3.1767 \(\int \frac{(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=329 \[ -\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e (a+b x) (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 x (a+b x) \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x^2 (a+b x) (-3 a B e+A b e+4 b B d)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^4 x^3 (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.825844, antiderivative size = 329, 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 \[ -\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e (a+b x) (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 x (a+b x) \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x^2 (a+b x) (-3 a B e+A b e+4 b B d)}{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^4 x^3 (a+b x)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 56.0123, size = 345, normalized size = 1.05 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{5}}{6 b e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{4} \left (3 A b e - 5 B a e + 2 B b d\right )}{12 b^{2} e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 \left (d + e x\right )^{3} \left (3 A b e - 5 B a e + 2 B b d\right )}{3 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e \left (2 a + 2 b x\right ) \left (d + e x\right )^{2} \left (3 A b e - 5 B a e + 2 B b d\right )}{2 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{2 e^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - 5 B a e + 2 B b d\right )}{b^{6}} + \frac{2 e \left (a + b x\right ) \left (a e - b d\right )^{2} \left (3 A b e - 5 B a e + 2 B b d\right ) \log{\left (a + b x \right )}}{b^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.454141, size = 373, normalized size = 1.13 \[ \frac{-3 A b \left (-7 a^4 e^4-2 a^3 b e^3 (e x-10 d)+a^2 b^2 e^2 \left (-18 d^2+16 d e x+11 e^2 x^2\right )+4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x-8 d e^3 x^3-e^4 x^4\right )\right )+B \left (-27 a^5 e^4+6 a^4 b e^3 (14 d+e x)+3 a^3 b^2 e^2 \left (-30 d^2+8 d e x+21 e^2 x^2\right )+4 a^2 b^3 e \left (9 d^3-18 d^2 e x-33 d e^2 x^2+5 e^3 x^3\right )+a b^4 \left (-3 d^4+48 d^3 e x+72 d^2 e^2 x^2-48 d e^3 x^3-5 e^4 x^4\right )+2 b^5 x \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )\right )+12 e (a+b x)^2 (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{6 b^6 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(-3*A*b*(-7*a^4*e^4 - 2*a^3*b*e^3*(-10*d + e*x) + a^2*b^2*e^2*(-18*d^2 + 16*d*e*
x + 11*e^2*x^2) + 4*a*b^3*e*(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + b^4*(d^4
+ 8*d^3*e*x - 8*d*e^3*x^3 - e^4*x^4)) + B*(-27*a^5*e^4 + 6*a^4*b*e^3*(14*d + e*
x) + 3*a^3*b^2*e^2*(-30*d^2 + 8*d*e*x + 21*e^2*x^2) + 4*a^2*b^3*e*(9*d^3 - 18*d^
2*e*x - 33*d*e^2*x^2 + 5*e^3*x^3) + a*b^4*(-3*d^4 + 48*d^3*e*x + 72*d^2*e^2*x^2
- 48*d*e^3*x^3 - 5*e^4*x^4) + 2*b^5*x*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e
