3.120 \(\int (d x)^m \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx\)

Optimal. Leaf size=97 \[ \frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]

[Out]

(a*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d*(1 + m)*(a + b*x^3)) + (b*(
d*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^4*(4 + m)*(a + b*x^3))

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Rubi [A]  time = 0.0928917, 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 \[ \frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(a*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d*(1 + m)*(a + b*x^3)) + (b*(
d*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^4*(4 + m)*(a + b*x^3))

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Rubi in Sympy [A]  time = 10.8222, size = 80, normalized size = 0.82 \[ \frac{3 a \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{d \left (a + b x^{3}\right ) \left (m + 1\right ) \left (m + 4\right )} + \frac{\left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{d \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b**2*x**6+2*a*b*x**3+a**2)**(1/2),x)

[Out]

3*a*(d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(d*(a + b*x**3)*(m + 1)*(
m + 4)) + (d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(d*(m + 4))

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Mathematica [A]  time = 0.0437938, size = 47, normalized size = 0.48 \[ \frac{\sqrt{\left (a+b x^3\right )^2} (d x)^m \left (\frac{a x}{m+1}+\frac{b x^4}{m+4}\right )}{a+b x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

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

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Maple [A]  time = 0.005, size = 56, normalized size = 0.6 \[{\frac{ \left ( bm{x}^{3}+b{x}^{3}+am+4\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 4+m \right ) \left ( 1+m \right ) \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b^2*x^6+2*a*b*x^3+a^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.795317, size = 47, normalized size = 0.48 \[ \frac{{\left (b d^{m}{\left (m + 1\right )} x^{4} + a d^{m}{\left (m + 4\right )} x\right )} x^{m}}{m^{2} + 5 \, m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*(d*x)^m,x, algorithm="maxima")

[Out]

(b*d^m*(m + 1)*x^4 + a*d^m*(m + 4)*x)*x^m/(m^2 + 5*m + 4)

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Fricas [A]  time = 0.275705, size = 47, normalized size = 0.48 \[ \frac{{\left ({\left (b m + b\right )} x^{4} +{\left (a m + 4 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*(d*x)^m,x, algorithm="fricas")

[Out]

((b*m + b)*x^4 + (a*m + 4*a)*x)*(d*x)^m/(m^2 + 5*m + 4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b**2*x**6+2*a*b*x**3+a**2)**(1/2),x)

[Out]

Integral((d*x)**m*sqrt((a + b*x**3)**2), x)

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GIAC/XCAS [A]  time = 0.314577, size = 123, normalized size = 1.27 \[ \frac{b m x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + b x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + a m x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 4 \, a x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right )}{m^{2} + 5 \, m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*(d*x)^m,x, algorithm="giac")

[Out]

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