e^{2} x -7168 B \,b^{6} d^{6} e x +3003 A \,a^{6} e^{7}-36036 A \,a^{5} b d \,e^{6}+120120 A \,a^{4} b^{2} d^{2} e^{5}-192192 A \,a^{3} b^{3} d^{3} e^{4}+164736 A \,a^{2} b^{4} d^{4} e^{3}-73216 A a \,b^{5} d^{5} e^{2}+13312 A \,b^{6} d^{6} e -6006 B \,a^{6} d \,e^{6}+48048 B \,a^{5} b \,d^{2} e^{5}-144144 B \,a^{4} b^{2} d^{3} e^{4}+219648 B \,a^{3} b^{3} d^{4} e^{3}-183040 B \,a^{2} b^{4} d^{5} e^{2}+79872 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{3003 \sqrt {e x +d}\, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3003*(-231*B*b^6*e^7*x^7-273*A*b^6*e^7*x^6-1638*B*a*b^5*e^7*x^6+294*B*b^6*d*e^6*x^6-2002*A*a*b^5*e^7*x^5+36
4*A*b^6*d*e^6*x^5-5005*B*a^2*b^4*e^7*x^5+2184*B*a*b^5*d*e^6*x^5-392*B*b^6*d^2*e^5*x^5-6435*A*a^2*b^4*e^7*x^4+2
860*A*a*b^5*d*e^6*x^4-520*A*b^6*d^2*e^5*x^4-8580*B*a^3*b^3*e^7*x^4+7150*B*a^2*b^4*d*e^6*x^4-3120*B*a*b^5*d^2*e
^5*x^4+560*B*b^6*d^3*e^4*x^4-12012*A*a^3*b^3*e^7*x^3+10296*A*a^2*b^4*d*e^6*x^3-4576*A*a*b^5*d^2*e^5*x^3+832*A*
b^6*d^3*e^4*x^3-9009*B*a^4*b^2*e^7*x^3+13728*B*a^3*b^3*d*e^6*x^3-11440*B*a^2*b^4*d^2*e^5*x^3+4992*B*a*b^5*d^3*
e^4*x^3-896*B*b^6*d^4*e^3*x^3-15015*A*a^4*b^2*e^7*x^2+24024*A*a^3*b^3*d*e^6*x^2-20592*A*a^2*b^4*d^2*e^5*x^2+91
52*A*a*b^5*d^3*e^4*x^2-1664*A*b^6*d^4*e^3*x^2-6006*B*a^5*b*e^7*x^2+18018*B*a^4*b^2*d*e^6*x^2-27456*B*a^3*b^3*d
^2*e^5*x^2+22880*B*a^2*b^4*d^3*e^4*x^2-9984*B*a*b^5*d^4*e^3*x^2+1792*B*b^6*d^5*e^2*x^2-18018*A*a^5*b*e^7*x+600
60*A*a^4*b^2*d*e^6*x-96096*A*a^3*b^3*d^2*e^5*x+82368*A*a^2*b^4*d^3*e^4*x-36608*A*a*b^5*d^4*e^3*x+6656*A*b^6*d^
5*e^2*x-3003*B*a^6*e^7*x+24024*B*a^5*b*d*e^6*x-72072*B*a^4*b^2*d^2*e^5*x+109824*B*a^3*b^3*d^3*e^4*x-91520*B*a^
2*b^4*d^4*e^3*x+39936*B*a*b^5*d^5*e^2*x-7168*B*b^6*d^6*e*x+3003*A*a^6*e^7-36036*A*a^5*b*d*e^6+120120*A*a^4*b^2
*d^2*e^5-192192*A*a^3*b^3*d^3*e^4+164736*A*a^2*b^4*d^4*e^3-73216*A*a*b^5*d^5*e^2+13312*A*b^6*d^6*e-6006*B*a^6*
d*e^6+48048*B*a^5*b*d^2*e^5-144144*B*a^4*b^2*d^3*e^4+219648*B*a^3*b^3*d^4*e^3-183040*B*a^2*b^4*d^5*e^2+79872*B
*a*b^5*d^6*e-14336*B*b^6*d^7)/(e*x+d)^(1/2)/e^8

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maxima [B]  time = 0.77, size = 775, normalized size = 2.58 \begin {gather*} \frac {2 \, {\left (\frac {231 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{6} - 273 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (7 \, B b^{6} d^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 2145 \, {\left (7 \, B b^{6} d^{3} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{2} - {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 3003 \, {\left (7 \, B b^{6} d^{4} - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 3003 \, {\left (7 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 3003 \, {\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} \sqrt {e x + d}}{e^{7}} + \frac {3003 \, {\left (B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}\right )}}{\sqrt {e x + d} e^{7}}\right )}}{3003 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*((231*(e*x + d)^(13/2)*B*b^6 - 273*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(11/2) + 1001*(7*B*b^6
*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(9/2) - 2145*(7*B*b^6*d^3 - 3*(6*B
*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*x + d)^(7/2) +
3003*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3 + 3*A*
a^2*b^4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^(5/2) - 3003*(7*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*
d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 5*(3*B*a^4*b^2 + 4*A*a
^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*(e*x + d)^(3/2) + 3003*(7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*b^6)*d^5
*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3
*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*sqrt(e*x + d))/e^7 + 3003*(B*b^6*
d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b
)*d*e^6)/(sqrt(e*x + d)*e^7))/e

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mupad [B]  time = 1.94, size = 438, normalized size = 1.46 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{11\,e^8}-\frac {-2\,B\,a^6\,d\,e^6+2\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5-12\,A\,a^5\,b\,d\,e^6-30\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3-40\,A\,a^3\,b^3\,d^3\,e^4-30\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3+12\,B\,a\,b^5\,d^6\,e-12\,A\,a\,b^5\,d^5\,e^2-2\,B\,b^6\,d^7+2\,A\,b^6\,d^6\,e}{e^8\,\sqrt {d+e\,x}}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}\,\left (6\,A\,b\,e+B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,A\,b\,e+2\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {2\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{3\,e^8}+\frac {2\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}\,\left (4\,A\,b\,e+3\,B\,a\,e-7\,B\,b\,d\right )}{e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{7\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(11/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(11*e^8) - (2*A*a^6*e^7 - 2*B*b^6*d^7 + 2*A*b^6*d^6
*e - 2*B*a^6*d*e^6 - 12*A*a*b^5*d^5*e^2 + 12*B*a^5*b*d^2*e^5 + 30*A*a^2*b^4*d^4*e^3 - 40*A*a^3*b^3*d^3*e^4 + 3
0*A*a^4*b^2*d^2*e^5 - 30*B*a^2*b^4*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 - 30*B*a^4*b^2*d^3*e^4 - 12*A*a^5*b*d*e^6 +
12*B*a*b^5*d^6*e)/(e^8*(d + e*x)^(1/2)) + (2*(a*e - b*d)^5*(d + e*x)^(1/2)*(6*A*b*e + B*a*e - 7*B*b*d))/e^8 +
(2*B*b^6*(d + e*x)^(13/2))/(13*e^8) + (2*b*(a*e - b*d)^4*(d + e*x)^(3/2)*(5*A*b*e + 2*B*a*e - 7*B*b*d))/e^8 +
(2*b^4*(a*e - b*d)*(d + e*x)^(9/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d))/(3*e^8) + (2*b^2*(a*e - b*d)^3*(d + e*x)^(5/
2)*(4*A*b*e + 3*B*a*e - 7*B*b*d))/e^8 + (10*b^3*(a*e - b*d)^2*(d + e*x)^(7/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(
7*e^8)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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