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3.1775 \(\int \frac{(A+B x) (d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=373 \[ \frac{5 e^3 (a+b x) (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 e^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{3 b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x (a+b x) (-5 a B e+A b e+5 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^5 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.978648, antiderivative size = 373, 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 \[ \frac{5 e^3 (a+b x) (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 e^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{3 b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x (a+b x) (-5 a B e+A b e+5 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^5 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.4324, size = 377, normalized size = 1.01 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{6}}{4 b e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{5} \left (- A b e + B \left (3 a e - 2 b d\right )\right )}{8 b^{2} e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{5 \left (d + e x\right )^{4} \left (- A b e + B \left (3 a e - 2 b d\right )\right )}{12 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5 e \left (2 a + 2 b x\right ) \left (d + e x\right )^{3} \left (- A b e + B \left (3 a e - 2 b d\right )\right )}{12 b^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5 e^{2} \left (d + e x\right )^{2} \left (- A b e + B \left (3 a e - 2 b d\right )\right )}{2 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{5 e^{4} \left (- A b e + B \left (3 a e - 2 b d\right )\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{7}} + \frac{5 e^{3} \left (a + b x\right ) \left (a e - b d\right ) \left (- A b e + B \left (3 a e - 2 b d\right )\right ) \log{\left (a + b x \right )}}{b^{7} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**6/(4*b*e*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) + (2*a
+ 2*b*x)*(d + e*x)**5*(-A*b*e + B*(3*a*e - 2*b*d))/(8*b**2*e*(a**2 + 2*a*b*x + b
**2*x**2)**(5/2)) + 5*(d + e*x)**4*(-A*b*e + B*(3*a*e - 2*b*d))/(12*b**3*(a**2 +
 2*a*b*x + b**2*x**2)**(3/2)) + 5*e*(2*a + 2*b*x)*(d + e*x)**3*(-A*b*e + B*(3*a*
e - 2*b*d))/(12*b**4*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + 5*e**2*(d + e*x)**2*
(-A*b*e + B*(3*a*e - 2*b*d))/(2*b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - 5*e**4*
(-A*b*e + B*(3*a*e - 2*b*d))*sqrt(a**2 + 2*a*b*x + b**2*x**2)/b**7 + 5*e**3*(a +
