3.1231 \(\int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=102 \[ 6 \sqrt{c} d^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d^4*(b + 2*c*x)^3)/Sqrt[a + b*x + c*x^2] + 12*c*d^4*(b + 2*c*x)*Sqrt[a + b*x
+ c*x^2] + 6*Sqrt[c]*(b^2 - 4*a*c)*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])]

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Rubi [A]  time = 0.159546, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ 6 \sqrt{c} d^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+12 c d^4 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{2 d^4 (b+2 c x)^3}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*d^4*(b + 2*c*x)^3)/Sqrt[a + b*x + c*x^2] + 12*c*d^4*(b + 2*c*x)*Sqrt[a + b*x
+ c*x^2] + 6*Sqrt[c]*(b^2 - 4*a*c)*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])]

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Rubi in Sympy [A]  time = 31.6992, size = 100, normalized size = 0.98 \[ 6 \sqrt{c} d^{4} \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} + 12 c d^{4} \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} - \frac{2 d^{4} \left (b + 2 c x\right )^{3}}{\sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

6*sqrt(c)*d**4*(-4*a*c + b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**
2))) + 12*c*d**4*(b + 2*c*x)*sqrt(a + b*x + c*x**2) - 2*d**4*(b + 2*c*x)**3/sqrt
(a + b*x + c*x**2)

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Mathematica [A]  time = 0.258188, size = 89, normalized size = 0.87 \[ d^4 \left (6 \sqrt{c} \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-\frac{2 (b+2 c x) \left (-2 c \left (3 a+c x^2\right )+b^2-2 b c x\right )}{\sqrt{a+x (b+c x)}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.021, size = 340, normalized size = 3.3 \[ -6\,{\frac{c{d}^{4}{b}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-3\,{\frac{{d}^{4}{b}^{5}}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+8\,{\frac{{d}^{4}{c}^{3}{x}^{3}}{\sqrt{c{x}^{2}+bx+a}}}+12\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{\sqrt{c{x}^{2}+bx+a}}}-6\,{\frac{{d}^{4}x{b}^{2}c}{\sqrt{c{x}^{2}+bx+a}}}-5\,{\frac{{d}^{4}{b}^{3}}{\sqrt{c{x}^{2}+bx+a}}}+6\,{d}^{4}\sqrt{c}{b}^{2}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) +12\,{\frac{c{d}^{4}ba}{\sqrt{c{x}^{2}+bx+a}}}+24\,{\frac{{c}^{2}{d}^{4}{b}^{2}ax}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+12\,{\frac{c{d}^{4}{b}^{3}a}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+24\,{\frac{{d}^{4}a{c}^{2}x}{\sqrt{c{x}^{2}+bx+a}}}-24\,{d}^{4}{c}^{3/2}a\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x)

[Out]

-6*d^4*c*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3*d^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+
a)^(1/2)+8*d^4*c^3*x^3/(c*x^2+b*x+a)^(1/2)+12*d^4*c^2*b*x^2/(c*x^2+b*x+a)^(1/2)-
6*d^4*c*b^2*x/(c*x^2+b*x+a)^(1/2)-5*d^4*b^3/(c*x^2+b*x+a)^(1/2)+6*d^4*c^(1/2)*b^
2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+12*d^4*c*b*a/(c*x^2+b*x+a)^(1/2)+2
4*d^4*c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+12*d^4*c*b^3*a/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)+24*d^4*c^2*a*x/(c*x^2+b*x+a)^(1/2)-24*d^4*c^(3/2)*a*ln((1/2*b+c
*x)/c^(1/2)+(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*d*x + b*d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.339035, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} +{\left (b^{3} - 4 \, a b c\right )} d^{4} x +{\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 2 \,{\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x -{\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{c x^{2} + b x + a}, \frac{2 \,{\left (3 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} +{\left (b^{3} - 4 \, a b c\right )} d^{4} x +{\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt{-c} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) +{\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x -{\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{4} \left (\int \frac{b^{4}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{16 c^{4} x^{4}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{32 b c^{3} x^{3}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{24 b^{2} c^{2} x^{2}}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx + \int \frac{8 b^{3} c x}{a \sqrt{a + b x + c x^{2}} + b x \sqrt{a + b x + c x^{2}} + c x^{2} \sqrt{a + b x + c x^{2}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

d**4*(Integral(b**4/(a*sqrt(a + b*x + c*x**2) + b*x*sqrt(a + b*x + c*x**2) + c*x
**2*sqrt(a + b*x + c*x**2)), x) + Integral(16*c**4*x**4/(a*sqrt(a + b*x + c*x**2
) + b*x*sqrt(a + b*x + c*x**2) + c*x**2*sqrt(a + b*x + c*x**2)), x) + Integral(3
2*b*c**3*x**3/(a*sqrt(a + b*x + c*x**2) + b*x*sqrt(a + b*x + c*x**2) + c*x**2*sq
rt(a + b*x + c*x**2)), x) + Integral(24*b**2*c**2*x**2/(a*sqrt(a + b*x + c*x**2)
+ b*x*sqrt(a + b*x + c*x**2) + c*x**2*sqrt(a + b*x + c*x**2)), x) + Integral(8*
b**3*c*x/(a*sqrt(a + b*x + c*x**2) + b*x*sqrt(a + b*x + c*x**2) + c*x**2*sqrt(a
+ b*x + c*x**2)), x))

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GIAC/XCAS [A]  time = 0.238048, size = 336, normalized size = 3.29 \[ -\frac{6 \,{\left (b^{2} c d^{4} - 4 \, a c^{2} d^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} + \frac{2 \,{\left (2 \,{\left ({\left (\frac{2 \,{\left (b^{2} c^{5} d^{4} - 4 \, a c^{6} d^{4}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac{3 \,{\left (b^{3} c^{4} d^{4} - 4 \, a b c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac{6 \,{\left (a b^{2} c^{4} d^{4} - 4 \, a^{2} c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac{b^{5} c^{2} d^{4} - 10 \, a b^{3} c^{3} d^{4} + 24 \, a^{2} b c^{4} d^{4}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

-6*(b^2*c*d^4 - 4*a*c^2*d^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(
c) - b))/sqrt(c) + 2*(2*((2*(b^2*c^5*d^4 - 4*a*c^6*d^4)*x/(b^2*c^2 - 4*a*c^3) +
3*(b^3*c^4*d^4 - 4*a*b*c^5*d^4)/(b^2*c^2 - 4*a*c^3))*x + 6*(a*b^2*c^4*d^4 - 4*a^
2*c^5*d^4)/(b^2*c^2 - 4*a*c^3))*x - (b^5*c^2*d^4 - 10*a*b^3*c^3*d^4 + 24*a^2*b*c
^4*d^4)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a)