3.16.87 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=304 \[ -\frac {2 b^5 (d+e x)^{7/2} (-6 a B e-A b e+7 b B d)}{7 e^8}+\frac {6 b^4 (d+e x)^{5/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac {10 b^3 (d+e x)^{3/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {10 b^2 \sqrt {d+e x} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac {6 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 \sqrt {d+e x}}-\frac {2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^{3/2}}+\frac {2 (b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\frac {2 b^6 B (d+e x)^{9/2}}{9 e^8} \]

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Rubi [A]  time = 0.15, antiderivative size = 304, 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)^{7/2} (-6 a B e-A b e+7 b B d)}{7 e^8}+\frac {6 b^4 (d+e x)^{5/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac {10 b^3 (d+e x)^{3/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {10 b^2 \sqrt {d+e x} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac {6 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 \sqrt {d+e x}}-\frac {2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^{3/2}}+\frac {2 (b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\frac {2 b^6 B (d+e x)^{9/2}}{9 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)^(7/2),x]

[Out]

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

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Mathematica [A]  time = 0.19, size = 259, normalized size = 0.85 \begin {gather*} \frac {2 \left (-45 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+189 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-525 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+1575 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)+945 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)-105 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)+63 (b d-a e)^6 (B d-A e)+35 b^6 B (d+e x)^7\right )}{315 e^8 (d+e x)^{5/2}} \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)^(7/2),x]

[Out]

(2*(63*(b*d - a*e)^6*(B*d - A*e) - 105*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x) + 945*b*(b*d - a*e)
^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 1575*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^
3 - 525*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 189*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e -
5*a*B*e)*(d + e*x)^5 - 45*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 35*b^6*B*(d + e*x)^7))/(315*e^8*(d +
e*x)^(5/2))

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IntegrateAlgebraic [B]  time = 0.23, size = 1069, normalized size = 3.52 \begin {gather*} \frac {2 \left (63 b^6 B d^7-63 A b^6 e d^6-378 a b^5 B e d^6-735 b^6 B (d+e x) d^6+378 a A b^5 e^2 d^5+945 a^2 b^4 B e^2 d^5+6615 b^6 B (d+e x)^2 d^5+630 A b^6 e (d+e x) d^5+3780 a b^5 B e (d+e x) d^5-945 a^2 A b^4 e^3 d^4-1260 a^3 b^3 B e^3 d^4+11025 b^6 B (d+e x)^3 d^4-4725 A b^6 e (d+e x)^2 d^4-28350 a b^5 B e (d+e x)^2 d^4-3150 a A b^5 e^2 (d+e x) d^4-7875 a^2 b^4 B e^2 (d+e x) d^4+1260 a^3 A b^3 e^4 d^3+945 a^4 b^2 B e^4 d^3-3675 b^6 B (d+e x)^4 d^3-6300 A b^6 e (d+e x)^3 d^3-37800 a b^5 B e (d+e x)^3 d^3+18900 a A b^5 e^2 (d+e x)^2 d^3+47250 a^2 b^4 B e^2 (d+e x)^2 d^3+6300 a^2 A b^4 e^3 (d+e x) d^3+8400 a^3 b^3 B e^3 (d+e x) d^3-945 a^4 A b^2 e^5 d^2-378 a^5 b B e^5 d^2+1323 b^6 B (d+e x)^5 d^2+1575 A b^6 e (d+e x)^4 d^2+9450 a b^5 B e (d+e x)^4 d^2+18900 a A b^5 e^2 (d+e x)^3 d^2+47250 a^2 b^4 B e^2 (d+e x)^3 d^2-28350 a^2 A b^4 e^3 (d+e x)^2 d^2-37800 a^3 b^3 B e^3 (d+e x)^2 d^2-6300 a^3 A b^3 e^4 (d+e x) d^2-4725 a^4 b^2 B e^4 (d+e x) d^2+378 a^5 A b e^6 d+63 a^6 B e^6 d-315 b^6 B (d+e x)^6 d-378 A b^6 e (d+e x)^5 d-2268 a b^5 B e (d+e x)^5 d-3150 a A b^5 e^2 (d+e x)^4 d-7875 a^2 b^4 B e^2 (d+e x)^4 d-18900 a^2 A b^4 e^3 (d+e x)^3 d-25200 a^3 b^3 B e^3 (d+e x)^3 d+18900 a^3 A b^3 e^4 (d+e x)^2 d+14175 a^4 b^2 B e^4 (d+e x)^2 d+3150 a^4 A b^2 e^5 (d+e x) d+1260 a^5 b B e^5 (d+e x) d-63 a^6 A e^7+35 b^6 B (d+e x)^7+45 A b^6 e (d+e x)^6+270 a b^5 B e (d+e x)^6+378 a A b^5 e^2 (d+e x)^5+945 a^2 b^4 B e^2 (d+e x)^5+1575 a^2 A b^4 e^3 (d+e x)^4+2100 a^3 b^3 B e^3 (d+e x)^4+6300 a^3 A b^3 e^4 (d+e x)^3+4725 a^4 b^2 B e^4 (d+e x)^3-4725 a^4 A b^2 e^5 (d+e x)^2-1890 a^5 b B e^5 (d+e x)^2-630 a^5 A b e^6 (d+e x)-105 a^6 B e^6 (d+e x)\right )}{315 e^8 (d+e x)^{5/2}} \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)^(7/2),x]

