3.1584 \(\int \frac{1}{(d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=231 \[ \frac{b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{b (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}+\frac{a+b x}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

[Out]

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

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Rubi [A]  time = 0.270459, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{b (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}+\frac{a+b x}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 59.0183, size = 218, normalized size = 0.94 \[ \frac{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{b^{2} e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{4}} + \frac{b \left (2 a + 2 b x\right )}{4 \left (d + e x\right )^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{2 a + 2 b x}{6 \left (d + e x\right )^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.131028, size = 124, normalized size = 0.54 \[ \frac{(a+b x) \left (6 b^3 (d+e x)^3 \log (a+b x)+6 b^2 (d+e x)^2 (b d-a e)+3 b (d+e x) (b d-a e)^2+2 (b d-a e)^3-6 b^3 (d+e x)^3 \log (d+e x)\right )}{6 \sqrt{(a+b x)^2} (d+e x)^3 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.024, size = 256, normalized size = 1.1 \[{\frac{ \left ( bx+a \right ) \left ( 6\,\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{3}-6\,\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{3}+18\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}-18\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}+18\,\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}e-18\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e-6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ){b}^{3}{d}^{3}-6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}+3\,x{a}^{2}b{e}^{3}-18\,xa{b}^{2}d{e}^{2}+15\,x{b}^{3}{d}^{2}e-2\,{a}^{3}{e}^{3}+9\,{a}^{2}bd{e}^{2}-18\,a{b}^{2}{d}^{2}e+11\,{b}^{3}{d}^{3} \right ) }{6\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/((b*x+a)^2)^(1/2),x)

[Out]

1/6*(b*x+a)*(6*ln(b*x+a)*x^3*b^3*e^3-6*ln(e*x+d)*x^3*b^3*e^3+18*ln(b*x+a)*x^2*b^
3*d*e^2-18*ln(e*x+d)*x^2*b^3*d*e^2+18*ln(b*x+a)*x*b^3*d^2*e-18*ln(e*x+d)*x*b^3*d
^2*e-6*x^2*a*b^2*e^3+6*x^2*b^3*d*e^2+6*ln(b*x+a)*b^3*d^3-6*ln(e*x+d)*b^3*d^3+3*x
*a^2*b*e^3-18*x*a*b^2*d*e^2+15*x*b^3*d^2*e-2*a^3*e^3+9*a^2*b*d*e^2-18*a*b^2*d^2*
e+11*b^3*d^3)/((b*x+a)^2)^(1/2)/(a*e-b*d)^4/(e*x+d)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217529, size = 574, normalized size = 2.48 \[ \frac{11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (b^{4} d^{7} - 4 \, a b^{3} d^{6} e + 6 \, a^{2} b^{2} d^{5} e^{2} - 4 \, a^{3} b d^{4} e^{3} + a^{4} d^{3} e^{4} +{\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{3} + 3 \,{\left (b^{4} d^{5} e^{2} - 4 \, a b^{3} d^{4} e^{3} + 6 \, a^{2} b^{2} d^{3} e^{4} - 4 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{2} + 3 \,{\left (b^{4} d^{6} e - 4 \, a b^{3} d^{5} e^{2} + 6 \, a^{2} b^{2} d^{4} e^{3} - 4 \, a^{3} b d^{3} e^{4} + a^{4} d^{2} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

1/6*(11*b^3*d^3 - 18*a*b^2*d^2*e + 9*a^2*b*d*e^2 - 2*a^3*e^3 + 6*(b^3*d*e^2 - a*
b^2*e^3)*x^2 + 3*(5*b^3*d^2*e - 6*a*b^2*d*e^2 + a^2*b*e^3)*x + 6*(b^3*e^3*x^3 +
3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(b*x + a) - 6*(b^3*e^3*x^3 + 3*b^3
*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(e*x + d))/(b^4*d^7 - 4*a*b^3*d^6*e + 6
*a^2*b^2*d^5*e^2 - 4*a^3*b*d^4*e^3 + a^4*d^3*e^4 + (b^4*d^4*e^3 - 4*a*b^3*d^3*e^
4 + 6*a^2*b^2*d^2*e^5 - 4*a^3*b*d*e^6 + a^4*e^7)*x^3 + 3*(b^4*d^5*e^2 - 4*a*b^3*
d^4*e^3 + 6*a^2*b^2*d^3*e^4 - 4*a^3*b*d^2*e^5 + a^4*d*e^6)*x^2 + 3*(b^4*d^6*e -
4*a*b^3*d^5*e^2 + 6*a^2*b^2*d^4*e^3 - 4*a^3*b*d^3*e^4 + a^4*d^2*e^5)*x)

