3.1960 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{d+e x} \, dx\)

Optimal. Leaf size=122 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^3 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.195408, antiderivative size = 122, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^3 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.4229, size = 102, normalized size = 0.84 \[ \frac{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2}} + \frac{\left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.054353, size = 61, normalized size = 0.5 \[ \frac{\sqrt{(a+b x)^2} \left (b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)\right )}{2 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.017, size = 102, normalized size = 0.8 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ({x}^{2}{b}^{2}{e}^{2}+2\,\ln \left ( bex+bd \right ){a}^{2}{e}^{2}-4\,\ln \left ( bex+bd \right ) abde+2\,\ln \left ( bex+bd \right ){b}^{2}{d}^{2}+4\,xab{e}^{2}-2\,x{b}^{2}de+3\,{a}^{2}{e}^{2}-2\,abde \right ) }{2\,{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*csgn(b*x+a)*(x^2*b^2*e^2+2*ln(b*e*x+b*d)*a^2*e^2-4*ln(b*e*x+b*d)*a*b*d*e+2*l
n(b*e*x+b*d)*b^2*d^2+4*x*a*b*e^2-2*x*b^2*d*e+3*a^2*e^2-2*a*b*d*e)/e^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(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283, size = 84, normalized size = 0.69 \[ \frac{b^{2} e^{2} x^{2} - 2 \,{\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.54567, size = 44, normalized size = 0.36 \[ \frac{b^{2} x^{2}}{2 e} + \frac{x \left (2 a b e - b^{2} d\right )}{e^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286278, size = 131, normalized size = 1.07 \[{\left (b^{2} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b d e{\rm sign}\left (b x + a\right ) + a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{2} x^{2} e{\rm sign}\left (b x + a\right ) - 2 \, b^{2} d x{\rm sign}\left (b x + a\right ) + 4 \, a b x e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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