3.1223 \(\int \frac{(b d+2 c d x)^4}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{3}{4} d^4 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}+\frac{3 d^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.174171, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{3}{4} d^4 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}+\frac{3 d^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{2} d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.597, size = 112, normalized size = 0.96 \[ \frac{d^{4} \left (b + 2 c x\right )^{3} \sqrt{a + b x + c x^{2}}}{2} + \frac{3 d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{4} + \frac{3 d^{4} \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{c}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.151388, size = 98, normalized size = 0.84 \[ d^4 \left (\frac{1}{4} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (2 c x^2-3 a\right )+5 b^2+8 b c x\right )+\frac{3 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{8 \sqrt{c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^4*(((b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(5*b^2 + 8*b*c*x + 4*c*(-3*a + 2*c*x^2))
)/4 + (3*(b^2 - 4*a*c)^2*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*Sq
rt[c]))

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Maple [B]  time = 0.021, size = 242, normalized size = 2.1 \[{\frac{3\,{d}^{4}{b}^{4}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+4\,{d}^{4}{c}^{3}{x}^{3}\sqrt{c{x}^{2}+bx+a}+6\,{d}^{4}{c}^{2}b{x}^{2}\sqrt{c{x}^{2}+bx+a}+{\frac{9\,{d}^{4}x{b}^{2}c}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{d}^{4}{b}^{3}}{4}\sqrt{c{x}^{2}+bx+a}}-3\,{d}^{4}\sqrt{c}{b}^{2}a\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) -3\,{d}^{4}cba\sqrt{c{x}^{2}+bx+a}-6\,{d}^{4}{c}^{2}ax\sqrt{c{x}^{2}+bx+a}+6\,{d}^{4}{c}^{3/2}{a}^{2}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) \]

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)^(1/2),x)

[Out]

3/8*d^4*b^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+4*d^4*c^3*x^3*(c
*x^2+b*x+a)^(1/2)+6*d^4*c^2*b*x^2*(c*x^2+b*x+a)^(1/2)+9/2*d^4*c*b^2*x*(c*x^2+b*x
+a)^(1/2)+5/4*d^4*b^3*(c*x^2+b*x+a)^(1/2)-3*d^4*c^(1/2)*b^2*a*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))-3*d^4*c*b*a*(c*x^2+b*x+a)^(1/2)-6*d^4*c^2*a*x*(c*x^2+b
*x+a)^(1/2)+6*d^4*c^(3/2)*a^2*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/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300925, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{4} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \,{\left (16 \, c^{3} d^{4} x^{3} + 24 \, b c^{2} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c - 4 \, a c^{2}\right )} d^{4} x +{\left (5 \, b^{3} - 12 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a} \sqrt{c}}{16 \, \sqrt{c}}, \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{4} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + 2 \,{\left (16 \, c^{3} d^{4} x^{3} + 24 \, b c^{2} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c - 4 \, a c^{2}\right )} d^{4} x +{\left (5 \, b^{3} - 12 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c}}{8 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^4*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)) + 4*(16*c^3*d^4*x^3 + 24
*b*c^2*d^4*x^2 + 6*(3*b^2*c - 4*a*c^2)*d^4*x + (5*b^3 - 12*a*b*c)*d^4)*sqrt(c*x^
2 + b*x + a)*sqrt(c))/sqrt(c), 1/8*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^4*arctan(
1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) + 2*(16*c^3*d^4*x^3 + 24*b*c
^2*d^4*x^2 + 6*(3*b^2*c - 4*a*c^2)*d^4*x + (5*b^3 - 12*a*b*c)*d^4)*sqrt(c*x^2 +
b*x + a)*sqrt(-c))/sqrt(-c)]

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

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)**(1/2),x)

[Out]

d**4*(Integral(b**4/sqrt(a + b*x + c*x**2), x) + Integral(16*c**4*x**4/sqrt(a +
b*x + c*x**2), x) + Integral(32*b*c**3*x**3/sqrt(a + b*x + c*x**2), x) + Integra
l(24*b**2*c**2*x**2/sqrt(a + b*x + c*x**2), x) + Integral(8*b**3*c*x/sqrt(a + b*
x + c*x**2), x))

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GIAC/XCAS [A]  time = 0.242857, size = 215, normalized size = 1.84 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c^{3} d^{4} x + 3 \, b c^{2} d^{4}\right )} x + \frac{3 \,{\left (3 \, b^{2} c^{4} d^{4} - 4 \, a c^{5} d^{4}\right )}}{c^{3}}\right )} x + \frac{5 \, b^{3} c^{3} d^{4} - 12 \, a b c^{4} d^{4}}{c^{3}}\right )} - \frac{3 \,{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x + a)*(2*(4*(2*c^3*d^4*x + 3*b*c^2*d^4)*x + 3*(3*b^2*c^4*d^4
- 4*a*c^5*d^4)/c^3)*x + (5*b^3*c^3*d^4 - 12*a*b*c^4*d^4)/c^3) - 3/8*(b^4*d^4 -
8*a*b^2*c*d^4 + 16*a^2*c^2*d^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sq
rt(c) - b))/sqrt(c)