3.343 \(\int \frac{\left (a+b x^2\right )^p}{c+d x^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c} \]

[Out]

(x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(c*(1 +
(b*x^2)/a)^p)

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Rubi [A]  time = 0.0711738, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(c*(1 +
(b*x^2)/a)^p)

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Rubi in Sympy [A]  time = 29.9045, size = 42, normalized size = 0.74 \[ \frac{x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- p,\frac{3}{2},- \frac{d x^{2}}{c},- \frac{b x^{2}}{a} \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*appellf1(1/2, 1, -p, 3/2, -d*x**2/c, -b*x
**2/a)/c

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Mathematica [B]  time = 0.307129, size = 162, normalized size = 2.84 \[ -\frac{3 a c x \left (a+b x^2\right )^p F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (2 x^2 \left (a d F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-b c p F_1\left (\frac{3}{2};1-p,1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-3 a c F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*a*c*x*(a + b*x^2)^p*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/(
(c + d*x^2)*(-3*a*c*AppellF1[1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x^
2*(-(b*c*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]) + a*d*Appel
lF1[3/2, -p, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p}}{d{x}^{2}+c}}\, dx \]

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. \[ \int \frac{{\left (b x^{2} + a\right )}^{p}}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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