Optimal. Leaf size=95 \[ -\frac{(d+e x)^4 (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,2 p+5;p+5;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(p+4) \left (c d^2-a e^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.215635, antiderivative size = 124, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{\left (c d^2-a e^2\right )^3 (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p-3,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c^4 d^4 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 67.7333, size = 122, normalized size = 1.28 \[ - \frac{\left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{- p} \left (a e + c d x\right )^{- p} \left (a e + c d x\right )^{p + 1} \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 3, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{c^{4} d^{4} \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 3.49068, size = 533, normalized size = 5.61 \[ \frac{d ((d+e x) (a e+c d x))^p \left (\frac{5 a e^5 x^4 F_1\left (4;-p,-p;5;-\frac{c d x}{a e},-\frac{e x}{d}\right )}{p x \left (c d^2 F_1\left (5;1-p,-p;6;-\frac{c d x}{a e},-\frac{e x}{d}\right )+a e^2 F_1\left (5;-p,1-p;6;-\frac{c d x}{a e},-\frac{e x}{d}\right )\right )+5 a d e F_1\left (4;-p,-p;5;-\frac{c d x}{a e},-\frac{e x}{d}\right )}+\frac{16 a d e^4 x^3 F_1\left (3;-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )}{p x \left (c d^2 F_1\left (4;1-p,-p;5;-\frac{c d x}{a e},-\frac{e x}{d}\right )+a e^2 F_1\left (4;-p,1-p;5;-\frac{c d x}{a e},-\frac{e x}{d}\right )\right )+4 a d e F_1\left (3;-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )}+\frac{18 a d^2 e^3 x^2 F_1\left (2;-p,-p;3;-\frac{c d x}{a e},-\frac{e x}{d}\right )}{p x \left (c d^2 F_1\left (3;1-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )+a e^2 F_1\left (3;-p,1-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )\right )+3 a d e F_1\left (2;-p,-p;3;-\frac{c d x}{a e},-\frac{e x}{d}\right )}+\frac{4 d^2 (d+e x) \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{p+1}\right )}{4 e} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.153, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{3} \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p,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)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p,x, algorithm="giac")
[Out]