3.1807 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx\)

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

[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)

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Rubi [A]  time = 0.484657, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{2 b^5 (d+e x)^{7/2} (-6 a B e-A b e+7 b B d)}{7 e^8}+\frac{6 b^4 (d+e x)^{5/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{3/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac{10 b^2 \sqrt{d+e x} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac{6 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 \sqrt{d+e x}}-\frac{2 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^{3/2}}+\frac{2 (b d-a e)^6 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\frac{2 b^6 B (d+e x)^{9/2}}{9 e^8} \]

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)

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Rubi in Sympy [A]  time = 157.03, size = 313, normalized size = 1.03 \[ \frac{2 B b^{6} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{8}} + \frac{2 b^{5} \left (d + e x\right )^{\frac{7}{2}} \left (A b e + 6 B a e - 7 B b d\right )}{7 e^{8}} + \frac{6 b^{4} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{5 e^{8}} + \frac{10 b^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right )}{3 e^{8}} + \frac{10 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right )}{e^{8}} - \frac{6 b \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right )}{e^{8} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{3 e^{8} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{6}}{5 e^{8} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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) + 2*b**5*(d + e*x)**(7/2)*(A*b*e + 6*B*a*e -
7*B*b*d)/(7*e**8) + 6*b**4*(d + e*x)**(5/2)*(a*e - b*d)*(2*A*b*e + 5*B*a*e - 7*B
*b*d)/(5*e**8) + 10*b**3*(d + e*x)**(3/2)*(a*e - b*d)**2*(3*A*b*e + 4*B*a*e - 7*
B*b*d)/(3*e**8) + 10*b**2*sqrt(d + e*x)*(a*e - b*d)**3*(4*A*b*e + 3*B*a*e - 7*B*
b*d)/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)
) - 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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Mathematica [B]  time = 0.862267, size = 627, normalized size = 2.06 \[ -\frac{2 \left (21 a^6 e^6 (3 A e+2 B d+5 B e x)+126 a^5 b e^5 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )-315 a^4 b^2 e^4 \left (3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+420 a^3 b^3 e^3 \left (B \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-3 A e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )-315 a^2 b^4 e^2 \left (A e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+B \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )+18 a b^5 e \left (3 B \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )-7 A e \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )+b^6 \left (9 A e \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )-7 B \left (2048 d^7+5120 d^6 e x+3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5-10 d e^6 x^6+5 e^7 x^7\right )\right )\right )}{315 e^8 (d+e x)^{5/2}} \]

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 + 1
5*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 + 32
0*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 + 24*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]  time = 0.016, size = 913, normalized size = 3. \[ \text{result too large to display} \]

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)

[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+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+13824
0*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)/(e*x+d)^(5/2)/e^8

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Maxima [A]  time = 0.732011, size = 1046, normalized size = 3.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/(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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Fricas [A]  time = 0.311705, size = 1068, normalized size = 3.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/(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 + 806
4*(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)/((e^10*x^2 + 2*d*e^9*x + d^2*e^8)*sqrt(e*x + d)
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{6}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]

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]

Integral((A + B*x)*(a + b*x)**6/(d + e*x)**(7/2), x)

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GIAC/XCAS [A]  time = 0.304631, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done