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

Optimal. Leaf size=191 \[ \frac{x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,2;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2}+\frac{e^2 x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d^4}-\frac{c d e \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{(p+1) \left (a e^2+c d^2\right )^2} \]

[Out]

(x*(a + c*x^2)^p*AppellF1[1/2, -p, 2, 3/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^2*(1
+ (c*x^2)/a)^p) + (e^2*x^3*(a + c*x^2)^p*AppellF1[3/2, -p, 2, 5/2, -((c*x^2)/a)
, (e^2*x^2)/d^2])/(3*d^4*(1 + (c*x^2)/a)^p) - (c*d*e*(a + c*x^2)^(1 + p)*Hyperge
ometric2F1[2, 1 + p, 2 + p, (e^2*(a + c*x^2))/(c*d^2 + a*e^2)])/((c*d^2 + a*e^2)
^2*(1 + p))

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Rubi [A]  time = 0.489836, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,2;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2}+\frac{e^2 x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d^4}-\frac{c d e \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{(p+1) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + c*x^2)^p*AppellF1[1/2, -p, 2, 3/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^2*(1
+ (c*x^2)/a)^p) + (e^2*x^3*(a + c*x^2)^p*AppellF1[3/2, -p, 2, 5/2, -((c*x^2)/a)
, (e^2*x^2)/d^2])/(3*d^4*(1 + (c*x^2)/a)^p) - (c*d*e*(a + c*x^2)^(1 + p)*Hyperge
ometric2F1[2, 1 + p, 2 + p, (e^2*(a + c*x^2))/(c*d^2 + a*e^2)])/((c*d^2 + a*e^2)
^2*(1 + p))

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Rubi in Sympy [A]  time = 18.6176, size = 144, normalized size = 0.75 \[ - \frac{\left (\frac{e \left (\sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} \left (d + e x\right )}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 1} \operatorname{appellf_{1}}{\left (- 2 p + 1,- p,- p,- 2 p + 2,\frac{d - \frac{e \sqrt{- a}}{\sqrt{c}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{c}}}{d + e x} \right )}}{e \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(e*(sqrt(c)*x + sqrt(-a))/(sqrt(c)*(d + e*x)))**(-p)*(-e*(-sqrt(c)*x + sqrt(-a)
)/(sqrt(c)*(d + e*x)))**(-p)*(a + c*x**2)**p*(1/(d + e*x))**(2*p)*(1/(d + e*x))*
*(-2*p + 1)*appellf1(-2*p + 1, -p, -p, -2*p + 2, (d - e*sqrt(-a)/sqrt(c))/(d + e
*x), (d + e*sqrt(-a)/sqrt(c))/(d + e*x))/(e*(-2*p + 1))

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Mathematica [A]  time = 0.13537, size = 141, normalized size = 0.74 \[ \frac{\left (a+c x^2\right )^p \left (\frac{e \left (x-\sqrt{-\frac{a}{c}}\right )}{d+e x}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{d+e x}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{c}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{c}} e}{d+e x}\right )}{e (2 p-1) (d+e x)} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^p/(e*x+d)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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