3.19.7 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{7/2}} \, dx\) [1807]

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

[Out]

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

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Rubi [A]
time = 0.10, 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, 78} \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 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]))))

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 [B] Leaf count is larger than twice the leaf count of optimal. \(627\) vs. \(2(304)=608\).
time = 0.53, size = 627, normalized size = 2.06 \begin {gather*} -\frac {2 \left (21 a^6 e^6 (2 B d+3 A e+5 B e x)+126 a^5 b e^5 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )-315 a^4 b^2 e^4 \left (-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )+3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+420 a^3 b^3 e^3 \left (-3 A e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )-315 a^2 b^4 e^2 \left (A e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+B \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )+18 a b^5 e \left (-7 A e \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )+3 B \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )+b^6 \left (9 A e \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )-7 B \left (2048 d^7+5120 d^6 e x+3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5-10 d e^6 x^6+5 e^7 x^7\right )\right )\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*(21*a^6*e^6*(2*B*d + 3*A*e + 5*B*e*x) + 126*a^5*b*e^5*(A*e*(2*d + 5*e*x) + B*(8*d^2 + 20*d*e*x + 15*e^2*x^
2)) - 315*a^4*b^2*e^4*(-(A*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^
3*x^3)) + 420*a^3*b^3*e^3*(-3*A*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 + 320*d^3*e*x
+ 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)) - 315*a^2*b^4*e^2*(A*e*(-128*d^4 - 320*d^3*e*x - 240*d^2*e^2*x^
2 - 40*d*e^3*x^3 + 5*e^4*x^4) + B*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3
*e^5*x^5)) + 18*a*b^5*e*(-7*A*e*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e
^5*x^5) + 3*B*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d*e^5*x^5 -
5*e^6*x^6)) + b^6*(9*A*e*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 12*d
*e^5*x^5 - 5*e^6*x^6) - 7*B*(2048*d^7 + 5120*d^6*e*x + 3840*d^5*e^2*x^2 + 640*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 2
4*d^2*e^5*x^5 - 10*d*e^6*x^6 + 5*e^7*x^7))))/(315*e^8*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(939\) vs. \(2(276)=552\).
time = 1.05, size = 940, normalized size = 3.09 Too large to display

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,method=_RETURNVERBOSE)

[Out]

2/e^8*(-1/5*(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b^3*d^3*e^4+15*A*a^2*b^4*d^4*e^3-6*A*a*b^
5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*a^4*b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d^5
*e^2+6*B*a*b^5*d^6*e-B*b^6*d^7)/(e*x+d)^(5/2)+21/5*B*b^6*d^2*(e*x+d)^(5/2)-35/3*B*b^6*d^3*(e*x+d)^(3/2)+1/7*A*
b^6*e*(e*x+d)^(7/2)+35*B*b^6*d^4*(e*x+d)^(1/2)-10*A*a*b^5*d*e^2*(e*x+d)^(3/2)-25*B*a^2*b^4*d*e^2*(e*x+d)^(3/2)
-20*A*b^6*d^3*e*(e*x+d)^(1/2)+15*B*a^4*b^2*e^4*(e*x+d)^(1/2)+5*A*a^2*b^4*e^3*(e*x+d)^(3/2)+6/7*B*a*b^5*e*(e*x+
d)^(7/2)+5*A*b^6*d^2*e*(e*x+d)^(3/2)+20/3*B*a^3*b^3*e^3*(e*x+d)^(3/2)-1/3*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60
*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b*d*e^5+45*B*a^4*b
^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a*b^5*d^5*e+7*B*b^6*d^6)/(e*x+d)^(3/2)+30*B*a*b^5*d^
2*e*(e*x+d)^(3/2)+60*A*a*b^5*d^2*e^2*(e*x+d)^(1/2)+150*B*a^2*b^4*d^2*e^2*(e*x+d)^(1/2)-60*A*a^2*b^4*d*e^3*(e*x
+d)^(1/2)-120*B*a*b^5*d^3*e*(e*x+d)^(1/2)-80*B*a^3*b^3*d*e^3*(e*x+d)^(1/2)-36/5*B*a*b^5*d*e*(e*x+d)^(5/2)+20*A
*a^3*b^3*e^4*(e*x+d)^(1/2)+6/5*A*a*b^5*e^2*(e*x+d)^(5/2)-6/5*A*b^6*d*e*(e*x+d)^(5/2)+3*B*a^2*b^4*e^2*(e*x+d)^(
5/2)-B*b^6*d*(e*x+d)^(7/2)+1/9*B*b^6*(e*x+d)^(9/2)-3*b*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-
20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3-50*B*a^2*b^3*d^3*e^2+30*B*a
*b^4*d^4*e-7*B*b^5*d^5)/(e*x+d)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (295) = 590\).
time = 0.30, size = 806, normalized size = 2.65 \begin {gather*} \frac {2}{315} \, {\left ({\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{6} - 45 \, {\left (7 \, B b^{6} d - 6 \, B a b^{5} e - A b^{6} e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 189 \, {\left (7 \, B b^{6} d^{2} + 5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2} - 2 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 525 \, {\left (7 \, B b^{6} d^{3} - 4 \, B a^{3} b^{3} e^{3} - 3 \, A a^{2} b^{4} e^{3} - 3 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{2} + 3 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 1575 \, {\left (7 \, B b^{6} d^{4} + 3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4} - 4 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{3} + 6 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{2} - 4 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-7\right )} + \frac {21 \, {\left (3 \, B b^{6} d^{7} - 3 \, A a^{6} e^{7} - 3 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 9 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} - 15 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 15 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 45 \, {\left (7 \, B b^{6} d^{5} - 2 \, B a^{5} b e^{5} - 5 \, A a^{4} b^{2} e^{5} - 5 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{4} + 10 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{3} - 10 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{2} + 5 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d\right )} {\left (x e + d\right )}^{2} - 9 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} - 5 \, {\left (7 \, B b^{6} d^{6} + B a^{6} e^{6} + 6 \, A a^{5} b e^{6} - 6 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{5} + 15 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{4} - 20 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{3} + 15 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{2} - 6 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d\right )} {\left (x e + d\right )} + 3 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d\right )} e^{\left (-7\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\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="maxima")