 b*x)*(a*e - b*d)*(-A*b*e + B*(3*a*e - 2*b*d))*log(a + b*x)/(b**7*sqrt(a**2 + 2*
a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.667319, size = 513, normalized size = 1.38 \[ \frac{-A b \left (77 a^5 e^5+a^4 b e^4 (248 e x-125 d)+2 a^3 b^2 e^3 \left (15 d^2-220 d e x+126 e^2 x^2\right )+2 a^2 b^3 e^2 \left (5 d^3+60 d^2 e x-270 d e^2 x^2+24 e^3 x^3\right )+a b^4 e \left (5 d^4+40 d^3 e x+180 d^2 e^2 x^2-240 d e^3 x^3-48 e^4 x^4\right )+b^5 \left (3 d^5+20 d^4 e x+60 d^3 e^2 x^2+120 d^2 e^3 x^3-12 e^5 x^5\right )\right )+B \left (171 a^6 e^5+7 a^5 b e^4 (72 e x-55 d)+2 a^4 b^2 e^3 \left (125 d^2-620 d e x+198 e^2 x^2\right )-2 a^3 b^3 e^2 \left (15 d^3-440 d^2 e x+630 d e^2 x^2+48 e^3 x^3\right )-a^2 b^4 e \left (5 d^4+120 d^3 e x-1080 d^2 e^2 x^2+240 d e^3 x^3+204 e^4 x^4\right )-a b^5 \left (d^5+20 d^4 e x+180 d^3 e^2 x^2-480 d^2 e^3 x^3-240 d e^4 x^4+36 e^5 x^5\right )+2 b^6 x \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )\right )+60 e^3 (a+b x)^4 (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{12 b^7 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(A*b*(77*a^5*e^5 + a^4*b*e^4*(-125*d + 248*e*x) + 2*a^3*b^2*e^3*(15*d^2 - 220*
d*e*x + 126*e^2*x^2) + 2*a^2*b^3*e^2*(5*d^3 + 60*d^2*e*x - 270*d*e^2*x^2 + 24*e^
3*x^3) + a*b^4*e*(5*d^4 + 40*d^3*e*x + 180*d^2*e^2*x^2 - 240*d*e^3*x^3 - 48*e^4*
x^4) + b^5*(3*d^5 + 20*d^4*e*x + 60*d^3*e^2*x^2 + 120*d^2*e^3*x^3 - 12*e^5*x^5))
) + B*(171*a^6*e^5 + 7*a^5*b*e^4*(-55*d + 72*e*x) + 2*a^4*b^2*e^3*(125*d^2 - 620
*d*e*x + 198*e^2*x^2) - 2*a^3*b^3*e^2*(15*d^3 - 440*d^2*e*x + 630*d*e^2*x^2 + 48
*e^3*x^3) - a^2*b^4*e*(5*d^4 + 120*d^3*e*x - 1080*d^2*e^2*x^2 + 240*d*e^3*x^3 +
204*e^4*x^4) + 2*b^6*x*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*
e^5*x^5) - a*b^5*(d^5 + 20*d^4*e*x + 180*d^3*e^2*x^2 - 480*d^2*e^3*x^3 - 240*d*e
^4*x^4 + 36*e^5*x^5)) + 60*e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)
^4*Log[a + b*x])/(12*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.032, size = 1153, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(-125*A*a^4*b^2*d*e^4+60*A*ln(b*x+a)*a^5*b*e^5+36*B*x^5*a*b^5*e^5-60*B*x^5
*b^6*d*e^4+20*A*x*b^6*d^4*e-504*B*x*a^5*b*e^5+30*B*x^2*b^6*d^4*e+248*A*x*a^4*b^2
*e^5+60*A*x^2*b^6*d^3*e^2+48*A*x^3*a^2*b^4*e^5+77*A*a^5*b*e^5+5*a^2*B*b^4*d^4*e+
5*a*A*b^5*d^4*e+3*A*b^6*d^5-171*B*a^6*e^5+1260*B*x^2*a^3*b^3*d*e^4+4*B*x*b^6*d^5
-6*B*x^6*b^6*e^5-12*A*x^5*b^6*e^5-180*B*ln(b*x+a)*a^6*e^5+30*A*a^3*b^3*d^2*e^3+3
00*B*ln(b*x+a)*a^5*b*d*e^4-120*B*ln(b*x+a)*a^4*b^2*d^2*e^3-1080*B*x^2*a^2*b^4*d^
