3.1290 \(\int \frac{(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=131 \[ -\frac{2 c d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{2 c d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{d \sqrt{b d+2 c d x}}{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.245908, antiderivative size = 131, 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 \[ -\frac{2 c d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{2 c d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{d \sqrt{b d+2 c d x}}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.5983, size = 131, normalized size = 1. \[ - \frac{2 c d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{4}}} - \frac{2 c d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{4}}} - \frac{d \sqrt{b d + 2 c d x}}{a + b x + c x^{2}} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.210752, size = 143, normalized size = 1.09 \[ -\frac{(d (b+2 c x))^{3/2} \left (\left (b^2-4 a c\right )^{3/4} \sqrt{b+2 c x}+2 c (a+x (b+c x)) \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+2 c (a+x (b+c x)) \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{\left (b^2-4 a c\right )^{3/4} (b+2 c x)^{3/2} (a+x (b+c x))} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 326, normalized size = 2.5 \[ -4\,{\frac{c{d}^{3}\sqrt{2\,cdx+bd}}{4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2}}}+{\frac{c{d}^{3}\sqrt{2}}{2}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{c{d}^{3}\sqrt{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{c{d}^{3}\sqrt{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \]

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)^2,x)

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231815, size = 633, normalized size = 4.83 \[ -\frac{4 \, \left (\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \arctan \left (-\frac{\left (\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}}{\sqrt{2 \, c d x + b d} c d + \sqrt{2 \, c^{3} d^{3} x + b c^{2} d^{3} + \sqrt{\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}}}\right ) + \left (\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \log \left (\sqrt{2 \, c d x + b d} c d + \left (\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) - \left (\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c x^{2} + b x + a\right )} \log \left (\sqrt{2 \, c d x + b d} c d - \left (\frac{c^{4} d^{6}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) + \sqrt{2 \, c d x + b d} d}{c x^{2} + b x + a} \]

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)^2,x, algorithm="fricas")

[Out]

-(4*(c^4*d^6/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(c*x^2 + b*
x + a)*arctan(-(c^4*d^6/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*
(b^2 - 4*a*c)/(sqrt(2*c*d*x + b*d)*c*d + sqrt(2*c^3*d^3*x + b*c^2*d^3 + sqrt(c^4
*d^6/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*(b^4 - 8*a*b^2*c + 16*a^2
*c^2)))) + (c^4*d^6/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(c*x
^2 + b*x + a)*log(sqrt(2*c*d*x + b*d)*c*d + (c^4*d^6/(b^6 - 12*a*b^4*c + 48*a^2*
b^2*c^2 - 64*a^3*c^3))^(1/4)*(b^2 - 4*a*c)) - (c^4*d^6/(b^6 - 12*a*b^4*c + 48*a^
2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(c*x^2 + b*x + a)*log(sqrt(2*c*d*x + b*d)*c*d - (
c^4*d^6/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(b^2 - 4*a*c)) +
sqrt(2*c*d*x + b*d)*d)/(c*x^2 + b*x + a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239039, size = 591, normalized size = 4.51 \[ \frac{4 \, \sqrt{2 \, c d x + b d} c d^{3}}{b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{2} - 4 \, a c} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{2} - 4 \, a c} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c d{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} \]

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)^2,x, algorithm="giac")

[Out]

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