3.731 \(\int (d+e x)^{-3-2 p} \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=270 \[ -\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.253369, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{c d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

-(e*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))) - (c*
d*(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 + Sqrt[-a]*e)*(c*d^2 + a*e^2)*(1 + 2*p)*(-(((Sq
rt[c]*d + Sqrt[-a]*e)*(Sqrt[-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 = 28.092, size = 226, normalized size = 0.84 \[ - \frac{c d \left (\frac{\left (\sqrt{c} d + e \sqrt{- a}\right ) \left (\sqrt{c} x + \sqrt{- a}\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (\sqrt{c} x - \sqrt{- a}\right )}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1} \left (- \sqrt{c} x + \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{2 \sqrt{c} \sqrt{- a} \left (d + e x\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (- \sqrt{c} x + \sqrt{- a}\right )}} \right )}}{\left (2 p + 1\right ) \left (a e^{2} + c d^{2}\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{- 2 p - 2}}{2 \left (p + 1\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-3-2*p)*(c*x**2+a)**p,x)

[Out]

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

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Mathematica [A]  time = 55.0809, size = 368, normalized size = 1.36 \[ \frac{2^{-2 p-3} \Gamma \left (-p-\frac{1}{2}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 (p+1)} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (1-\frac{d+e x}{e \sqrt{-\frac{a}{c}}+d}\right )^{p+1} \left (\Gamma (1-2 p) \Gamma (-p) \left (e \sqrt{-\frac{a}{c}}+d\right ) \left (e \left (2 p \sqrt{-\frac{a}{c}}+\sqrt{-\frac{a}{c}}+x\right )+2 d (p+1)\right ) \, _2F_1\left (1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d+\sqrt{-\frac{a}{c}} e\right ) \left (x+\sqrt{-\frac{a}{c}}\right )}\right )+\frac{2 e \Gamma (1-p) \Gamma (-2 p) \left (c x \sqrt{-\frac{a}{c}}+a\right ) (d+e x) \, _2F_1\left (2,1-p;1-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d+\sqrt{-\frac{a}{c}} e\right ) \left (x+\sqrt{-\frac{a}{c}}\right )}\right )}{c \left (\sqrt{-\frac{a}{c}}+x\right )}\right )}{\sqrt{\pi } e (p+1) \Gamma (1-2 p) \Gamma (-2 p) \left (e \sqrt{-\frac{a}{c}}+d\right )^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(-3 - 2*p)*(a + c*x^2)^p,x]

[Out]

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

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-3-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-3-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-3-2*p)*(c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3),x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 3), 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)**(-3-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 3), x)