[Out]

(2*(63*b^6*B*d^7 - 63*A*b^6*d^6*e - 378*a*b^5*B*d^6*e + 378*a*A*b^5*d^5*e^2 + 945*a^2*b^4*B*d^5*e^2 - 945*a^2*
A*b^4*d^4*e^3 - 1260*a^3*b^3*B*d^4*e^3 + 1260*a^3*A*b^3*d^3*e^4 + 945*a^4*b^2*B*d^3*e^4 - 945*a^4*A*b^2*d^2*e^
5 - 378*a^5*b*B*d^2*e^5 + 378*a^5*A*b*d*e^6 + 63*a^6*B*d*e^6 - 63*a^6*A*e^7 - 735*b^6*B*d^6*(d + e*x) + 630*A*
b^6*d^5*e*(d + e*x) + 3780*a*b^5*B*d^5*e*(d + e*x) - 3150*a*A*b^5*d^4*e^2*(d + e*x) - 7875*a^2*b^4*B*d^4*e^2*(
d + e*x) + 6300*a^2*A*b^4*d^3*e^3*(d + e*x) + 8400*a^3*b^3*B*d^3*e^3*(d + e*x) - 6300*a^3*A*b^3*d^2*e^4*(d + e
*x) - 4725*a^4*b^2*B*d^2*e^4*(d + e*x) + 3150*a^4*A*b^2*d*e^5*(d + e*x) + 1260*a^5*b*B*d*e^5*(d + e*x) - 630*a
^5*A*b*e^6*(d + e*x) - 105*a^6*B*e^6*(d + e*x) + 6615*b^6*B*d^5*(d + e*x)^2 - 4725*A*b^6*d^4*e*(d + e*x)^2 - 2
8350*a*b^5*B*d^4*e*(d + e*x)^2 + 18900*a*A*b^5*d^3*e^2*(d + e*x)^2 + 47250*a^2*b^4*B*d^3*e^2*(d + e*x)^2 - 283
50*a^2*A*b^4*d^2*e^3*(d + e*x)^2 - 37800*a^3*b^3*B*d^2*e^3*(d + e*x)^2 + 18900*a^3*A*b^3*d*e^4*(d + e*x)^2 + 1
4175*a^4*b^2*B*d*e^4*(d + e*x)^2 - 4725*a^4*A*b^2*e^5*(d + e*x)^2 - 1890*a^5*b*B*e^5*(d + e*x)^2 + 11025*b^6*B
*d^4*(d + e*x)^3 - 6300*A*b^6*d^3*e*(d + e*x)^3 - 37800*a*b^5*B*d^3*e*(d + e*x)^3 + 18900*a*A*b^5*d^2*e^2*(d +
e*x)^3 + 47250*a^2*b^4*B*d^2*e^2*(d + e*x)^3 - 18900*a^2*A*b^4*d*e^3*(d + e*x)^3 - 25200*a^3*b^3*B*d*e^3*(d +
e*x)^3 + 6300*a^3*A*b^3*e^4*(d + e*x)^3 + 4725*a^4*b^2*B*e^4*(d + e*x)^3 - 3675*b^6*B*d^3*(d + e*x)^4 + 1575*
A*b^6*d^2*e*(d + e*x)^4 + 9450*a*b^5*B*d^2*e*(d + e*x)^4 - 3150*a*A*b^5*d*e^2*(d + e*x)^4 - 7875*a^2*b^4*B*d*e
^2*(d + e*x)^4 + 1575*a^2*A*b^4*e^3*(d + e*x)^4 + 2100*a^3*b^3*B*e^3*(d + e*x)^4 + 1323*b^6*B*d^2*(d + e*x)^5
- 378*A*b^6*d*e*(d + e*x)^5 - 2268*a*b^5*B*d*e*(d + e*x)^5 + 378*a*A*b^5*e^2*(d + e*x)^5 + 945*a^2*b^4*B*e^2*(
d + e*x)^5 - 315*b^6*B*d*(d + e*x)^6 + 45*A*b^6*e*(d + e*x)^6 + 270*a*b^5*B*e*(d + e*x)^6 + 35*b^6*B*(d + e*x)
^7))/(315*e^8*(d + e*x)^(5/2))

