3.1278 \(\int \frac{(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=117 \[ -2 d^{3/2} \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{3/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+4 d \sqrt{b d+2 c d x} \]

[Out]

4*d*Sqrt[b*d + 2*c*d*x] - 2*(b^2 - 4*a*c)^(1/4)*d^(3/2)*ArcTan[Sqrt[d*(b + 2*c*x
)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^2 - 4*a*c)^(1/4)*d^(3/2)*ArcTanh[Sqrt[d
*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.250131, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -2 d^{3/2} \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{3/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+4 d \sqrt{b d+2 c d x} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2),x]

[Out]

4*d*Sqrt[b*d + 2*c*d*x] - 2*(b^2 - 4*a*c)^(1/4)*d^(3/2)*ArcTan[Sqrt[d*(b + 2*c*x
)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^2 - 4*a*c)^(1/4)*d^(3/2)*ArcTanh[Sqrt[d
*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi in Sympy [A]  time = 60.7493, size = 117, normalized size = 1. \[ - 2 d^{\frac{3}{2}} \sqrt [4]{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 2 d^{\frac{3}{2}} \sqrt [4]{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 4 d \sqrt{b d + 2 c d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

-2*d**(3/2)*(-4*a*c + b**2)**(1/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b
**2)**(1/4))) - 2*d**(3/2)*(-4*a*c + b**2)**(1/4)*atanh(sqrt(b*d + 2*c*d*x)/(sqr
t(d)*(-4*a*c + b**2)**(1/4))) + 4*d*sqrt(b*d + 2*c*d*x)

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Mathematica [A]  time = 0.110293, size = 111, normalized size = 0.95 \[ -\frac{2 (d (b+2 c x))^{3/2} \left (\sqrt [4]{b^2-4 a c} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+\sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-2 \sqrt{b+2 c x}\right )}{(b+2 c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2),x]

[Out]

(-2*(d*(b + 2*c*x))^(3/2)*(-2*Sqrt[b + 2*c*x] + (b^2 - 4*a*c)^(1/4)*ArcTan[Sqrt[
b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] + (b^2 - 4*a*c)^(1/4)*ArcTanh[Sqrt[b + 2*c*x]/(b
^2 - 4*a*c)^(1/4)]))/(b + 2*c*x)^(3/2)

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Maple [B]  time = 0.01, size = 582, normalized size = 5. \[ 4\,d\sqrt{2\,cdx+bd}-4\,{\frac{{d}^{3}\sqrt{2}ac}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\arctan \left ({\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }+{{d}^{3}\sqrt{2}{b}^{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+4\,{\frac{{d}^{3}\sqrt{2}ac}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\arctan \left ( -{\frac{\sqrt{2}\sqrt{2\,cdx+bd}}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}+1 \right ) }-{{d}^{3}\sqrt{2}{b}^{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-2\,{\frac{{d}^{3}\sqrt{2}ac}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{3/4}}\ln \left ({\frac{2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}{2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}} \right ) }+{\frac{{d}^{3}\sqrt{2}{b}^{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a),x)

[Out]

4*d*(2*c*d*x+b*d)^(1/2)-4*d^3/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*a*c+d^3/(4*a*c*d^2-b^2*d^2)^(3/4
)*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)*b^2+4*
d^3/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*a*c-d^3/(4*a*c*d^2-b^2*d^2)^(3/4)*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)*b^2-2*d^3/(4*a*c*d^2-b^2*d^2)^
(3/4)*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)))*a*c+1/2*d^3/(4*a*c*d^2-b^2*d^2)^(3
/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)))*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((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230631, size = 247, normalized size = 2.11 \[ 4 \, \sqrt{2 \, c d x + b d} d + 4 \, \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac{1}{4}} \arctan \left (\frac{\left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac{1}{4}}}{\sqrt{2 \, c d x + b d} d + \sqrt{2 \, c d^{3} x + b d^{3} + \sqrt{{\left (b^{2} - 4 \, a c\right )} d^{6}}}}\right ) - \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac{1}{4}} \log \left (\sqrt{2 \, c d x + b d} d + \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac{1}{4}}\right ) + \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac{1}{4}} \log \left (\sqrt{2 \, c d x + b d} d - \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac{1}{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

4*sqrt(2*c*d*x + b*d)*d + 4*((b^2 - 4*a*c)*d^6)^(1/4)*arctan(((b^2 - 4*a*c)*d^6)
^(1/4)/(sqrt(2*c*d*x + b*d)*d + sqrt(2*c*d^3*x + b*d^3 + sqrt((b^2 - 4*a*c)*d^6)
))) - ((b^2 - 4*a*c)*d^6)^(1/4)*log(sqrt(2*c*d*x + b*d)*d + ((b^2 - 4*a*c)*d^6)^
(1/4)) + ((b^2 - 4*a*c)*d^6)^(1/4)*log(sqrt(2*c*d*x + b*d)*d - ((b^2 - 4*a*c)*d^
6)^(1/4))

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Sympy [A]  time = 53.5724, size = 212, normalized size = 1.81 \[ - 16 a c d^{3} \operatorname{RootSum}{\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left ( t \mapsto t \log{\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt{b d + 2 c d x} \right )} \right )\right )} + 4 b^{2} d^{3} \operatorname{RootSum}{\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left ( t \mapsto t \log{\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt{b d + 2 c d x} \right )} \right )\right )} + 4 d \sqrt{b d + 2 c d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

-16*a*c*d**3*RootSum(_t**4*(16384*a**3*c**3*d**6 - 12288*a**2*b**2*c**2*d**6 + 3
072*a*b**4*c*d**6 - 256*b**6*d**6) + 1, Lambda(_t, _t*log(16*_t*a*c*d**2 - 4*_t*
b**2*d**2 + sqrt(b*d + 2*c*d*x)))) + 4*b**2*d**3*RootSum(_t**4*(16384*a**3*c**3*
d**6 - 12288*a**2*b**2*c**2*d**6 + 3072*a*b**4*c*d**6 - 256*b**6*d**6) + 1, Lamb
da(_t, _t*log(16*_t*a*c*d**2 - 4*_t*b**2*d**2 + sqrt(b*d + 2*c*d*x)))) + 4*d*sqr
t(b*d + 2*c*d*x)

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GIAC/XCAS [A]  time = 0.240918, size = 478, normalized size = 4.09 \[ -\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \frac{1}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} d{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{1}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} d{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 4 \, \sqrt{2 \, c d x + b d} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

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