3.1551 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{d+e x} \, dx\)

Optimal. Leaf size=199 \[ \frac{\left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 \sqrt{c} e^3}-\frac{(2 c d-b e) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3}-\frac{\sqrt{a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2} \]

[Out]

-((4*c*d - 3*b*e - 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(2*e^2) + ((8*c^2*d^2 + b^2*e
^2 - 4*c*e*(2*b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]
)/(4*Sqrt[c]*e^3) - ((2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(b*d - 2*
a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e
^3

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Rubi [A]  time = 0.599412, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 \sqrt{c} e^3}-\frac{(2 c d-b e) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3}-\frac{\sqrt{a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{2 e^2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x),x]

[Out]

-((4*c*d - 3*b*e - 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(2*e^2) + ((8*c^2*d^2 + b^2*e
^2 - 4*c*e*(2*b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]
)/(4*Sqrt[c]*e^3) - ((2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(b*d - 2*
a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e
^3

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Rubi in Sympy [A]  time = 79.8534, size = 190, normalized size = 0.95 \[ \frac{\sqrt{a + b x + c x^{2}} \left (3 b e - 4 c d + 2 c e x\right )}{2 e^{2}} - \frac{\left (b e - 2 c d\right ) \sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e^{3}} + \frac{\left (4 a c e^{2} + b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{4 \sqrt{c} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)

[Out]

sqrt(a + b*x + c*x**2)*(3*b*e - 4*c*d + 2*c*e*x)/(2*e**2) - (b*e - 2*c*d)*sqrt(a
*e**2 - b*d*e + c*d**2)*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b*x +
c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2)))/e**3 + (4*a*c*e**2 + b**2*e**2 - 8*b*c*d
*e + 8*c**2*d**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(4*sqrt(
c)*e**3)

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Mathematica [A]  time = 1.13827, size = 220, normalized size = 1.11 \[ \frac{\frac{\left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}+4 (b e-2 c d) \log (d+e x) \sqrt{e (a e-b d)+c d^2}+4 (2 c d-b e) \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+2 e \sqrt{a+x (b+c x)} (3 b e-4 c d+2 c e x)}{4 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x),x]

[Out]

(2*e*(-4*c*d + 3*b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)] + 4*(-2*c*d + b*e)*Sqrt[c*
d^2 + e*(-(b*d) + a*e)]*Log[d + e*x] + ((8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a
*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sqrt[c] + 4*(2*c*d - b*e)
*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*
d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(4*e^3)

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Maple [B]  time = 0.014, size = 1302, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d),x)

[Out]

1/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-2/e^2*(c
*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d+1/2/e*ln((1/
2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2))/c^(1/2)*b^2-2/e^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)
+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*b*d+
2/e^3*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(3/2)*d^2-1/e/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+
x))*a*b+2/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e
-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*c*d+1/e^2/((a*e^2-b*d*e+c*d^2)
/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2))/(d/e+x))*b^2*d-3/e^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d
^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*
e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b*d^2*c+2/e^4/((a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*
((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+
c*d^2)/e^2)^(1/2))/(d/e+x))*c^2*d^3+c/e*(c*x^2+b*x+a)^(1/2)*x+1/2/e*(c*x^2+b*x+a
)^(1/2)*b+c^(1/2)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/4/c^(1/2)/e*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2

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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)*(2*c*x + b)/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.5469, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d),x, algorithm="fricas")

[Out]

[-1/8*(4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d - b*e)*sqrt(c)*log((8*a*b*d*e - 8*a^
2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*
sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x
) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)
) - 4*(2*c*e^2*x - 4*c*d*e + 3*b*e^2)*sqrt(c*x^2 + b*x + a)*sqrt(c) - (8*c^2*d^2
- 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) -
(8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/(sqrt(c)*e^3), 1/8*(8*sqrt(-c*d^2
+ b*d*e - a*e^2)*(2*c*d - b*e)*sqrt(c)*arctan(-1/2*(b*d - 2*a*e + (2*c*d - b*e)
*x)/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a))) + 4*(2*c*e^2*x - 4*c*d
*e + 3*b*e^2)*sqrt(c*x^2 + b*x + a)*sqrt(c) + (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*
a*c)*e^2)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x +
b^2 + 4*a*c)*sqrt(c)))/(sqrt(c)*e^3), -1/4*(2*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d
- b*e)*sqrt(-c)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8
*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b
*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*
a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(2*c*e^2*x - 4*c*d*e + 3*b*e^2)*sqrt
(c*x^2 + b*x + a)*sqrt(-c) - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*arctan(
1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*e^3), 1/4*(4*sqrt
(-c*d^2 + b*d*e - a*e^2)*(2*c*d - b*e)*sqrt(-c)*arctan(-1/2*(b*d - 2*a*e + (2*c*
d - b*e)*x)/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a))) + 2*(2*c*e^2*x
- 4*c*d*e + 3*b*e^2)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + (8*c^2*d^2 - 8*b*c*d*e +
(b^2 + 4*a*c)*e^2)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(
sqrt(-c)*e^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)

[Out]

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError