3.2032 \(\int \frac{a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac{b^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{2 b e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}-\frac{e (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{3 b^2 e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{3 b^2 e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

[Out]

-(b^2/((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (e*(a + b*x))/(2*(b*d - a
*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b*e*(a + b*x))/((b*d - a*e
)^3*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e*(a + b*x)*Log[a + b*x])/
((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2*e*(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 = 0.362642, antiderivative size = 223, 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 \[ -\frac{b^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{2 b e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}-\frac{e (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{3 b^2 e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{3 b^2 e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2/((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (e*(a + b*x))/(2*(b*d - a
*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b*e*(a + b*x))/((b*d - a*e
)^3*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e*(a + b*x)*Log[a + b*x])/
((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2*e*(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 = 59.4472, size = 219, normalized size = 0.98 \[ - \frac{3 b^{2} e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} + \frac{3 b^{2} e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} + \frac{3 b e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{4}} - \frac{3 e \left (2 a + 2 b x\right )}{4 \left (d + e x\right )^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{1}{\left (d + e x\right )^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b**2*e*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*d)**
4) + 3*b**2*e*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(a*e - b*
d)**4) + 3*b*e**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*d)**4) -
3*e*(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
)) + 1/((d + e*x)**2*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.182349, size = 135, normalized size = 0.61 \[ \frac{-(b d-a e) \left (-a^2 e^2+a b e (5 d+3 e x)+b^2 \left (2 d^2+9 d e x+6 e^2 x^2\right )\right )-6 b^2 e (a+b x) (d+e x)^2 \log (a+b x)+6 b^2 e (a+b x) (d+e x)^2 \log (d+e x)}{2 \sqrt{(a+b x)^2} (d+e x)^2 (b d-a e)^4} \]

Antiderivative was successfully verified.

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

[Out]

(-((b*d - a*e)*(-(a^2*e^2) + a*b*e*(5*d + 3*e*x) + b^2*(2*d^2 + 9*d*e*x + 6*e^2*
x^2))) - 6*b^2*e*(a + b*x)*(d + e*x)^2*Log[a + b*x] + 6*b^2*e*(a + b*x)*(d + e*x
)^2*Log[d + e*x])/(2*(b*d - a*e)^4*Sqrt[(a + b*x)^2]*(d + e*x)^2)

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Maple [A]  time = 0.032, size = 331, normalized size = 1.5 \[ -{\frac{ \left ( 6\,\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{3}-6\,\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{3}+6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}-6\,\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{3}-12\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}+12\,\ln \left ( bx+a \right ) xa{b}^{2}d{e}^{2}+6\,\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}e-12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}-6\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e-6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e-3\,x{a}^{2}b{e}^{3}-6\,xa{b}^{2}d{e}^{2}+9\,x{b}^{3}{d}^{2}e+{a}^{3}{e}^{3}-6\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e+2\,{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) ^{2}}{2\, \left ( ex+d \right ) ^{2} \left ( ae-bd \right ) ^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-1/2*(6*ln(b*x+a)*x^3*b^3*e^3-6*ln(e*x+d)*x^3*b^3*e^3+6*ln(b*x+a)*x^2*a*b^2*e^3+
12*ln(b*x+a)*x^2*b^3*d*e^2-6*ln(e*x+d)*x^2*a*b^2*e^3-12*ln(e*x+d)*x^2*b^3*d*e^2+
12*ln(b*x+a)*x*a*b^2*d*e^2+6*ln(b*x+a)*x*b^3*d^2*e-12*ln(e*x+d)*x*a*b^2*d*e^2-6*
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)*a*b^2*d^2*e-6*
ln(e*x+d)*a*b^2*d^2*e-3*x*a^2*b*e^3-6*x*a*b^2*d*e^2+9*x*b^3*d^2*e+a^3*e^3-6*a^2*
b*d*e^2+3*a*b^2*d^2*e+2*b^3*d^3)*(b*x+a)^2/(e*x+d)^2/(a*e-b*d)^4/((b*x+a)^2)^(3/
2)

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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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295181, size = 668, normalized size = 3. \[ -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} +{\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} +{\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} +{\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d e^{5}\right )} x\right )}} \]

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)^(3/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-1/2*(2*b^3*d^3 + 3*a*b^2*d^2*e - 6*a^2*b*d*e^2 + a^3*e^3 + 6*(b^3*d*e^2 - a*b^2
*e^3)*x^2 + 3*(3*b^3*d^2*e - 2*a*b^2*d*e^2 - a^2*b*e^3)*x + 6*(b^3*e^3*x^3 + a*b
^2*d^2*e + (2*b^3*d*e^2 + a*b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*b^2*d*e^2)*x)*log(b*
x + a) - 6*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*b^3*d*e^2 + a*b^2*e^3)*x^2 + (b^3*d^2
*e + 2*a*b^2*d*e^2)*x)*log(e*x + d))/(a*b^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^
4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b
^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 + (2*b^5*d^5*e - 7*a*b^4*d^4*e^2 +
8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6
- 2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a
^5*d*e^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.308006, size = 622, normalized size = 2.79 \[ -\frac{3 \, a b^{2} e{\rm ln}\left ({\left | b + \frac{a}{x} \right |}\right )}{a b^{4} d^{4}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a^{2} b^{3} d^{3} e{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + 6 \, a^{3} b^{2} d^{2} e^{2}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a^{4} b d e^{3}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{5} e^{4}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{3 \, b^{2} d e{\rm ln}\left ({\left | \frac{d}{x} + e \right |}\right )}{b^{4} d^{5}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a b^{3} d^{4} e{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + 6 \, a^{2} b^{2} d^{3} e^{2}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a^{3} b d^{2} e^{3}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{4} d e^{4}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} - 6 \, a^{2} b^{2} d e^{4} + a^{3} b e^{5} + \frac{4 \, b^{4} d^{4} e + 2 \, a b^{3} d^{3} e^{2} - 3 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}}{x} + \frac{2 \,{\left (b^{4} d^{5} - a b^{3} d^{4} e + 3 \, a^{2} b^{2} d^{3} e^{2} - 4 \, a^{3} b d^{2} e^{3} + a^{4} d e^{4}\right )}}{x^{2}}}{2 \,{\left (b d - a e\right )}^{4} a{\left (b + \frac{a}{x}\right )} d^{2}{\left (\frac{d}{x} + e\right )}^{2}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} \]

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)^(3/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

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