3.1279 \(\int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}} \]

[Out]

(2*Sqrt[d]*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a
*c)^(1/4) - (2*Sqrt[d]*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
])/(b^2 - 4*a*c)^(1/4)

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Rubi [A]  time = 0.198029, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d]*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a
*c)^(1/4) - (2*Sqrt[d]*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
])/(b^2 - 4*a*c)^(1/4)

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Rubi in Sympy [A]  time = 45.2641, size = 100, normalized size = 0.99 \[ \frac{2 \sqrt{d} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt [4]{- 4 a c + b^{2}}} - \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt [4]{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(-4*a*c + b
**2)**(1/4) - 2*sqrt(d)*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4
)))/(-4*a*c + b**2)**(1/4)

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Mathematica [A]  time = 0.0551923, size = 87, normalized size = 0.86 \[ \frac{2 \sqrt{d (b+2 c x)} \left (\tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-\tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{\sqrt [4]{b^2-4 a c} \sqrt{b+2 c x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d*(b + 2*c*x)]*(ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - ArcTanh[Sq
rt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/((b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])

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Maple [B]  time = 0.01, size = 271, normalized size = 2.7 \[{\frac{d\sqrt{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{d\sqrt{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{d\sqrt{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227589, size = 298, normalized size = 2.95 \[ 4 \, \left (\frac{d^{2}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b^{2} - 4 \, a c\right )} \left (\frac{d^{2}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}}{\sqrt{2 \, c d x + b d} d + \sqrt{2 \, c d^{3} x + b d^{3} +{\left (b^{2} - 4 \, a c\right )} d^{2} \sqrt{\frac{d^{2}}{b^{2} - 4 \, a c}}}}\right ) - \left (\frac{d^{2}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}} \log \left ({\left (b^{2} - 4 \, a c\right )} \left (\frac{d^{2}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}} + \sqrt{2 \, c d x + b d} d\right ) + \left (\frac{d^{2}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}} \log \left (-{\left (b^{2} - 4 \, a c\right )} \left (\frac{d^{2}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}} + \sqrt{2 \, c d x + b d} d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(d^2/(b^2 - 4*a*c))^(1/4)*arctan(-(b^2 - 4*a*c)*(d^2/(b^2 - 4*a*c))^(3/4)/(sqr
t(2*c*d*x + b*d)*d + sqrt(2*c*d^3*x + b*d^3 + (b^2 - 4*a*c)*d^2*sqrt(d^2/(b^2 -
4*a*c))))) - (d^2/(b^2 - 4*a*c))^(1/4)*log((b^2 - 4*a*c)*(d^2/(b^2 - 4*a*c))^(3/
4) + sqrt(2*c*d*x + b*d)*d) + (d^2/(b^2 - 4*a*c))^(1/4)*log(-(b^2 - 4*a*c)*(d^2/
(b^2 - 4*a*c))^(3/4) + sqrt(2*c*d*x + b*d)*d)

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Sympy [A]  time = 8.87848, size = 65, normalized size = 0.64 \[ 4 d \operatorname{RootSum}{\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left ( t \mapsto t \log{\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt{b d + 2 c d x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*d*RootSum(_t**4*(1024*a*c*d**2 - 256*b**2*d**2) + 1, Lambda(_t, _t*log(256*_t*
*3*a*c*d**2 - 64*_t**3*b**2*d**2 + sqrt(b*d + 2*c*d*x))))

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GIAC/XCAS [A]  time = 0.242139, size = 531, normalized size = 5.26 \[ -\frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{2} d - 4 \, a c d} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{2} d - 4 \, a c d} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{2} d - 4 \, \sqrt{2} a c d} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{2} d - 4 \, \sqrt{2} a c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*
a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(b^2*d - 4
*a*c*d) - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^
2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/
(b^2*d - 4*a*c*d) + (-b^2*d^2 + 4*a*c*d^2)^(3/4)*ln(2*c*d*x + b*d + sqrt(2)*(-b^
2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt
(2)*b^2*d - 4*sqrt(2)*a*c*d) - (-b^2*d^2 + 4*a*c*d^2)^(3/4)*ln(2*c*d*x + b*d - s
qrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*
d^2))/(sqrt(2)*b^2*d - 4*sqrt(2)*a*c*d)