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fricas [B]  time = 0.44, size = 802, normalized size = 2.64 \begin {gather*} \frac {2 \, {\left (35 \, B b^{6} e^{7} x^{7} + 14336 \, B b^{6} d^{7} - 63 \, A a^{6} e^{7} - 9216 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 16128 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 13440 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5040 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 504 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} - 42 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} - 5 \, {\left (14 \, B b^{6} d e^{6} - 9 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 3 \, {\left (56 \, B b^{6} d^{2} e^{5} - 36 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 63 \, {\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} - 72 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 126 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} - 105 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 5 \, {\left (896 \, B b^{6} d^{4} e^{3} - 576 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 1008 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} - 840 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 315 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 15 \, {\left (1792 \, B b^{6} d^{5} e^{2} - 1152 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 2016 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} - 1680 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 630 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} - 63 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 5 \, {\left (7168 \, B b^{6} d^{6} e - 4608 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 8064 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} - 6720 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 2520 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} - 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} - 21 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^6*e^7*x^7 + 14336*B*b^6*d^7 - 63*A*a^6*e^7 - 9216*(6*B*a*b^5 + A*b^6)*d^6*e + 16128*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^5*e^2 - 13440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5040*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 -
504*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 42*(B*a^6 + 6*A*a^5*b)*d*e^6 - 5*(14*B*b^6*d*e^6 - 9*(6*B*a*b^5 + A*b
^6)*e^7)*x^6 + 3*(56*B*b^6*d^2*e^5 - 36*(6*B*a*b^5 + A*b^6)*d*e^6 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 5*
(112*B*b^6*d^3*e^4 - 72*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 126*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 105*(4*B*a^3*b^3 +
3*A*a^2*b^4)*e^7)*x^4 + 5*(896*B*b^6*d^4*e^3 - 576*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 1008*(5*B*a^2*b^4 + 2*A*a*b^
5)*d^2*e^5 - 840*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 315*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 15*(1792*B*b^6
*d^5*e^2 - 1152*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 2016*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 1680*(4*B*a^3*b^3 + 3*A
*a^2*b^4)*d^2*e^5 + 630*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 63*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 5*(7168*B*
b^6*d^6*e - 4608*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 8064*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 6720*(4*B*a^3*b^3 + 3*
A*a^2*b^4)*d^3*e^4 + 2520*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 252*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 - 21*(B*a^
6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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giac [B]  time = 0.33, size = 1103, normalized size = 3.63