[Out]

2/315*((35*(x*e + d)^(9/2)*B*b^6 - 45*(7*B*b^6*d - 6*B*a*b^5*e - A*b^6*e)*(x*e + d)^(7/2) + 189*(7*B*b^6*d^2 +
5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2 - 2*(6*B*a*b^5*e + A*b^6*e)*d)*(x*e + d)^(5/2) - 525*(7*B*b^6*d^3 - 4*B*a^3*b
^3*e^3 - 3*A*a^2*b^4*e^3 - 3*(6*B*a*b^5*e + A*b^6*e)*d^2 + 3*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d)*(x*e + d)^(3
/2) + 1575*(7*B*b^6*d^4 + 3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4 - 4*(6*B*a*b^5*e + A*b^6*e)*d^3 + 6*(5*B*a^2*b^4*e
^2 + 2*A*a*b^5*e^2)*d^2 - 4*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d)*sqrt(x*e + d))*e^(-7) + 21*(3*B*b^6*d^7 - 3
*A*a^6*e^7 - 3*(6*B*a*b^5*e + A*b^6*e)*d^6 + 9*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 - 15*(4*B*a^3*b^3*e^3 + 3
*A*a^2*b^4*e^3)*d^4 + 15*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^3 + 45*(7*B*b^6*d^5 - 2*B*a^5*b*e^5 - 5*A*a^4*b
^2*e^5 - 5*(6*B*a*b^5*e + A*b^6*e)*d^4 + 10*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^3 - 10*(4*B*a^3*b^3*e^3 + 3*A*
a^2*b^4*e^3)*d^2 + 5*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d)*(x*e + d)^2 - 9*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*
d^2 - 5*(7*B*b^6*d^6 + B*a^6*e^6 + 6*A*a^5*b*e^6 - 6*(6*B*a*b^5*e + A*b^6*e)*d^5 + 15*(5*B*a^2*b^4*e^2 + 2*A*a
*b^5*e^2)*d^4 - 20*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d^3 + 15*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^2 - 6*(2
*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d)*(x*e + d) + 3*(B*a^6*e^6 + 6*A*a^5*b*e^6)*d)*e^(-7)/(x*e + d)^(5/2))*e^(-1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (295) = 590\).
time = 1.72, size = 765, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left (14336 \, B b^{6} d^{7} + {\left (35 \, B b^{6} x^{7} - 63 \, A a^{6} + 45 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 189 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 525 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1575 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 945 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 105 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} - 2 \, {\left (35 \, B b^{6} d x^{6} + 54 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 315 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} + 2100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} - 4725 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 630 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x + 21 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 24 \, {\left (7 \, B b^{6} d^{2} x^{5} + 15 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 210 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} - 1050 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x - 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} - 80 \, {\left (7 \, B b^{6} d^{3} x^{4} + 36 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} - 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 420 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x - 63 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + 640 \, {\left (7 \, B b^{6} d^{4} x^{3} - 27 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 63 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x - 21 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 768 \, {\left (35 \, B b^{6} d^{5} x^{2} - 30 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + 21 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + 1024 \, {\left (35 \, B b^{6} d^{6} x - 9 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e\right )} \sqrt {x e + d}}{315 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + 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*(14336*B*b^6*d^7 + (35*B*b^6*x^7 - 63*A*a^6 + 45*(6*B*a*b^5 + A*b^6)*x^6 + 189*(5*B*a^2*b^4 + 2*A*a*b^5)
*x^5 + 525*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 1575*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 945*(2*B*a^5*b + 5*A*a^4*b
^2)*x^2 - 105*(B*a^6 + 6*A*a^5*b)*x)*e^7 - 2*(35*B*b^6*d*x^6 + 54*(6*B*a*b^5 + A*b^6)*d*x^5 + 315*(5*B*a^2*b^4
+ 2*A*a*b^5)*d*x^4 + 2100*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^3 - 4725*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^2 + 630*(2
*B*a^5*b + 5*A*a^4*b^2)*d*x + 21*(B*a^6 + 6*A*a^5*b)*d)*e^6 + 24*(7*B*b^6*d^2*x^5 + 15*(6*B*a*b^5 + A*b^6)*d^2
*x^4 + 210*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^3 - 1050*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*x^2 + 525*(3*B*a^4*b^2 + 4
*A*a^3*b^3)*d^2*x - 21*(2*B*a^5*b + 5*A*a^4*b^2)*d^2)*e^5 - 80*(7*B*b^6*d^3*x^4 + 36*(6*B*a*b^5 + A*b^6)*d^3*x
^3 - 378*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^2 + 420*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x - 63*(3*B*a^4*b^2 + 4*A*a^3
*b^3)*d^3)*e^4 + 640*(7*B*b^6*d^4*x^3 - 27*(6*B*a*b^5 + A*b^6)*d^4*x^2 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x -
21*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4)*e^3 + 768*(35*B*b^6*d^5*x^2 - 30*(6*B*a*b^5 + A*b^6)*d^5*x + 21*(5*B*a^2*b
^4 + 2*A*a*b^5)*d^5)*e^2 + 1024*(35*B*b^6*d^6*x - 9*(6*B*a*b^5 + A*b^6)*d^6)*e)*sqrt(x*e + d)/(x^3*e^11 + 3*d*
x^2*e^10 + 3*d^2*x*e^9 + d^3*e^8)

