3.797 \(\int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx\)

Optimal. Leaf size=74 \[ \frac{\left (a+b x^2\right ) (d x)^{m+1} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,\frac{1}{2} (m+4 p+3);\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0726291, antiderivative size = 77, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(d x)^{m+1} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{m+1}{2},-2 p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.4921, size = 66, normalized size = 0.89 \[ \frac{\left (d x\right )^{m + 1} \left (1 + \frac{b x^{2}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{d \left (m + 1\right )} \]

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)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0612803, size = 66, normalized size = 0.89 \[ \frac{x (d x)^m \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{m+1}{2},-2 p;\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p}\, dx \]

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)^p,x)

[Out]

int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p*(d*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^p*(d*x)^m, x)

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

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)**p,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p*(d*x)^m, x)