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)^(7/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*b^6*e^64 - 315*(x*e + d)^(7/2)*B*b^6*d*e^64 + 1323*(x*e + d)^(5/2)*B*b^6*d^2*e^64
- 3675*(x*e + d)^(3/2)*B*b^6*d^3*e^64 + 11025*sqrt(x*e + d)*B*b^6*d^4*e^64 + 270*(x*e + d)^(7/2)*B*a*b^5*e^65
+ 45*(x*e + d)^(7/2)*A*b^6*e^65 - 2268*(x*e + d)^(5/2)*B*a*b^5*d*e^65 - 378*(x*e + d)^(5/2)*A*b^6*d*e^65 + 945
0*(x*e + d)^(3/2)*B*a*b^5*d^2*e^65 + 1575*(x*e + d)^(3/2)*A*b^6*d^2*e^65 - 37800*sqrt(x*e + d)*B*a*b^5*d^3*e^6
5 - 6300*sqrt(x*e + d)*A*b^6*d^3*e^65 + 945*(x*e + d)^(5/2)*B*a^2*b^4*e^66 + 378*(x*e + d)^(5/2)*A*a*b^5*e^66
- 7875*(x*e + d)^(3/2)*B*a^2*b^4*d*e^66 - 3150*(x*e + d)^(3/2)*A*a*b^5*d*e^66 + 47250*sqrt(x*e + d)*B*a^2*b^4*
d^2*e^66 + 18900*sqrt(x*e + d)*A*a*b^5*d^2*e^66 + 2100*(x*e + d)^(3/2)*B*a^3*b^3*e^67 + 1575*(x*e + d)^(3/2)*A
*a^2*b^4*e^67 - 25200*sqrt(x*e + d)*B*a^3*b^3*d*e^67 - 18900*sqrt(x*e + d)*A*a^2*b^4*d*e^67 + 4725*sqrt(x*e +
d)*B*a^4*b^2*e^68 + 6300*sqrt(x*e + d)*A*a^3*b^3*e^68)*e^(-72) + 2/15*(315*(x*e + d)^2*B*b^6*d^5 - 35*(x*e + d
)*B*b^6*d^6 + 3*B*b^6*d^7 - 1350*(x*e + d)^2*B*a*b^5*d^4*e - 225*(x*e + d)^2*A*b^6*d^4*e + 180*(x*e + d)*B*a*b
^5*d^5*e + 30*(x*e + d)*A*b^6*d^5*e - 18*B*a*b^5*d^6*e - 3*A*b^6*d^6*e + 2250*(x*e + d)^2*B*a^2*b^4*d^3*e^2 +
900*(x*e + d)^2*A*a*b^5*d^3*e^2 - 375*(x*e + d)*B*a^2*b^4*d^4*e^2 - 150*(x*e + d)*A*a*b^5*d^4*e^2 + 45*B*a^2*b
^4*d^5*e^2 + 18*A*a*b^5*d^5*e^2 - 1800*(x*e + d)^2*B*a^3*b^3*d^2*e^3 - 1350*(x*e + d)^2*A*a^2*b^4*d^2*e^3 + 40
0*(x*e + d)*B*a^3*b^3*d^3*e^3 + 300*(x*e + d)*A*a^2*b^4*d^3*e^3 - 60*B*a^3*b^3*d^4*e^3 - 45*A*a^2*b^4*d^4*e^3
+ 675*(x*e + d)^2*B*a^4*b^2*d*e^4 + 900*(x*e + d)^2*A*a^3*b^3*d*e^4 - 225*(x*e + d)*B*a^4*b^2*d^2*e^4 - 300*(x
*e + d)*A*a^3*b^3*d^2*e^4 + 45*B*a^4*b^2*d^3*e^4 + 60*A*a^3*b^3*d^3*e^4 - 90*(x*e + d)^2*B*a^5*b*e^5 - 225*(x*
e + d)^2*A*a^4*b^2*e^5 + 60*(x*e + d)*B*a^5*b*d*e^5 + 150*(x*e + d)*A*a^4*b^2*d*e^5 - 18*B*a^5*b*d^2*e^5 - 45*
A*a^4*b^2*d^2*e^5 - 5*(x*e + d)*B*a^6*e^6 - 30*(x*e + d)*A*a^5*b*e^6 + 3*B*a^6*d*e^6 + 18*A*a^5*b*d*e^6 - 3*A*
a^6*e^7)*e^(-8)/(x*e + d)^(5/2)