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Sympy [A]
time = 163.67, size = 491, normalized size = 1.62 \begin {gather*} \frac {2 B b^{6} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{8}} - \frac {6 b \left (a e - b d\right )^{4} \cdot \left (5 A b e + 2 B a e - 7 B b d\right )}{e^{8} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A b^{6} e + 12 B a b^{5} e - 14 B b^{6} d\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (12 A a b^{5} e^{2} - 12 A b^{6} d e + 30 B a^{2} b^{4} e^{2} - 72 B a b^{5} d e + 42 B b^{6} d^{2}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (30 A a^{2} b^{4} e^{3} - 60 A a b^{5} d e^{2} + 30 A b^{6} d^{2} e + 40 B a^{3} b^{3} e^{3} - 150 B a^{2} b^{4} d e^{2} + 180 B a b^{5} d^{2} e - 70 B b^{6} d^{3}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (40 A a^{3} b^{3} e^{4} - 120 A a^{2} b^{4} d e^{3} + 120 A a b^{5} d^{2} e^{2} - 40 A b^{6} d^{3} e + 30 B a^{4} b^{2} e^{4} - 160 B a^{3} b^{3} d e^{3} + 300 B a^{2} b^{4} d^{2} e^{2} - 240 B a b^{5} d^{3} e + 70 B b^{6} d^{4}\right )}{e^{8}} - \frac {2 \left (a e - b d\right )^{5} \cdot \left (6 A b e + B a e - 7 B b d\right )}{3 e^{8} \left (d + e x\right )^{\frac {3}{2}}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{6}}{5 e^{8} \left (d + e x\right )^{\frac {5}{2}}} \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]

