3.1777 \(\int \frac{(A+B x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=227 \[ -\frac{(d+e x)^4 (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 B e^2 (b d-a e)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 B e (b d-a e)^2}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{B (b d-a e)^3}{3 b^5 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^3 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.434872, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(d+e x)^4 (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 B e^2 (b d-a e)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 B e (b d-a e)^2}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{B (b d-a e)^3}{3 b^5 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^3 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^3)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 47.1924, size = 246, normalized size = 1.08 \[ - \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{4}}{8 b e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{B \left (d + e x\right )^{3}}{3 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{B e \left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{B e^{2} \left (d + e x\right )}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{B e^{3} \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{4} \left (A e - B d\right )}{8 e \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.263099, size = 239, normalized size = 1.05 \[ \frac{-3 A b \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+B \left (25 a^4 e^3+a^3 b e^2 (88 e x-9 d)-3 a^2 b^2 e \left (d^2+12 d e x-36 e^2 x^2\right )-a b^3 \left (d^3+12 d^2 e x+54 d e^2 x^2-48 e^3 x^3\right )-2 b^4 d x \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )+12 B e^3 (a+b x)^4 \log (a+b x)}{12 b^5 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^3)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(B*(25*a^4*e^3 + a^3*b*e^2*(-9*d + 88*e*x) - 3*a^2*b^2*e*(d^2 + 12*d*e*x - 36*e^
2*x^2) - 2*b^4*d*x*(2*d^2 + 9*d*e*x + 18*e^2*x^2) - a*b^3*(d^3 + 12*d^2*e*x + 54
*d*e^2*x^2 - 48*e^3*x^3)) - 3*A*b*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^
2 + 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + 12
*B*e^3*(a + b*x)^4*Log[a + b*x])/(12*b^5*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.024, size = 385, normalized size = 1.7 \[ -{\frac{ \left ( 3\,A{a}^{3}b{e}^{3}+54\,B{x}^{2}a{b}^{3}d{e}^{2}+18\,A{x}^{2}{b}^{4}d{e}^{2}+3\,{b}^{2}B{a}^{2}{d}^{2}e+18\,B{x}^{2}{b}^{4}{d}^{2}e-108\,B{x}^{2}{a}^{2}{b}^{2}{e}^{3}-48\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{3}+36\,Bx{a}^{2}{b}^{2}d{e}^{2}+12\,Bxa{b}^{3}{d}^{2}e+12\,Ax{a}^{2}{b}^{2}{e}^{3}-48\,B\ln \left ( bx+a \right ) x{a}^{3}b{e}^{3}+3\,A{d}^{3}{b}^{4}+3\,Ad{a}^{2}{b}^{2}{e}^{2}-48\,B{x}^{3}a{b}^{3}{e}^{3}+18\,A{x}^{2}a{b}^{3}{e}^{3}-25\,B{e}^{3}{a}^{4}+12\,Ax{b}^{4}{d}^{2}e-88\,Bx{a}^{3}b{e}^{3}+12\,A{x}^{3}{b}^{4}{e}^{3}-12\,B\ln \left ( bx+a \right ){a}^{4}{e}^{3}+4\,Bx{b}^{4}{d}^{3}+Ba{b}^{3}{d}^{3}+12\,Axa{b}^{3}d{e}^{2}+9\,B{a}^{3}bd{e}^{2}-12\,B\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{3}+36\,B{x}^{3}{b}^{4}d{e}^{2}-72\,B\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{3}+3\,A{d}^{2}a{b}^{3}e \right ) \left ( bx+a \right ) }{12\,{b}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(3*A*a^3*b*e^3+54*B*x^2*a*b^3*d*e^2+18*A*x^2*b^4*d*e^2+3*b^2*B*a^2*d^2*e+1
8*B*x^2*b^4*d^2*e-108*B*x^2*a^2*b^2*e^3-48*B*ln(b*x+a)*x^3*a*b^3*e^3+36*B*x*a^2*
b^2*d*e^2+12*B*x*a*b^3*d^2*e+12*A*x*a^2*b^2*e^3-48*B*ln(b*x+a)*x*a^3*b*e^3+3*A*d
^3*b^4+3*A*d*a^2*b^2*e^2-48*B*x^3*a*b^3*e^3+18*A*x^2*a*b^3*e^3-25*B*e^3*a^4+12*A
*x*b^4*d^2*e-88*B*x*a^3*b*e^3+12*A*x^3*b^4*e^3-12*B*ln(b*x+a)*a^4*e^3+4*B*x*b^4*
d^3+B*a*b^3*d^3+12*A*x*a*b^3*d*e^2+9*B*a^3*b*d*e^2-12*B*ln(b*x+a)*x^4*b^4*e^3+36
*B*x^3*b^4*d*e^2-72*B*ln(b*x+a)*x^2*a^2*b^2*e^3+3*A*d^2*a*b^3*e)*(b*x+a)/b^5/((b
*x+a)^2)^(5/2)

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Maxima [A]  time = 0.756046, size = 849, normalized size = 3.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

1/12*B*e^3*((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) - 1/4*
B*d*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
)) - 1/12*A*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^3*(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)) - 1/4*A*d^2*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)) - 1/4*B*d^2*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)) - 1/4*A*d*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^3/((b^2)^(5/2)*(x + a/b)^4)

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Fricas [A]  time = 0.28803, size = 485, normalized size = 2.14 \[ -\frac{{\left (B a b^{3} + 3 \, A b^{4}\right )} d^{3} + 3 \,{\left (B a^{2} b^{2} + A a b^{3}\right )} d^{2} e + 3 \,{\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} d e^{2} -{\left (25 \, B a^{4} - 3 \, A a^{3} b\right )} e^{3} + 12 \,{\left (3 \, B b^{4} d e^{2} -{\left (4 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 18 \,{\left (B b^{4} d^{2} e +{\left (3 \, B a b^{3} + A b^{4}\right )} d e^{2} -{\left (6 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 4 \,{\left (B b^{4} d^{3} + 3 \,{\left (B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + A a b^{3}\right )} d e^{2} -{\left (22 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x - 12 \,{\left (B b^{4} e^{3} x^{4} + 4 \, B a b^{3} e^{3} x^{3} + 6 \, B a^{2} b^{2} e^{3} x^{2} + 4 \, B a^{3} b e^{3} x + B a^{4} e^{3}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12*((B*a*b^3 + 3*A*b^4)*d^3 + 3*(B*a^2*b^2 + A*a*b^3)*d^2*e + 3*(3*B*a^3*b +
A*a^2*b^2)*d*e^2 - (25*B*a^4 - 3*A*a^3*b)*e^3 + 12*(3*B*b^4*d*e^2 - (4*B*a*b^3 -
A*b^4)*e^3)*x^3 + 18*(B*b^4*d^2*e + (3*B*a*b^3 + A*b^4)*d*e^2 - (6*B*a^2*b^2 -
A*a*b^3)*e^3)*x^2 + 4*(B*b^4*d^3 + 3*(B*a*b^3 + A*b^4)*d^2*e + 3*(3*B*a^2*b^2 +
A*a*b^3)*d*e^2 - (22*B*a^3*b - 3*A*a^2*b^2)*e^3)*x - 12*(B*b^4*e^3*x^4 + 4*B*a*b
^3*e^3*x^3 + 6*B*a^2*b^2*e^3*x^2 + 4*B*a^3*b*e^3*x + B*a^4*e^3)*log(b*x + a))/(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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**3/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.651567, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x