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maple [B]  time = 0.05, size = 913, normalized size = 3.00 \begin {gather*} -\frac {2 \left (-35 B \,b^{6} x^{7} e^{7}-45 A \,b^{6} e^{7} x^{6}-270 B a \,b^{5} e^{7} x^{6}+70 B \,b^{6} d \,e^{6} x^{6}-378 A a \,b^{5} e^{7} x^{5}+108 A \,b^{6} d \,e^{6} x^{5}-945 B \,a^{2} b^{4} e^{7} x^{5}+648 B a \,b^{5} d \,e^{6} x^{5}-168 B \,b^{6} d^{2} e^{5} x^{5}-1575 A \,a^{2} b^{4} e^{7} x^{4}+1260 A a \,b^{5} d \,e^{6} x^{4}-360 A \,b^{6} d^{2} e^{5} x^{4}-2100 B \,a^{3} b^{3} e^{7} x^{4}+3150 B \,a^{2} b^{4} d \,e^{6} x^{4}-2160 B a \,b^{5} d^{2} e^{5} x^{4}+560 B \,b^{6} d^{3} e^{4} x^{4}-6300 A \,a^{3} b^{3} e^{7} x^{3}+12600 A \,a^{2} b^{4} d \,e^{6} x^{3}-10080 A a \,b^{5} d^{2} e^{5} x^{3}+2880 A \,b^{6} d^{3} e^{4} x^{3}-4725 B \,a^{4} b^{2} e^{7} x^{3}+16800 B \,a^{3} b^{3} d \,e^{6} x^{3}-25200 B \,a^{2} b^{4} d^{2} e^{5} x^{3}+17280 B a \,b^{5} d^{3} e^{4} x^{3}-4480 B \,b^{6} d^{4} e^{3} x^{3}+4725 A \,a^{4} b^{2} e^{7} x^{2}-37800 A \,a^{3} b^{3} d \,e^{6} x^{2}+75600 A \,a^{2} b^{4} d^{2} e^{5} x^{2}-60480 A a \,b^{5} d^{3} e^{4} x^{2}+17280 A \,b^{6} d^{4} e^{3} x^{2}+1890 B \,a^{5} b \,e^{7} x^{2}-28350 B \,a^{4} b^{2} d \,e^{6} x^{2}+100800 B \,a^{3} b^{3} d^{2} e^{5} x^{2}-151200 B \,a^{2} b^{4} d^{3} e^{4} x^{2}+103680 B a \,b^{5} d^{4} e^{3} x^{2}-26880 B \,b^{6} d^{5} e^{2} x^{2}+630 A \,a^{5} b \,e^{7} x +6300 A \,a^{4} b^{2} d \,e^{6} x -50400 A \,a^{3} b^{3} d^{2} e^{5} x +100800 A \,a^{2} b^{4} d^{3} e^{4} x -80640 A a \,b^{5} d^{4} e^{3} x +23040 A \,b^{6} d^{5} e^{2} x +105 B \,a^{6} e^{7} x +2520 B \,a^{5} b d \,e^{6} x -37800 B \,a^{4} b^{2} d^{2} e^{5} x +134400 B \,a^{3} b^{3} d^{3} e^{4} x -201600 B \,a^{2} b^{4} d^{4} e^{3} x +138240 B a \,b^{5} d^{5} e^{2} x -35840 B \,b^{6} d^{6} e x +63 A \,a^{6} e^{7}+252 A \,a^{5} b d \,e^{6}+2520 A \,a^{4} b^{2} d^{2} e^{5}-20160 A \,a^{3} b^{3} d^{3} e^{4}+40320 A \,a^{2} b^{4} d^{4} e^{3}-32256 A a \,b^{5} d^{5} e^{2}+9216 A \,b^{6} d^{6} e +42 B \,a^{6} d \,e^{6}+1008 B \,a^{5} b \,d^{2} e^{5}-15120 B \,a^{4} b^{2} d^{3} e^{4}+53760 B \,a^{3} b^{3} d^{4} e^{3}-80640 B \,a^{2} b^{4} d^{5} e^{2}+55296 B a \,b^{5} d^{6} e -14336 B \,b^{6} d^{7}\right )}{315 \left (e x +d \right )^{\frac {5}{2}} 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)^(7/2),x)

[Out]