2*e^3+60*A*ln(b*x+a)*x^4*a*b^5*e^5-60*A*ln(b*x+a)*x^4*b^6*d*e^4-180*B*ln(b*x+a)*
x^4*a^2*b^4*e^5-120*B*ln(b*x+a)*x^4*b^6*d^2*e^3+240*A*ln(b*x+a)*x^3*a^2*b^4*e^5-
720*B*ln(b*x+a)*x^3*a^3*b^3*e^5+360*A*ln(b*x+a)*x^2*a^3*b^3*e^5-1080*B*ln(b*x+a)
*x^2*a^4*b^2*e^5-540*A*x^2*a^2*b^4*d*e^4+180*A*x^2*a*b^5*d^2*e^3-440*A*x*a^3*b^3
*d*e^4+120*A*x*a^2*b^4*d^2*e^3-880*B*x*a^3*b^3*d^2*e^3+120*B*x*a^2*b^4*d^3*e^2+2
0*B*x*a*b^5*d^4*e-240*A*x^3*a*b^5*d*e^4-480*B*x^3*a*b^5*d^2*e^3-240*B*x^4*a*b^5*
d*e^4+240*B*x^3*a^2*b^4*d*e^4+180*B*x^2*a*b^5*d^3*e^2+40*A*x*a*b^5*d^3*e^2+1240*
B*x*a^4*b^2*d*e^4-720*B*ln(b*x+a)*x*a^5*b*e^5-60*A*ln(b*x+a)*a^4*b^2*d*e^4+240*A
*ln(b*x+a)*x*a^4*b^2*e^5+300*B*ln(b*x+a)*x^4*a*b^5*d*e^4-240*A*ln(b*x+a)*x^3*a*b
^5*d*e^4+1200*B*ln(b*x+a)*x^3*a^2*b^4*d*e^4-480*B*ln(b*x+a)*x^3*a*b^5*d^2*e^3-36
0*A*ln(b*x+a)*x^2*a^2*b^4*d*e^4+1800*B*ln(b*x+a)*x^2*a^3*b^3*d*e^4-720*B*ln(b*x+
a)*x^2*a^2*b^4*d^2*e^3-240*A*ln(b*x+a)*x*a^3*b^3*d*e^4+1200*B*ln(b*x+a)*x*a^4*b^
2*d*e^4-480*B*ln(b*x+a)*x*a^3*b^3*d^2*e^3+385*B*a^5*b*d*e^4+120*B*x^3*b^6*d^3*e^
2+120*A*x^3*b^6*d^2*e^3+96*B*x^3*a^3*b^3*e^5+204*B*x^4*a^2*b^4*e^5-48*A*x^4*a*b^
5*e^5+252*A*x^2*a^3*b^3*e^5-396*B*x^2*a^4*b^2*e^5+30*B*a^3*b^3*d^3*e^2+10*A*a^2*
b^4*d^3*e^2-250*B*a^4*b^2*d^2*e^3+B*a*b^5*d^5)*(b*x+a)/b^7/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 0.82615, size = 1493, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*e^5*((2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4
*b^2*x^2 + 168*a^5*b*x + 57*a^6)/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^
3*b^8*x + a^4*b^7) + 60*a^2*log(b*x + a)/b^7) + 5/12*B*d*e^4*((12*b^5*x^5 + 48*a
*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5)/(b^10*x^4 +
4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6) - 60*a*log(b*x + a)/b^6) +
1/12*A*e^5*((12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*
a^4*b*x - 77*a^5)/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^
6) - 60*a*log(b*x + a)/b^6) + 5/6*B*d^2*e^3*((48*a*b^3*x^3 + 108*a^2*b^2*x^2 + 8
8*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b
^5) + 12*log(b*x + a)/b^5) + 5/12*A*d*e^4*((48*a*b^3*x^3 + 108*a^2*b^2*x^2 + 88*
a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5
) + 12*log(b*x + a)/b^5) - 5/6*B*d^3*e^2*(12*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2
)*b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^4) + 3*a^3*b/((b^2)^(9/2)*(x +
 a/b)^4) - 8*a^2/((b^2)^(7/2)*(x + a/b)^3) + 6*a/((b^2)^(5/2)*b*(x + a/b)^2) - 6
