3.2030 \(\int \frac{a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=120 \[ -\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac{e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.226981, antiderivative size = 120, 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 \[ -\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac{e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.0942, size = 112, normalized size = 0.93 \[ - \frac{e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{1}{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0682031, size = 57, normalized size = 0.48 \[ \frac{e (a+b x) \log (d+e x)-e (a+b x) \log (a+b x)+a e-b d}{\sqrt{(a+b x)^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.215, size = 77, normalized size = 0.6 \[ -{\frac{ \left ( \ln \left ( bx+a \right ) xbe-\ln \left ( ex+d \right ) xbe+\ln \left ( bx+a \right ) ae-\ln \left ( ex+d \right ) ae-ae+bd \right ) \left ( bx+a \right ) ^{2}}{ \left ( ae-bd \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289512, size = 126, normalized size = 1.05 \[ -\frac{b d - a e +{\left (b e x + a e\right )} \log \left (b x + a\right ) -{\left (b e x + a e\right )} \log \left (e x + d\right )}{a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x} \]

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)),x, algorithm="fricas")

[Out]

-(b*d - a*e + (b*e*x + a*e)*log(b*x + a) - (b*e*x + a*e)*log(e*x + d))/(a*b^2*d^
2 - 2*a^2*b*d*e + a^3*e^2 + (b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.29493, size = 274, normalized size = 2.28 \[ -\frac{a e{\rm ln}\left ({\left | b + \frac{a}{x} \right |}\right )}{a b^{2} d^{2}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 2 \, a^{2} b d e{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{3} e^{2}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{d e{\rm ln}\left ({\left | \frac{d}{x} + e \right |}\right )}{b^{2} d^{3}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 2 \, a b d^{2} e{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{2} d e^{2}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{b^{2} d - a b e}{{\left (b d - a e\right )}^{2} a{\left (b + \frac{a}{x}\right )}{\rm sign}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} \]

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)),x, algorithm="giac")

[Out]

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