3.2740 \(\int x^m \left (a+b x^{2+2 m}\right )^{3/2} \, dx\)

Optimal. Leaf size=104 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{8 \sqrt{b} (m+1)}+\frac{x^{m+1} \left (a+b x^{2 (m+1)}\right )^{3/2}}{4 (m+1)}+\frac{3 a x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{8 (m+1)} \]

[Out]

(3*a*x^(1 + m)*Sqrt[a + b*x^(2*(1 + m))])/(8*(1 + m)) + (x^(1 + m)*(a + b*x^(2*(
1 + m)))^(3/2))/(4*(1 + m)) + (3*a^2*ArcTanh[(Sqrt[b]*x^(1 + m))/Sqrt[a + b*x^(2
*(1 + m))]])/(8*Sqrt[b]*(1 + m))

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Rubi [A]  time = 0.0990939, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{8 \sqrt{b} (m+1)}+\frac{x^{m+1} \left (a+b x^{2 (m+1)}\right )^{3/2}}{4 (m+1)}+\frac{3 a x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{8 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(3*a*x^(1 + m)*Sqrt[a + b*x^(2*(1 + m))])/(8*(1 + m)) + (x^(1 + m)*(a + b*x^(2*(
1 + m)))^(3/2))/(4*(1 + m)) + (3*a^2*ArcTanh[(Sqrt[b]*x^(1 + m))/Sqrt[a + b*x^(2
*(1 + m))]])/(8*Sqrt[b]*(1 + m))

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Rubi in Sympy [A]  time = 6.89625, size = 58, normalized size = 0.56 \[ \frac{a x^{m + 1} \sqrt{a + b x^{2 m + 2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{\sqrt{1 + \frac{b x^{2 m + 2}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.190514, size = 112, normalized size = 1.08 \[ \frac{3 a^{5/2} \sqrt{\frac{b x^{2 m+2}}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )+\sqrt{b} x^{m+1} \left (5 a^2+7 a b x^{2 m+2}+2 b^2 x^{4 m+4}\right )}{8 \sqrt{b} (m+1) \sqrt{a+b x^{2 m+2}}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*x^(1 + m)*(5*a^2 + 7*a*b*x^(2 + 2*m) + 2*b^2*x^(4 + 4*m)) + 3*a^(5/2)*S
qrt[1 + (b*x^(2 + 2*m))/a]*ArcSinh[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(8*Sqrt[b]*(1 +
m)*Sqrt[a + b*x^(2 + 2*m)])

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b{x}^{2+2\,m} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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