3.1728 \(\int (d+e x)^m \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+2}}{e^2 (m+2) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+1}}{e^2 (m+1) (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.125554, antiderivative size = 101, 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 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+2}}{e^2 (m+2) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+1}}{e^2 (m+1) (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.626, size = 83, normalized size = 0.82 \[ \frac{\left (d + e x\right )^{m + 1} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2} \left (a + b x\right ) \left (m + 1\right ) \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0651639, size = 59, normalized size = 0.58 \[ \frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} (a e (m+2)-b d+b e (m+1) x)}{e^2 (m+1) (m+2) (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 62, normalized size = 0.6 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( bemx+aem+bex+2\,ae-bd \right ) }{ \left ( bx+a \right ){e}^{2} \left ({m}^{2}+3\,m+2 \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.766107, size = 84, normalized size = 0.83 \[ \frac{{\left (b e^{2}{\left (m + 1\right )} x^{2} + a d e{\left (m + 2\right )} - b d^{2} +{\left (a e^{2}{\left (m + 2\right )} + b d e m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221478, size = 112, normalized size = 1.11 \[ \frac{{\left (a d e m - b d^{2} + 2 \, a d e +{\left (b e^{2} m + b e^{2}\right )} x^{2} +{\left (2 \, a e^{2} +{\left (b d e + a e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{2} m^{2} + 3 \, e^{2} m + 2 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**m*sqrt((a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.223033, size = 270, normalized size = 2.67 \[ \frac{b m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}{\rm sign}\left (b x + a\right ) + b d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )}{\rm sign}\left (b x + a\right ) + a m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}{\rm sign}\left (b x + a\right ) + b x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}{\rm sign}\left (b x + a\right ) + a d m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )}{\rm sign}\left (b x + a\right ) - b d^{2} e^{\left (m{\rm ln}\left (x e + d\right )\right )}{\rm sign}\left (b x + a\right ) + 2 \, a x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}{\rm sign}\left (b x + a\right ) + 2 \, a d e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )}{\rm sign}\left (b x + a\right )}{m^{2} e^{2} + 3 \, m e^{2} + 2 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*m*x^2*e^(m*ln(x*e + d) + 2)*sign(b*x + a) + b*d*m*x*e^(m*ln(x*e + d) + 1)*sig
n(b*x + a) + a*m*x*e^(m*ln(x*e + d) + 2)*sign(b*x + a) + b*x^2*e^(m*ln(x*e + d)
+ 2)*sign(b*x + a) + a*d*m*e^(m*ln(x*e + d) + 1)*sign(b*x + a) - b*d^2*e^(m*ln(x
*e + d))*sign(b*x + a) + 2*a*x*e^(m*ln(x*e + d) + 2)*sign(b*x + a) + 2*a*d*e^(m*
ln(x*e + d) + 1)*sign(b*x + a))/(m^2*e^2 + 3*m*e^2 + 2*e^2)