3.734 \(\int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=559 \[ -\frac{c^2 \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^4}+\frac{c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac{c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(5 + 2*p))) + (c
*e*(3*a*e^2*(2 + p) - c*d^2*(18 + 11*p + 2*p^2))*(d + e*x)^(-3 - 2*p)*(a + c*x^2
)^(1 + p))/((c*d^2 + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) + (c^2*d*e*(3 + p)*(a
*e^2*(8 + 5*p) - c*d^2*(8 + 7*p + 2*p^2))*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^
4*(1 + p)*(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 + p))) - (c*d*e*(4 + p)*(a
+ c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2 + p)*(5 + 2*p)*(d + e*x)^(2*(2 + p))) -
(c^2*(3*a^2*e^4 - 6*a*c*d^2*e^2*(5 + 2*p) + c^2*d^4*(15 + 16*p + 4*p^2))*(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] - S
qrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)^4*(1 + 2*p)*(3 + 2*p)*(5
+ 2*p)*(-(((Sqrt[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 = 1.8861, antiderivative size = 559, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{c^2 \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^4}+\frac{c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac{c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-6 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)*(5 + 2*p))) + (c
*e*(3*a*e^2*(2 + p) - c*d^2*(18 + 11*p + 2*p^2))*(d + e*x)^(-3 - 2*p)*(a + c*x^2
)^(1 + p))/((c*d^2 + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) + (c^2*d*e*(3 + p)*(a
*e^2*(8 + 5*p) - c*d^2*(8 + 7*p + 2*p^2))*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^
4*(1 + p)*(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 + p))) - (c*d*e*(4 + p)*(a
+ c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2 + p)*(5 + 2*p)*(d + e*x)^(2*(2 + p))) -
(c^2*(3*a^2*e^4 - 6*a*c*d^2*e^2*(5 + 2*p) + c^2*d^4*(15 + 16*p + 4*p^2))*(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] - S
qrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c*d^2 + a*e^2)^4*(1 + 2*p)*(3 + 2*p)*(5
+ 2*p)*(-(((Sqrt[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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-6-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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Mathematica [F]  time = 179.999, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^(-6 - 2*p)*(a + c*x^2)^p,x]

[Out]

$Aborted

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-6-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-6-2*p)*(c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 6), 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)**(-6-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 6), x)