^4*x^4)) + 12*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^2*Log[a +
b*x])/(6*b^6*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.03, size = 858, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/6*(36*A*ln(b*x+a)*a^4*b*e^4+6*A*x*a^3*b^2*e^4+63*B*x^2*a^3*b^2*e^4-5*B*x^4*a*b
^4*e^4+12*B*x^4*b^5*d*e^3-12*A*x^3*a*b^4*e^4-3*A*b^5*d^4+144*B*ln(b*x+a)*x^2*a^2
*b^3*d*e^3-108*B*ln(b*x+a)*x^2*a*b^4*d^2*e^2-144*A*ln(b*x+a)*x*a^2*b^3*d*e^3+72*
A*ln(b*x+a)*x*a*b^4*d^2*e^2+288*B*ln(b*x+a)*x*a^3*b^2*d*e^3-216*B*ln(b*x+a)*x*a^
2*b^3*d^2*e^2+48*B*ln(b*x+a)*x*a*b^4*d^3*e-48*A*x*a^2*b^3*d*e^3+21*A*a^4*b*e^4-1
2*A*a*b^4*d^3*e-3*B*a*b^4*d^4+36*B*a^2*b^3*d^3*e-6*B*x*b^5*d^4+2*B*x^5*b^5*e^4+3
*A*x^4*b^5*e^4-60*B*ln(b*x+a)*a^5*e^4-27*B*a^5*e^4-72*A*ln(b*x+a)*x^2*a*b^4*d*e^
3+24*A*x^3*b^5*d*e^3+20*B*x^3*a^2*b^3*e^4+36*B*x^3*b^5*d^2*e^2+6*B*x*a^4*b*e^4-3
3*A*x^2*a^2*b^3*e^4-24*A*x*b^5*d^3*e-48*B*x^3*a*b^4*d*e^3+48*A*x^2*a*b^4*d*e^3+7
2*A*x*a*b^4*d^2*e^2+48*B*x*a*b^4*d^3*e+36*A*ln(b*x+a)*x^2*a^2*b^3*e^4+36*A*ln(b*
x+a)*x^2*b^5*d^2*e^2-60*B*ln(b*x+a)*x^2*a^3*b^2*e^4+24*B*ln(b*x+a)*x^2*b^5*d^3*e
+72*A*ln(b*x+a)*x*a^3*b^2*e^4-120*B*ln(b*x+a)*x*a^4*b*e^4-72*A*ln(b*x+a)*a^3*b^2
*d*e^3+36*A*ln(b*x+a)*a^2*b^3*d^2*e^2+144*B*ln(b*x+a)*a^4*b*d*e^3-108*B*ln(b*x+a
)*a^3*b^2*d^2*e^2+24*B*ln(b*x+a)*a^2*b^3*d^3*e+24*B*x*a^3*b^2*d*e^3-72*B*x*a^2*b
^3*d^2*e^2-132*B*x^2*a^2*b^3*d*e^3+72*B*x^2*a*b^4*d^2*e^2-90*B*a^3*b^2*d^2*e^2-6
0*A*a^3*b^2*d*e^3+84*B*a^4*b*d*e^3+54*A*a^2*b^3*d^2*e^2)*(b*x+a)/b^6/((b*x+a)^2)
^(3/2)

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Maxima [A]  time = 0.72799, size = 1235, normalized size = 3.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

1/3*B*e^4*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 7/6*B*a*e^4*x^3/(sqrt(b^2*x^
2 + 2*a*b*x + a^2)*b^3) + 9/2*B*a^2*e^4*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4)
- 10*B*a^3*e^4*log(x + a/b)/((b^2)^(3/2)*b^3) - 15*B*a^5*e^4/((b^2)^(7/2)*b*(x +
a/b)^2) - 20*B*a^4*e^4*x/((b^2)^(5/2)*b^2*(x + a/b)^2) + 9*B*a^4*e^4/(sqrt(b^2*
x^2 + 2*a*b*x + a^2)*b^6) - 1/2*A*d^4/((b^2)^(3/2)*(x + a/b)^2) - 9/2*B*a^5*e^4/
((b^2)^(3/2)*b^5*(x + a/b)^2) + 1/2*(4*B*d*e^3 + A*e^4)*x^3/(sqrt(b^2*x^2 + 2*a*
b*x + a^2)*b^2) - 5/2*(4*B*d*e^3 + A*e^4)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b
^3) + 2*(3*B*d^2*e^2 + 2*A*d*e^3)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 2*(2
*B*d^3*e + 3*A*d^2*e^2)*log(x + a/b)/(b^2)^(3/2) + 6*(4*B*d*e^3 + A*e^4)*a^2*log
(x + a/b)/((b^2)^(3/2)*b^2) - 6*(3*B*d^2*e^2 + 2*A*d*e^3)*a*log(x + a/b)/((b^2)^
(3/2)*b) + 9*(4*B*d*e^3 + A*e^4)*a^4/((b^2)^(7/2)*(x + a/b)^2) - 9*(3*B*d^2*e^2
+ 2*A*d*e^3)*a^3*b/((b^2)^(7/2)*(x + a/b)^2) + 3*(2*B*d^3*e + 3*A*d^2*e^2)*a^2*b
^2/((b^2)^(7/2)*(x + a/b)^2) - 12*(3*B*d^2*e^2 + 2*A*d*e^3)*a^2*x/((b^2)^(5/2)*(
x + a/b)^2) + 12*(4*B*d*e^3 + A*e^4)*a^3*x/((b^2)^(5/2)*b*(x + a/b)^2) + 4*(2*B*
d^3*e + 3*A*d^2*e^2)*a*b*x/((b^2)^(5/2)*(x + a/b)^2) - 5*(4*B*d*e^3 + A*e^4)*a^3