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Sympy [A]  time = 6.8736, size = 570, normalized size = 2.47 \[ - \frac{b^{3} \log{\left (x + \frac{- \frac{a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac{5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} - \frac{10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} + \frac{10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} - \frac{5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e + \frac{b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac{b^{3} \log{\left (x + \frac{\frac{a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac{5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} + \frac{10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} - \frac{10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} + \frac{5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e - \frac{b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} - \frac{2 a^{2} e^{2} - 7 a b d e + 11 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} + 15 b^{2} d e\right )}{6 a^{3} d^{3} e^{3} - 18 a^{2} b d^{4} e^{2} + 18 a b^{2} d^{5} e - 6 b^{3} d^{6} + x^{3} \left (6 a^{3} e^{6} - 18 a^{2} b d e^{5} + 18 a b^{2} d^{2} e^{4} - 6 b^{3} d^{3} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{5} - 54 a^{2} b d^{2} e^{4} + 54 a b^{2} d^{3} e^{3} - 18 b^{3} d^{4} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{4} - 54 a^{2} b d^{3} e^{3} + 54 a b^{2} d^{4} e^{2} - 18 b^{3} d^{5} e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

-b**3*log(x + (-a**5*b**3*e**5/(a*e - b*d)**4 + 5*a**4*b**4*d*e**4/(a*e - b*d)**
4 - 10*a**3*b**5*d**2*e**3/(a*e - b*d)**4 + 10*a**2*b**6*d**3*e**2/(a*e - b*d)**
4 - 5*a*b**7*d**4*e/(a*e - b*d)**4 + a*b**3*e + b**8*d**5/(a*e - b*d)**4 + b**4*
d)/(2*b**4*e))/(a*e - b*d)**4 + b**3*log(x + (a**5*b**3*e**5/(a*e - b*d)**4 - 5*
a**4*b**4*d*e**4/(a*e - b*d)**4 + 10*a**3*b**5*d**2*e**3/(a*e - b*d)**4 - 10*a**
2*b**6*d**3*e**2/(a*e - b*d)**4 + 5*a*b**7*d**4*e/(a*e - b*d)**4 + a*b**3*e - b*
*8*d**5/(a*e - b*d)**4 + b**4*d)/(2*b**4*e))/(a*e - b*d)**4 - (2*a**2*e**2 - 7*a
*b*d*e + 11*b**2*d**2 + 6*b**2*e**2*x**2 + x*(-3*a*b*e**2 + 15*b**2*d*e))/(6*a**
3*d**3*e**3 - 18*a**2*b*d**4*e**2 + 18*a*b**2*d**5*e - 6*b**3*d**6 + x**3*(6*a**
3*e**6 - 18*a**2*b*d*e**5 + 18*a*b**2*d**2*e**4 - 6*b**3*d**3*e**3) + x**2*(18*a
**3*d*e**5 - 54*a**2*b*d**2*e**4 + 54*a*b**2*d**3*e**3 - 18*b**3*d**4*e**2) + x*
(18*a**3*d**2*e**4 - 54*a**2*b*d**3*e**3 + 54*a*b**2*d**4*e**2 - 18*b**3*d**5*e)
)

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GIAC/XCAS [A]  time = 0.215399, size = 332, normalized size = 1.44 \[ \frac{1}{6} \,{\left (\frac{6 \, b^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} - \frac{6 \, b^{3} e{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac{11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{{\left (b d - a e\right )}^{4}{\left (x e + d\right )}^{3}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="giac")

[Out]

1/6*(6*b^4*ln(abs(b*x + a))/(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3
*b^2*d*e^3 + a^4*b*e^4) - 6*b^3*e*ln(abs(x*e + d))/(b^4*d^4*e - 4*a*b^3*d^3*e^2
+ 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5) + (11*b^3*d^3 - 18*a*b^2*d^2*e +
9*a^2*b*d*e^2 - 2*a^3*e^3 + 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 3*(5*b^3*d^2*e - 6*a
*b^2*d*e^2 + a^2*b*e^3)*x)/((b*d - a*e)^4*(x*e + d)^3))*sign(b*x + a)