3.793 \(\int (d x)^m \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=97 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]

[Out]

(a*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*(1 + m)*(a + b*x^2)) + (b*(
d*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(3 + m)*(a + b*x^2))

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Rubi [A]  time = 0.103666, antiderivative size = 97, 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 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(a*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*(1 + m)*(a + b*x^2)) + (b*(
d*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(3 + m)*(a + b*x^2))

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Rubi in Sympy [A]  time = 10.6697, size = 80, normalized size = 0.82 \[ \frac{2 a \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{d \left (a + b x^{2}\right ) \left (m + 1\right ) \left (m + 3\right )} + \frac{\left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{d \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**(1/2),x)

[Out]

2*a*(d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(d*(a + b*x**2)*(m + 1)*(
m + 3)) + (d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(d*(m + 3))

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Mathematica [A]  time = 0.0434534, size = 47, normalized size = 0.48 \[ \frac{\sqrt{\left (a+b x^2\right )^2} (d x)^m \left (\frac{a x}{m+1}+\frac{b x^3}{m+3}\right )}{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

((d*x)^m*Sqrt[(a + b*x^2)^2]*((a*x)/(1 + m) + (b*x^3)/(3 + m)))/(a + b*x^2)

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Maple [A]  time = 0.005, size = 56, normalized size = 0.6 \[{\frac{ \left ( bm{x}^{2}+b{x}^{2}+am+3\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 3+m \right ) \left ( 1+m \right ) \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.698239, size = 47, normalized size = 0.48 \[ \frac{{\left (b d^{m}{\left (m + 1\right )} x^{3} + a d^{m}{\left (m + 3\right )} x\right )} x^{m}}{m^{2} + 4 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^m,x, algorithm="maxima")

[Out]

(b*d^m*(m + 1)*x^3 + a*d^m*(m + 3)*x)*x^m/(m^2 + 4*m + 3)

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Fricas [A]  time = 0.285496, size = 47, normalized size = 0.48 \[ \frac{{\left ({\left (b m + b\right )} x^{3} +{\left (a m + 3 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{2} + 4 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^m,x, algorithm="fricas")

[Out]

((b*m + b)*x^3 + (a*m + 3*a)*x)*(d*x)^m/(m^2 + 4*m + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.267416, size = 123, normalized size = 1.27 \[ \frac{b m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + b x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + a m x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right )}{m^{2} + 4 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^m,x, algorithm="giac")

[Out]

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