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

Optimal. Leaf size=91 \[ -\frac{d \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x}-\frac{e \left (a+c x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{c x^2}{a}+1\right )}{2 a (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.124242, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{d \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x}-\frac{e \left (a+c x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{c x^2}{a}+1\right )}{2 a (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.2181, size = 71, normalized size = 0.78 \[ - \frac{d \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{x} - \frac{e \left (a + c x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{c x^{2}}{a}} \right )}}{2 a \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0749403, size = 93, normalized size = 1.02 \[ \frac{1}{2} \left (a+c x^2\right )^p \left (\frac{e \left (\frac{a}{c x^2}+1\right )^{-p} \, _2F_1\left (-p,-p;1-p;-\frac{a}{c x^2}\right )}{p}-\frac{2 d \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{c x^2}{a}\right )}{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a)^p/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 43.5539, size = 68, normalized size = 0.75 \[ - \frac{a^{p} d{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - p \\ \frac{1}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac{c^{p} e x^{2 p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{a e^{i \pi }}{c x^{2}}} \right )}}{2 \Gamma \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**p*d*hyper((-1/2, -p), (1/2,), c*x**2*exp_polar(I*pi)/a)/x - c**p*e*x**(2*p)*
gamma(-p)*hyper((-p, -p), (-p + 1,), a*exp_polar(I*pi)/(c*x**2))/(2*gamma(-p + 1
))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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