3.732 \(\int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=347 \[ \frac{c \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (a e^2-c d^2 (2 p+3)\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac{c d e (p+2) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{(p+1) (2 p+3) \left (a e^2+c d^2\right )^2} \]

[Out]

-((e*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(3 + 2*p))) - (c
*d*e*(2 + p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(1 + p)*(3 + 2*p)*(d + e*x)
^(2*(1 + p))) + (c*(a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^(-
1 - 2*p)*(a + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[c]
*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sq
rt[-a]*e)*(c*d^2 + a*e^2)^2*(1 + 2*p)*(3 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sq
rt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^p)

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Rubi [A]  time = 0.503732, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{c \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (a e^2-c d^2 (2 p+3)\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac{c d e (p+2) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{(p+1) (2 p+3) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((e*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(3 + 2*p))) - (c
*d*e*(2 + p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(1 + p)*(3 + 2*p)*(d + e*x)
^(2*(1 + p))) + (c*(a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^(-
1 - 2*p)*(a + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[c]
*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sq
rt[-a]*e)*(c*d^2 + a*e^2)^2*(1 + 2*p)*(3 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sq
rt[-a] + Sqrt[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^p)

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Rubi in Sympy [A]  time = 60.6452, size = 296, normalized size = 0.85 \[ - \frac{c d e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{- 2 p - 2} \left (p + 2\right )}{\left (p + 1\right ) \left (2 p + 3\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \left (\frac{\left (\sqrt{c} d + e \sqrt{- a}\right ) \left (\sqrt{c} x + \sqrt{- a}\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (\sqrt{c} x - \sqrt{- a}\right )}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1} \left (a e^{2} - c d^{2} \left (2 p + 3\right )\right ) \left (- \sqrt{c} x + \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{2 \sqrt{c} \sqrt{- a} \left (d + e x\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (- \sqrt{c} x + \sqrt{- a}\right )}} \right )}}{\left (2 p + 1\right ) \left (2 p + 3\right ) \left (a e^{2} + c d^{2}\right )^{2} \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{- 2 p - 3}}{\left (2 p + 3\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-4-2*p)*(c*x**2+a)**p,x)

[Out]

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

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Mathematica [B]  time = 138.515, size = 1439, normalized size = 4.15 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(-4 - 2*p)*(a + c*x^2)^p,x]

[Out]

(-2*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 +
p)*Gamma[-2*(1 + p)]*((d + Sqrt[-(a/c)]*e)^3*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*
Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-
(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*(d + Sqrt[-(a/c)]*e)^3*p*(Sqrt[-(a/c)] + x)*G
amma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x)
)/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 2*(d + Sqrt[-(a/c)]*e)^3*p^2*(Sqr
t[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-
(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + (d + Sqrt[-(a/c)]
*e)^2*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1,
-p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))]
+ 2*(d + Sqrt[-(a/c)]*e)^2*p*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[1 - 2*p]*Gamma[
-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]
*e)*(Sqrt[-(a/c)] + x))] + (d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x)*(d + e*x)^2*G
amma[1 - 2*p]*Gamma[-p]*Hypergeometric2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x)
)/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 3*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*
e)^2*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*
Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 4*Sqrt[-(a/
c)]*(d + Sqrt[-(a/c)]*e)^2*p*(d + e*x)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F
1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/
c)] + x))] + 4*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^2*Gamma[1 - p]*Gamm
a[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sq
rt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*Sqrt[-(a/c)]*(d + e*x)^3*Gamma[1 - p]*Gam
ma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + S
qrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*(d + e
*x)*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*S
qrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] - 2*Sqrt[-(a/c
)]*(d + Sqrt[-(a/c)]*e)*(d + e*x)^2*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{
2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sq
rt[-(a/c)] + x))] + Sqrt[-(a/c)]*(d + e*x)^3*Gamma[1 - p]*Gamma[-2*p]*Hypergeome
tricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c
)]*e)*(Sqrt[-(a/c)] + x))]))/(e*(d + Sqrt[-(a/c)]*e)^3*(3 + 2*p)*((e*(Sqrt[-(a/c
)] - x))/(d + Sqrt[-(a/c)]*e))^p*(Sqrt[-(a/c)] + x)*Gamma[1 - 2*p]*Gamma[-2*p]*G
amma[-p])

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Maple [F]  time = 0.12, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-4-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-4-2*p)*(c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4),x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), 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((e*x+d)**(-4-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 4), x)