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

Optimal. Leaf size=73 \[ \frac{\left (a+b x^3\right ) (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

((d*x)^(1 + m)*(a + b*x^3)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/
a)])/(a*d*(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.0883383, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\left (a+b x^3\right ) (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^(1 + m)*(a + b*x^3)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/
a)])/(a*d*(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi in Sympy [A]  time = 16.506, size = 61, normalized size = 0.84 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a d \left (a + b x^{3}\right ) \left (m + 1\right )} \]

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]

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

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Mathematica [A]  time = 0.0487395, size = 62, normalized size = 0.85 \[ \frac{x \left (a+b x^3\right ) (d x)^m \, _2F_1\left (1,\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{a (m+1) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d*x)^m*(a + b*x^3)*Hypergeometric2F1[1, (1 + m)/3, 1 + (1 + m)/3, -((b*x^3)/
a)])/(a*(1 + m)*Sqrt[(a + b*x^3)^2])

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{{b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2}}}}}\, dx \]

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]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/sqrt(b^2*x^6 + 2*a*b*x^3 + a^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{\sqrt{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x)^m/sqrt(b^2*x^6 + 2*a*b*x^3 + a^2), x)

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

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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/sqrt(b^2*x^6 + 2*a*b*x^3 + a^2), x)