3.274 \(\int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^p \, dx\)

Optimal. Leaf size=140 \[ \frac{B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)}-\frac{x^{m+1} \left (\frac{c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p (b B (m+2 p+1)-A c (m+4 p+3)) \, _2F_1\left (-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right )}{c (m+2 p+1) (m+4 p+3)} \]

[Out]

(B*x^(-1 + m)*(b*x^2 + c*x^4)^(1 + p))/(c*(3 + m + 4*p)) - ((b*B*(1 + m + 2*p) -
A*c*(3 + m + 4*p))*x^(1 + m)*(b*x^2 + c*x^4)^p*Hypergeometric2F1[-p, (1 + m + 2
*p)/2, (3 + m + 2*p)/2, -((c*x^2)/b)])/(c*(1 + m + 2*p)*(3 + m + 4*p)*(1 + (c*x^
2)/b)^p)

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Rubi [A]  time = 0.265858, antiderivative size = 126, normalized size of antiderivative = 0.9, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ x^{m+1} \left (\frac{c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p \left (\frac{A}{m+2 p+1}-\frac{b B}{c (m+4 p+3)}\right ) \, _2F_1\left (-p,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\frac{c x^2}{b}\right )+\frac{B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p,x]

[Out]

(B*x^(-1 + m)*(b*x^2 + c*x^4)^(1 + p))/(c*(3 + m + 4*p)) + ((A/(1 + m + 2*p) - (
b*B)/(c*(3 + m + 4*p)))*x^(1 + m)*(b*x^2 + c*x^4)^p*Hypergeometric2F1[-p, (1 + m
+ 2*p)/2, (3 + m + 2*p)/2, -((c*x^2)/b)])/(1 + (c*x^2)/b)^p

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Rubi in Sympy [A]  time = 36.9992, size = 131, normalized size = 0.94 \[ \frac{B x^{m - 1} \left (b x^{2} + c x^{4}\right )^{p + 1}}{c \left (m + 4 p + 3\right )} + \frac{x^{m} x^{- m - 2 p} x^{m + 2 p + 1} \left (1 + \frac{c x^{2}}{b}\right )^{- p} \left (b x^{2} + c x^{4}\right )^{p} \left (A c \left (m + 4 p + 3\right ) - B b \left (m + 2 p + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + p + \frac{1}{2} \\ \frac{m}{2} + p + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{c \left (m + 2 p + 1\right ) \left (m + 4 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(B*x**2+A)*(c*x**4+b*x**2)**p,x)

[Out]

B*x**(m - 1)*(b*x**2 + c*x**4)**(p + 1)/(c*(m + 4*p + 3)) + x**m*x**(-m - 2*p)*x
**(m + 2*p + 1)*(1 + c*x**2/b)**(-p)*(b*x**2 + c*x**4)**p*(A*c*(m + 4*p + 3) - B
*b*(m + 2*p + 1))*hyper((-p, m/2 + p + 1/2), (m/2 + p + 3/2,), -c*x**2/b)/(c*(m
+ 2*p + 1)*(m + 4*p + 3))

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Mathematica [A]  time = 0.139331, size = 135, normalized size = 0.96 \[ \frac{x^{m+1} \left (x^2 \left (b+c x^2\right )\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \left (A (m+2 p+3) \, _2F_1\left (-p,\frac{m}{2}+p+\frac{1}{2};\frac{m}{2}+p+\frac{3}{2};-\frac{c x^2}{b}\right )+B x^2 (m+2 p+1) \, _2F_1\left (-p,\frac{m}{2}+p+\frac{3}{2};\frac{m}{2}+p+\frac{5}{2};-\frac{c x^2}{b}\right )\right )}{(m+2 p+1) (m+2 p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p,x]

[Out]

(x^(1 + m)*(x^2*(b + c*x^2))^p*(A*(3 + m + 2*p)*Hypergeometric2F1[-p, 1/2 + m/2
+ p, 3/2 + m/2 + p, -((c*x^2)/b)] + B*(1 + m + 2*p)*x^2*Hypergeometric2F1[-p, 3/
2 + m/2 + p, 5/2 + m/2 + p, -((c*x^2)/b)]))/((1 + m + 2*p)*(3 + m + 2*p)*(1 + (c
*x^2)/b)^p)

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( B{x}^{2}+A \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x)

[Out]

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(c*x^4 + b*x^2)^p*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)*(c*x^4 + b*x^2)^p*x^m, x)

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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*(B*x**2+A)*(c*x**4+b*x**2)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*(c*x^4 + b*x^2)^p*x^m, x)