Optimal. Leaf size=809 \[ -\frac{e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac{e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 4.34609, antiderivative size = 809, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac{e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]
[Out]
_______________________________________________________________________________________
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+b*x+a)**p,x)
[Out]
_______________________________________________________________________________________
Mathematica [F] time = 180., size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In] Integrate[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.229, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-6-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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+b*x+a)**p,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="giac")
[Out]