/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5) + 4*(3*B*d^2*e^2 + 2*A*d*e^3)*a^2/(sqrt(b^2
*x^2 + 2*a*b*x + a^2)*b^4) - (B*d^4 + 4*A*d^3*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*
b^2) + 5/2*(4*B*d*e^3 + A*e^4)*a^4/((b^2)^(3/2)*b^4*(x + a/b)^2) - 2*(3*B*d^2*e^
2 + 2*A*d*e^3)*a^3/((b^2)^(3/2)*b^3*(x + a/b)^2) + 1/2*(B*d^4 + 4*A*d^3*e)*a/((b
^2)^(3/2)*b*(x + a/b)^2)

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Fricas [A]  time = 0.283995, size = 902, normalized size = 2.74 \[ \frac{2 \, B b^{5} e^{4} x^{5} - 3 \,{\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \,{\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \,{\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} +{\left (12 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \,{\left (9 \, B b^{5} d^{2} e^{2} - 6 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} +{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \,{\left (24 \, B a b^{4} d^{2} e^{2} - 4 \,{\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} +{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \,{\left (B b^{5} d^{4} - 4 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} -{\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (2 \, B a^{2} b^{3} d^{3} e - 3 \,{\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} -{\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} +{\left (2 \, B b^{5} d^{3} e - 3 \,{\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} -{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \,{\left (2 \, B a b^{4} d^{3} e - 3 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} -{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*e^4*x^5 - 3*(B*a*b^4 + A*b^5)*d^4 + 12*(3*B*a^2*b^3 - A*a*b^4)*d^3*
e - 18*(5*B*a^3*b^2 - 3*A*a^2*b^3)*d^2*e^2 + 12*(7*B*a^4*b - 5*A*a^3*b^2)*d*e^3
- 3*(9*B*a^5 - 7*A*a^4*b)*e^4 + (12*B*b^5*d*e^3 - (5*B*a*b^4 - 3*A*b^5)*e^4)*x^4
+ 4*(9*B*b^5*d^2*e^2 - 6*(2*B*a*b^4 - A*b^5)*d*e^3 + (5*B*a^2*b^3 - 3*A*a*b^4)*
e^4)*x^3 + 3*(24*B*a*b^4*d^2*e^2 - 4*(11*B*a^2*b^3 - 4*A*a*b^4)*d*e^3 + (21*B*a^
3*b^2 - 11*A*a^2*b^3)*e^4)*x^2 - 6*(B*b^5*d^4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 12
*(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B*a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (B*a^4*b +
A*a^3*b^2)*e^4)*x + 12*(2*B*a^2*b^3*d^3*e - 3*(3*B*a^3*b^2 - A*a^2*b^3)*d^2*e^2
+ 6*(2*B*a^4*b - A*a^3*b^2)*d*e^3 - (5*B*a^5 - 3*A*a^4*b)*e^4 + (2*B*b^5*d^3*e -
3*(3*B*a*b^4 - A*b^5)*d^2*e^2 + 6*(2*B*a^2*b^3 - A*a*b^4)*d*e^3 - (5*B*a^3*b^2
- 3*A*a^2*b^3)*e^4)*x^2 + 2*(2*B*a*b^4*d^3*e - 3*(3*B*a^2*b^3 - A*a*b^4)*d^2*e^2
+ 6*(2*B*a^3*b^2 - A*a^2*b^3)*d*e^3 - (5*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*log(b*x
+ a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.616928, 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)^4/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x