-2/315*(-35*B*b^6*e^7*x^7-45*A*b^6*e^7*x^6-270*B*a*b^5*e^7*x^6+70*B*b^6*d*e^6*x^6-378*A*a*b^5*e^7*x^5+108*A*b^
6*d*e^6*x^5-945*B*a^2*b^4*e^7*x^5+648*B*a*b^5*d*e^6*x^5-168*B*b^6*d^2*e^5*x^5-1575*A*a^2*b^4*e^7*x^4+1260*A*a*
b^5*d*e^6*x^4-360*A*b^6*d^2*e^5*x^4-2100*B*a^3*b^3*e^7*x^4+3150*B*a^2*b^4*d*e^6*x^4-2160*B*a*b^5*d^2*e^5*x^4+5
60*B*b^6*d^3*e^4*x^4-6300*A*a^3*b^3*e^7*x^3+12600*A*a^2*b^4*d*e^6*x^3-10080*A*a*b^5*d^2*e^5*x^3+2880*A*b^6*d^3
*e^4*x^3-4725*B*a^4*b^2*e^7*x^3+16800*B*a^3*b^3*d*e^6*x^3-25200*B*a^2*b^4*d^2*e^5*x^3+17280*B*a*b^5*d^3*e^4*x^
3-4480*B*b^6*d^4*e^3*x^3+4725*A*a^4*b^2*e^7*x^2-37800*A*a^3*b^3*d*e^6*x^2+75600*A*a^2*b^4*d^2*e^5*x^2-60480*A*
a*b^5*d^3*e^4*x^2+17280*A*b^6*d^4*e^3*x^2+1890*B*a^5*b*e^7*x^2-28350*B*a^4*b^2*d*e^6*x^2+100800*B*a^3*b^3*d^2*
e^5*x^2-151200*B*a^2*b^4*d^3*e^4*x^2+103680*B*a*b^5*d^4*e^3*x^2-26880*B*b^6*d^5*e^2*x^2+630*A*a^5*b*e^7*x+6300
*A*a^4*b^2*d*e^6*x-50400*A*a^3*b^3*d^2*e^5*x+100800*A*a^2*b^4*d^3*e^4*x-80640*A*a*b^5*d^4*e^3*x+23040*A*b^6*d^
5*e^2*x+105*B*a^6*e^7*x+2520*B*a^5*b*d*e^6*x-37800*B*a^4*b^2*d^2*e^5*x+134400*B*a^3*b^3*d^3*e^4*x-201600*B*a^2
*b^4*d^4*e^3*x+138240*B*a*b^5*d^5*e^2*x-35840*B*b^6*d^6*e*x+63*A*a^6*e^7+252*A*a^5*b*d*e^6+2520*A*a^4*b^2*d^2*
e^5-20160*A*a^3*b^3*d^3*e^4+40320*A*a^2*b^4*d^4*e^3-32256*A*a*b^5*d^5*e^2+9216*A*b^6*d^6*e+42*B*a^6*d*e^6+1008
*B*a^5*b*d^2*e^5-15120*B*a^4*b^2*d^3*e^4+53760*B*a^3*b^3*d^4*e^3-80640*B*a^2*b^4*d^5*e^2+55296*B*a*b^5*d^6*e-1
4336*B*b^6*d^7)/(e*x+d)^(5/2)/e^8

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maxima [B]  time = 0.63, size = 775, normalized size = 2.55 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{6} - 45 \, {\left (7 \, B b^{6} d - {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\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 {5}{2}} - 525 \, {\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 {3}{2}} + 1575 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {21 \, {\left (3 \, B b^{6} d^{7} - 3 \, A a^{6} e^{7} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 9 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 3 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 45 \, {\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 )}^{2} - 5 \, {\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 )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{7}}\right )}}{315 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*B*b^6 - 45*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(7/2) + 189*(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)^(5/2) - 525*(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)^(3/2) + 1575*(
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)*sqrt(e*x + d))/e^7 + 21*(3*B*b^6*d^7 - 3*A*a^6*e^7 - 3*(6*B*a*b^5 +
A*b^6)*d^6*e + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d^3*e^4 - 9*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 3*(B*a^6 + 6*A*a^5*b)*d*e^6 + 45*(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)^2 - 5*(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)*(e*x + d))/((e*x
+ d)^(5/2)*e^7))/e

