3.1574 \(\int \frac{(b+2 c x) (d+e x)^2}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{6 c^2}+\frac{2}{3} (d+e x)^2 \sqrt{a+b x+c x^2} \]

[Out]

(2*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/3 + ((8*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(3*b*d
+ 4*a*e) + 2*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(6*c^2) + ((b^2 - 4*a*
c)*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(4*c^
(5/2))

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Rubi [A]  time = 0.447978, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{6 c^2}+\frac{2}{3} (d+e x)^2 \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/3 + ((8*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(3*b*d
+ 4*a*e) + 2*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(6*c^2) + ((b^2 - 4*a*
c)*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(4*c^
(5/2))

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Rubi in Sympy [A]  time = 46.2733, size = 144, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{2} \sqrt{a + b x + c x^{2}}}{3} + \frac{\sqrt{a + b x + c x^{2}} \left (- 8 a c e^{2} + 3 b^{2} e^{2} - 6 b c d e + 8 c^{2} d^{2} - 2 c e x \left (b e - 2 c d\right )\right )}{6 c^{2}} - \frac{e \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{4 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**2*sqrt(a + b*x + c*x**2)/3 + sqrt(a + b*x + c*x**2)*(-8*a*c*e**2 +
3*b**2*e**2 - 6*b*c*d*e + 8*c**2*d**2 - 2*c*e*x*(b*e - 2*c*d))/(6*c**2) - e*(-4*
a*c + b**2)*(b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/
(4*c**(5/2))

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Mathematica [A]  time = 0.173095, size = 123, normalized size = 0.83 \[ \frac{\sqrt{a+x (b+c x)} \left (-2 c e (4 a e+3 b d+b e x)+3 b^2 e^2+4 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{6 c^2}-\frac{e \left (b^2-4 a c\right ) (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{4 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(3*b^2*e^2 - 2*c*e*(3*b*d + 4*a*e + b*e*x) + 4*c^2*(3*d^2
+ 3*d*e*x + e^2*x^2)))/(6*c^2) - ((b^2 - 4*a*c)*e*(-2*c*d + b*e)*Log[b + 2*c*x
+ 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(4*c^(5/2))

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Maple [B]  time = 0.015, size = 280, normalized size = 1.9 \[ -{\frac{b{e}^{2}x}{3\,c}\sqrt{c{x}^{2}+bx+a}}+2\,dex\sqrt{c{x}^{2}+bx+a}+{\frac{{b}^{2}{e}^{2}}{2\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{bde}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{3}{e}^{2}}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}de}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{ab{e}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{aed}{\sqrt{c}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }+2\,{d}^{2}\sqrt{c{x}^{2}+bx+a}+{\frac{2\,{e}^{2}{x}^{2}}{3}\sqrt{c{x}^{2}+bx+a}}-{\frac{4\,a{e}^{2}}{3\,c}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*x/c*(c*x^2+b*x+a)^(1/2)*b*e^2+2*d*e*x*(c*x^2+b*x+a)^(1/2)+1/2*b^2/c^2*(c*x^
2+b*x+a)^(1/2)*e^2-b/c*(c*x^2+b*x+a)^(1/2)*d*e-1/4*b^3/c^(5/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*e^2+1/2*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))*d*e+a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b*e^2-2*a/c
^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e+2*d^2*(c*x^2+b*x+a)^(1/2)
+2/3*x^2*e^2*(c*x^2+b*x+a)^(1/2)-4/3*e^2/c*a*(c*x^2+b*x+a)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.424906, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (4 \, c^{2} e^{2} x^{2} + 12 \, c^{2} d^{2} - 6 \, b c d e +{\left (3 \, b^{2} - 8 \, a c\right )} e^{2} + 2 \,{\left (6 \, c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e -{\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{24 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (4 \, c^{2} e^{2} x^{2} + 12 \, c^{2} d^{2} - 6 \, b c d e +{\left (3 \, b^{2} - 8 \, a c\right )} e^{2} + 2 \,{\left (6 \, c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 3 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e -{\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{12 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(4*(4*c^2*e^2*x^2 + 12*c^2*d^2 - 6*b*c*d*e + (3*b^2 - 8*a*c)*e^2 + 2*(6*c^
2*d*e - b*c*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(2*(b^2*c - 4*a*c^2)*d*e -
(b^3 - 4*a*b*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)))/c^(5/2), 1/12*(2*(4*c^2*e^2*x^2 + 12*c^2*d^2
- 6*b*c*d*e + (3*b^2 - 8*a*c)*e^2 + 2*(6*c^2*d*e - b*c*e^2)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(-c) + 3*(2*(b^2*c - 4*a*c^2)*d*e - (b^3 - 4*a*b*c)*e^2)*arctan(1/2*(2*
c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.289336, size = 197, normalized size = 1.33 \[ \frac{1}{6} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (2 \, x e^{2} + \frac{6 \, c^{2} d e - b c e^{2}}{c^{2}}\right )} x + \frac{12 \, c^{2} d^{2} - 6 \, b c d e + 3 \, b^{2} e^{2} - 8 \, a c e^{2}}{c^{2}}\right )} - \frac{{\left (2 \, b^{2} c d e - 8 \, a c^{2} d e - b^{3} e^{2} + 4 \, a b c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(c*x^2 + b*x + a)*(2*(2*x*e^2 + (6*c^2*d*e - b*c*e^2)/c^2)*x + (12*c^2*d
^2 - 6*b*c*d*e + 3*b^2*e^2 - 8*a*c*e^2)/c^2) - 1/4*(2*b^2*c*d*e - 8*a*c^2*d*e -
b^3*e^2 + 4*a*b*c*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b
))/c^(5/2)