3.1578 \(\int \frac{(d+e x)^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=124 \[ \frac{e x (a+b x) (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^2 \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.141593, antiderivative size = 124, 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 \[ \frac{e x (a+b x) (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^2 \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.5321, size = 114, normalized size = 0.92 \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{e \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{3}} + \frac{\left (a + b x\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0640763, size = 59, normalized size = 0.48 \[ \frac{(a+b x) \left (b e x (-2 a e+4 b d+b e x)+2 (b d-a e)^2 \log (a+b x)\right )}{2 b^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 87, normalized size = 0.7 \[{\frac{ \left ( bx+a \right ) \left ({x}^{2}{b}^{2}{e}^{2}+2\,\ln \left ( bx+a \right ){a}^{2}{e}^{2}-4\,\ln \left ( bx+a \right ) abde+2\,\ln \left ( bx+a \right ){b}^{2}{d}^{2}-2\,xab{e}^{2}+4\,x{b}^{2}de \right ) }{2\,{b}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.689673, size = 153, normalized size = 1.23 \[ \frac{a^{2} b^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{a b e^{2} x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{e^{2} x^{2}}{2 \, \sqrt{b^{2}}} + \sqrt{\frac{1}{b^{2}}} d^{2} \log \left (x + \frac{a}{b}\right ) - \frac{2 \, a \sqrt{\frac{1}{b^{2}}} d e \log \left (x + \frac{a}{b}\right )}{b} + \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d e}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*b^2*e^2*log(x + a/b)/(b^2)^(5/2) - a*b*e^2*x/(b^2)^(3/2) + 1/2*e^2*x^2/sqrt(
b^2) + sqrt(b^(-2))*d^2*log(x + a/b) - 2*a*sqrt(b^(-2))*d*e*log(x + a/b)/b + 2*s
qrt(b^2*x^2 + 2*a*b*x + a^2)*d*e/b^2

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Fricas [A]  time = 0.20564, size = 85, normalized size = 0.69 \[ \frac{b^{2} e^{2} x^{2} + 2 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.54308, size = 44, normalized size = 0.35 \[ \frac{e^{2} x^{2}}{2 b} - \frac{x \left (a e^{2} - 2 b d e\right )}{b^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213133, size = 128, normalized size = 1.03 \[ \frac{b x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, b d x e{\rm sign}\left (b x + a\right ) - 2 \, a x e^{2}{\rm sign}\left (b x + a\right )}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b d e{\rm sign}\left (b x + a\right ) + a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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