Optimal. Leaf size=248 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}+\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.642339, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}+\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*Sqrt[a + b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 57.0931, size = 253, normalized size = 1.02 \[ \frac{e \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5 c} + \frac{e \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 8 a c e^{2} + \frac{35 b^{2} e^{2}}{4} - \frac{75 b c d e}{2} + 48 c^{2} d^{2} - \frac{21 c e x \left (b e - 2 c d\right )}{2}\right )}{60 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (- 12 a c e^{2} + 7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{128 c^{4}} + \frac{\left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \left (- 12 a c e^{2} + 7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.462088, size = 292, normalized size = 1.18 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 c^2 \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )-4 b^2 c e \left (c \left (180 d^2+75 d e x+14 e^2 x^2\right )-115 a e^2\right )+8 b c^2 \left (6 c \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )-a e^2 (195 d+29 e x)\right )-105 b^4 e^3+10 b^3 c e^2 (45 d+7 e x)\right )+15 \left (b^2-4 a c\right ) (b e-2 c d) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3840 c^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*Sqrt[a + b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.016, size = 795, normalized size = 3.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+b*x+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.294337, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{3} \sqrt{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.220314, size = 513, normalized size = 2.07 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x e^{3} + \frac{30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac{240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3} + 16 \, a c^{3} e^{3}}{c^{4}}\right )} x + \frac{480 \, c^{4} d^{3} + 240 \, b c^{3} d^{2} e - 150 \, b^{2} c^{2} d e^{2} + 360 \, a c^{3} d e^{2} + 35 \, b^{3} c e^{3} - 116 \, a b c^{2} e^{3}}{c^{4}}\right )} x + \frac{480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 1920 \, a c^{3} d^{2} e + 450 \, b^{3} c d e^{2} - 1560 \, a b c^{2} d e^{2} - 105 \, b^{4} e^{3} + 460 \, a b^{2} c e^{3} - 256 \, a^{2} c^{2} e^{3}}{c^{4}}\right )} + \frac{{\left (32 \, b^{2} c^{3} d^{3} - 128 \, a c^{4} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 192 \, a b c^{3} d^{2} e + 30 \, b^{4} c d e^{2} - 144 \, a b^{2} c^{2} d e^{2} + 96 \, a^{2} c^{3} d e^{2} - 7 \, b^{5} e^{3} + 40 \, a b^{3} c e^{3} - 48 \, a^{2} b c^{2} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="giac")
[Out]