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mupad [B]  time = 1.96, size = 675, normalized size = 2.22 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^6\,e-14\,B\,b^6\,d+12\,B\,a\,b^5\,e\right )}{7\,e^8}-\frac {{\left (d+e\,x\right )}^2\,\left (12\,B\,a^5\,b\,e^5-90\,B\,a^4\,b^2\,d\,e^4+30\,A\,a^4\,b^2\,e^5+240\,B\,a^3\,b^3\,d^2\,e^3-120\,A\,a^3\,b^3\,d\,e^4-300\,B\,a^2\,b^4\,d^3\,e^2+180\,A\,a^2\,b^4\,d^2\,e^3+180\,B\,a\,b^5\,d^4\,e-120\,A\,a\,b^5\,d^3\,e^2-42\,B\,b^6\,d^5+30\,A\,b^6\,d^4\,e\right )+\left (d+e\,x\right )\,\left (\frac {2\,B\,a^6\,e^6}{3}-8\,B\,a^5\,b\,d\,e^5+4\,A\,a^5\,b\,e^6+30\,B\,a^4\,b^2\,d^2\,e^4-20\,A\,a^4\,b^2\,d\,e^5-\frac {160\,B\,a^3\,b^3\,d^3\,e^3}{3}+40\,A\,a^3\,b^3\,d^2\,e^4+50\,B\,a^2\,b^4\,d^4\,e^2-40\,A\,a^2\,b^4\,d^3\,e^3-24\,B\,a\,b^5\,d^5\,e+20\,A\,a\,b^5\,d^4\,e^2+\frac {14\,B\,b^6\,d^6}{3}-4\,A\,b^6\,d^5\,e\right )+\frac {2\,A\,a^6\,e^7}{5}-\frac {2\,B\,b^6\,d^7}{5}+\frac {2\,A\,b^6\,d^6\,e}{5}-\frac {2\,B\,a^6\,d\,e^6}{5}-\frac {12\,A\,a\,b^5\,d^5\,e^2}{5}+\frac {12\,B\,a^5\,b\,d^2\,e^5}{5}+6\,A\,a^2\,b^4\,d^4\,e^3-8\,A\,a^3\,b^3\,d^3\,e^4+6\,A\,a^4\,b^2\,d^2\,e^5-6\,B\,a^2\,b^4\,d^5\,e^2+8\,B\,a^3\,b^3\,d^4\,e^3-6\,B\,a^4\,b^2\,d^3\,e^4-\frac {12\,A\,a^5\,b\,d\,e^6}{5}+\frac {12\,B\,a\,b^5\,d^6\,e}{5}}{e^8\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,B\,b^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {6\,b^4\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e+5\,B\,a\,e-7\,B\,b\,d\right )}{5\,e^8}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}\,\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 )}^{3/2}\,\left (3\,A\,b\,e+4\,B\,a\,e-7\,B\,b\,d\right )}{3\,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)^(7/2),x)

[Out]

((d + e*x)^(7/2)*(2*A*b^6*e - 14*B*b^6*d + 12*B*a*b^5*e))/(7*e^8) - ((d + e*x)^2*(12*B*a^5*b*e^5 - 42*B*b^6*d^
5 + 30*A*b^6*d^4*e + 30*A*a^4*b^2*e^5 - 120*A*a*b^5*d^3*e^2 - 120*A*a^3*b^3*d*e^4 - 90*B*a^4*b^2*d*e^4 + 180*A
*a^2*b^4*d^2*e^3 - 300*B*a^2*b^4*d^3*e^2 + 240*B*a^3*b^3*d^2*e^3 + 180*B*a*b^5*d^4*e) + (d + e*x)*((2*B*a^6*e^
6)/3 + (14*B*b^6*d^6)/3 + 4*A*a^5*b*e^6 - 4*A*b^6*d^5*e + 20*A*a*b^5*d^4*e^2 - 20*A*a^4*b^2*d*e^5 - 40*A*a^2*b
^4*d^3*e^3 + 40*A*a^3*b^3*d^2*e^4 + 50*B*a^2*b^4*d^4*e^2 - (160*B*a^3*b^3*d^3*e^3)/3 + 30*B*a^4*b^2*d^2*e^4 -
24*B*a*b^5*d^5*e - 8*B*a^5*b*d*e^5) + (2*A*a^6*e^7)/5 - (2*B*b^6*d^7)/5 + (2*A*b^6*d^6*e)/5 - (2*B*a^6*d*e^6)/
5 - (12*A*a*b^5*d^5*e^2)/5 + (12*B*a^5*b*d^2*e^5)/5 + 6*A*a^2*b^4*d^4*e^3 - 8*A*a^3*b^3*d^3*e^4 + 6*A*a^4*b^2*
d^2*e^5 - 6*B*a^2*b^4*d^5*e^2 + 8*B*a^3*b^3*d^4*e^3 - 6*B*a^4*b^2*d^3*e^4 - (12*A*a^5*b*d*e^6)/5 + (12*B*a*b^5
*d^6*e)/5)/(e^8*(d + e*x)^(5/2)) + (2*B*b^6*(d + e*x)^(9/2))/(9*e^8) + (6*b^4*(a*e - b*d)*(d + e*x)^(5/2)*(2*A
*b*e + 5*B*a*e - 7*B*b*d))/(5*e^8) + (10*b^2*(a*e - b*d)^3*(d + e*x)^(1/2)*(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)^(3/2)*(3*A*b*e + 4*B*a*e - 7*B*b*d))/(3*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)**(7/2),x)

[Out]

Timed out

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