*a^3/((b^2)^(5/2)*b^3*(x + a/b)^4)) - 5/6*A*d^2*e^3*(12*x^2/((b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)*b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^4) + 3*a^3*b/((b^2)
^(9/2)*(x + a/b)^4) - 8*a^2/((b^2)^(7/2)*(x + a/b)^3) + 6*a/((b^2)^(5/2)*b*(x +
a/b)^2) - 6*a^3/((b^2)^(5/2)*b^3*(x + a/b)^4)) - 1/12*B*d^5*(4/((b^2*x^2 + 2*a*b
*x + a^2)^(3/2)*b^2) - 3*a/((b^2)^(5/2)*b*(x + a/b)^4)) - 5/12*A*d^4*e*(4/((b^2*
x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/((b^2)^(5/2)*b*(x + a/b)^4)) - 5/12*B*d^4*
e*(3*a^2*b^2/((b^2)^(9/2)*(x + a/b)^4) - 8*a*b/((b^2)^(7/2)*(x + a/b)^3) + 6/((b
^2)^(5/2)*(x + a/b)^2)) - 5/6*A*d^3*e^2*(3*a^2*b^2/((b^2)^(9/2)*(x + a/b)^4) - 8
*a*b/((b^2)^(7/2)*(x + a/b)^3) + 6/((b^2)^(5/2)*(x + a/b)^2)) - 1/4*A*d^5/((b^2)
^(5/2)*(x + a/b)^4)

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Fricas [A]  time = 0.294958, size = 1234, normalized size = 3.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(6*B*b^6*e^5*x^6 - (B*a*b^5 + 3*A*b^6)*d^5 - 5*(B*a^2*b^4 + A*a*b^5)*d^4*e
- 10*(3*B*a^3*b^3 + A*a^2*b^4)*d^3*e^2 + 10*(25*B*a^4*b^2 - 3*A*a^3*b^3)*d^2*e^3
 - 5*(77*B*a^5*b - 25*A*a^4*b^2)*d*e^4 + (171*B*a^6 - 77*A*a^5*b)*e^5 + 12*(5*B*
b^6*d*e^4 - (3*B*a*b^5 - A*b^6)*e^5)*x^5 + 12*(20*B*a*b^5*d*e^4 - (17*B*a^2*b^4
- 4*A*a*b^5)*e^5)*x^4 - 24*(5*B*b^6*d^3*e^2 - 5*(4*B*a*b^5 - A*b^6)*d^2*e^3 + 10
*(B*a^2*b^4 - A*a*b^5)*d*e^4 + 2*(2*B*a^3*b^3 + A*a^2*b^4)*e^5)*x^3 - 6*(5*B*b^6
*d^4*e + 10*(3*B*a*b^5 + A*b^6)*d^3*e^2 - 30*(6*B*a^2*b^4 - A*a*b^5)*d^2*e^3 + 3
0*(7*B*a^3*b^3 - 3*A*a^2*b^4)*d*e^4 - 6*(11*B*a^4*b^2 - 7*A*a^3*b^3)*e^5)*x^2 -
4*(B*b^6*d^5 + 5*(B*a*b^5 + A*b^6)*d^4*e + 10*(3*B*a^2*b^4 + A*a*b^5)*d^3*e^2 -
10*(22*B*a^3*b^3 - 3*A*a^2*b^4)*d^2*e^3 + 10*(31*B*a^4*b^2 - 11*A*a^3*b^3)*d*e^4
 - 2*(63*B*a^5*b - 31*A*a^4*b^2)*e^5)*x + 60*(2*B*a^4*b^2*d^2*e^3 - (5*B*a^5*b -
 A*a^4*b^2)*d*e^4 + (3*B*a^6 - A*a^5*b)*e^5 + (2*B*b^6*d^2*e^3 - (5*B*a*b^5 - A*
b^6)*d*e^4 + (3*B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 4*(2*B*a*b^5*d^2*e^3 - (5*B*a^2*
b^4 - A*a*b^5)*d*e^4 + (3*B*a^3*b^3 - A*a^2*b^4)*e^5)*x^3 + 6*(2*B*a^2*b^4*d^2*e
^3 - (5*B*a^3*b^3 - A*a^2*b^4)*d*e^4 + (3*B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 4*(2
*B*a^3*b^3*d^2*e^3 - (5*B*a^4*b^2 - A*a^3*b^3)*d*e^4 + (3*B*a^5*b - A*a^4*b^2)*e
^5)*x)*log(b*x + a))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^
4*b^7)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.617365, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x

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