2*B*b**6*(d + e*x)**(9/2)/(9*e**8) - 6*b*(a*e - b*d)**4*(5*A*b*e + 2*B*a*e - 7*B*b*d)/(e**8*sqrt(d + e*x)) + (
d + e*x)**(7/2)*(2*A*b**6*e + 12*B*a*b**5*e - 14*B*b**6*d)/(7*e**8) + (d + e*x)**(5/2)*(12*A*a*b**5*e**2 - 12*
A*b**6*d*e + 30*B*a**2*b**4*e**2 - 72*B*a*b**5*d*e + 42*B*b**6*d**2)/(5*e**8) + (d + e*x)**(3/2)*(30*A*a**2*b*
*4*e**3 - 60*A*a*b**5*d*e**2 + 30*A*b**6*d**2*e + 40*B*a**3*b**3*e**3 - 150*B*a**2*b**4*d*e**2 + 180*B*a*b**5*
d**2*e - 70*B*b**6*d**3)/(3*e**8) + sqrt(d + e*x)*(40*A*a**3*b**3*e**4 - 120*A*a**2*b**4*d*e**3 + 120*A*a*b**5
*d**2*e**2 - 40*A*b**6*d**3*e + 30*B*a**4*b**2*e**4 - 160*B*a**3*b**3*d*e**3 + 300*B*a**2*b**4*d**2*e**2 - 240
*B*a*b**5*d**3*e + 70*B*b**6*d**4)/e**8 - 2*(a*e - b*d)**5*(6*A*b*e + B*a*e - 7*B*b*d)/(3*e**8*(d + e*x)**(3/2
)) + 2*(-A*e + B*d)*(a*e - b*d)**6/(5*e**8*(d + e*x)**(5/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (295) = 590\).
time = 1.25, size = 1103, normalized size = 3.63 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{6} e^{64} - 315 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{6} d e^{64} + 1323 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{6} d^{2} e^{64} - 3675 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{6} d^{3} e^{64} + 11025 \, \sqrt {x e + d} B b^{6} d^{4} e^{64} + 270 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{5} e^{65} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{6} e^{65} - 2268 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{5} d e^{65} - 378 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{6} d e^{65} + 9450 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{5} d^{2} e^{65} + 1575 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{6} d^{2} e^{65} - 37800 \, \sqrt {x e + d} B a b^{5} d^{3} e^{65} - 6300 \, \sqrt {x e + d} A b^{6} d^{3} e^{65} + 945 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{4} e^{66} + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{5} e^{66} - 7875 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{4} d e^{66} - 3150 \, {\left (x e + d\right )}^{\frac {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 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{3} e^{67} + 1575 \, {\left (x e + d\right )}^{\frac {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}\right )} e^{\left (-72\right )} + \frac {2 \, {\left (315 \, {\left (x e + d\right )}^{2} B b^{6} d^{5} - 35 \, {\left (x e + d\right )} B b^{6} d^{6} + 3 \, B b^{6} d^{7} - 1350 \, {\left (x e + d\right )}^{2} B a b^{5} d^{4} e - 225 \, {\left (x e + d\right )}^{2} A b^{6} d^{4} e + 180 \, {\left (x e + d\right )} B a b^{5} d^{5} e + 30 \, {\left (x e + d\right )} A b^{6} d^{5} e - 18 \, B a b^{5} d^{6} e - 3 \, A b^{6} d^{6} e + 2250 \, {\left (x e + d\right )}^{2} B a^{2} b^{4} d^{3} e^{2} + 900 \, {\left (x e + d\right )}^{2} A a b^{5} d^{3} e^{2} - 375 \, {\left (x e + d\right )} B a^{2} b^{4} d^{4} e^{2} - 150 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )}^{2} B a^{3} b^{3} d^{2} e^{3} - 1350 \, {\left (x e + d\right )}^{2} A a^{2} b^{4} d^{2} e^{3} + 400 \, {\left (x e + d\right )} B a^{3} b^{3} d^{3} e^{3} + 300 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )}^{2} B a^{4} b^{2} d e^{4} + 900 \, {\left (x e + d\right )}^{2} A a^{3} b^{3} d e^{4} - 225 \, {\left (x e + d\right )} B a^{4} b^{2} d^{2} e^{4} - 300 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )}^{2} B a^{5} b e^{5} - 225 \, {\left (x e + d\right )}^{2} A a^{4} b^{2} e^{5} + 60 \, {\left (x e + d\right )} B a^{5} b d e^{5} + 150 \, {\left (x e + d\right )} 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 \, {\left (x e + d\right )} B a^{6} e^{6} - 30 \, {\left (x e + d\right )} 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}\right )} e^{\left (-8\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \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="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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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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