3.341 \(\int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx\)

Optimal. Leaf size=93 \[ \frac{d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} (a d-b c (2 p+3)) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b (2 p+3)} \]

[Out]

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

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Rubi [A]  time = 0.0961799, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c-\frac{a d}{2 b p+3 b}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+\frac{d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7575, size = 73, normalized size = 0.78 \[ \frac{d x \left (a + b x^{2}\right )^{p + 1}}{b \left (2 p + 3\right )} - \frac{x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (a d - b c \left (2 p + 3\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{b \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**p*(d*x**2+c),x)

[Out]

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

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Mathematica [A]  time = 0.030472, size = 75, normalized size = 0.81 \[ \frac{1}{3} x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 c \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+d x^2 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^2)^p*(3*c*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + d*x^2*Hype
rgeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)]))/(3*(1 + (b*x^2)/a)^p)

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p*(d*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 39.9354, size = 53, normalized size = 0.57 \[ a^{p} c x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} + \frac{a^{p} d x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**p*(d*x**2+c),x)

[Out]

a**p*c*x*hyper((1/2, -p), (3/2,), b*x**2*exp_polar(I*pi)/a) + a**p*d*x**3*hyper(
(3/2, -p), (5/2,), b*x**2*exp_polar(I*pi)/a)/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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