3.1241 \(\int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ 16 c^{3/2} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d^4*(b + 2*c*x)^3)/(3*(a + b*x + c*x^2)^(3/2)) - (8*c*d^4*(b + 2*c*x))/Sqrt[
a + b*x + c*x^2] + 16*c^(3/2)*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
c*x^2])]

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Rubi [A]  time = 0.143249, antiderivative size = 96, 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 \[ 16 c^{3/2} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d^4*(b + 2*c*x)^3)/(3*(a + b*x + c*x^2)^(3/2)) - (8*c*d^4*(b + 2*c*x))/Sqrt[
a + b*x + c*x^2] + 16*c^(3/2)*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 = 31.0807, size = 94, normalized size = 0.98 \[ 16 c^{\frac{3}{2}} d^{4} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} - \frac{8 c d^{4} \left (b + 2 c x\right )}{\sqrt{a + b x + c x^{2}}} - \frac{2 d^{4} \left (b + 2 c x\right )^{3}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.246396, size = 84, normalized size = 0.88 \[ d^4 \left (16 c^{3/2} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-\frac{2 (b+2 c x) \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )}{3 (a+x (b+c x))^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

d^4*((-2*(b + 2*c*x)*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)))/(3*(a + x*(b + c*x)
)^(3/2)) + 16*c^(3/2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])

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Maple [B]  time = 0.015, size = 531, normalized size = 5.5 \[ -8\,{\frac{{c}^{2}{d}^{4}{b}^{2}ax}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}-18\,{\frac{{d}^{4}x{b}^{2}c}{ \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}+8\,{\frac{{b}^{5}{d}^{4}c}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}-16\,{\frac{c{d}^{4}ba}{ \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}+8\,{\frac{c{d}^{4}{b}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-24\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}-{\frac{16\,{d}^{4}{c}^{3}{x}^{3}}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{5}{d}^{4}}{4\,ac-{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}-16\,{\frac{{c}^{2}{d}^{4}x}{\sqrt{c{x}^{2}+bx+a}}}+8\,{\frac{c{d}^{4}b}{\sqrt{c{x}^{2}+bx+a}}}+16\,{d}^{4}{c}^{3/2}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) +{\frac{{d}^{4}{b}^{3}}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}-64\,{\frac{{b}^{2}{d}^{4}{c}^{3}ax}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{c{d}^{4}{b}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}+16\,{\frac{{b}^{4}{d}^{4}{c}^{2}x}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{c{d}^{4}{b}^{3}a}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}-32\,{\frac{{b}^{3}{d}^{4}{c}^{2}a}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}+16\,{\frac{{c}^{2}{d}^{4}{b}^{2}x}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \]

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)^(5/2),x)

[Out]

-8*d^4*c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-18*d^4*c*b^2*x/(c*x^2+b*x+a)^
(3/2)+8*d^4*c*b^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-16*d^4*c*b*a/(c*x^2+b*x+a)^(
3/2)+8*d^4*c*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-24*d^4*c^2*b*x^2/(c*x^2+b*x+a)^
(3/2)-16/3*d^4*c^3*x^3/(c*x^2+b*x+a)^(3/2)+d^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)-16*d^4*c^2*x/(c*x^2+b*x+a)^(1/2)+8*d^4*c*b/(c*x^2+b*x+a)^(1/2)+16*d^4*c^(3/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/3*d^4*b^3/(c*x^2+b*x+a)^(3/2)-64*
d^4*c^3*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+2*d^4*c*b^4/(4*a*c-b^2)/(c*x^2
+b*x+a)^(3/2)*x+16*d^4*c^2*b^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-4*d^4*c*b^3*a
/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-32*d^4*c^2*b^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)+16*d^4*c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.358532, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (12 \,{\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} +{\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x +{\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac{2 \,{\left (24 \,{\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} +{\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) -{\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x +{\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \]

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)^(5/2),x, algorithm="fricas")

[Out]

[2/3*(12*(c^3*d^4*x^4 + 2*b*c^2*d^4*x^3 + 2*a*b*c*d^4*x + a^2*c*d^4 + (b^2*c + 2
*a*c^2)*d^4*x^2)*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)*sqrt(c) - 4*a*c) - (32*c^3*d^4*x^3 + 48*b*c^2*d^4*x^2 + 6*(3*b^2*c
+ 4*a*c^2)*d^4*x + (b^3 + 12*a*b*c)*d^4)*sqrt(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*
c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), 2/3*(24*(c^3*d^4*x^4 + 2*b*c^2*d^4*x
^3 + 2*a*b*c*d^4*x + a^2*c*d^4 + (b^2*c + 2*a*c^2)*d^4*x^2)*sqrt(-c)*arctan(1/2*
(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) - (32*c^3*d^4*x^3 + 48*b*c^2*d^4*x
^2 + 6*(3*b^2*c + 4*a*c^2)*d^4*x + (b^3 + 12*a*b*c)*d^4)*sqrt(c*x^2 + b*x + a))/
(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)]

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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)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238384, size = 458, normalized size = 4.77 \[ -\frac{8 \, d^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} - \frac{2 \,{\left (8 \,{\left (\frac{2 \,{\left (b^{4} c^{3} d^{4} - 8 \, a b^{2} c^{4} d^{4} + 16 \, a^{2} c^{5} d^{4}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (b^{5} c^{2} d^{4} - 8 \, a b^{3} c^{3} d^{4} + 16 \, a^{2} b c^{4} d^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (3 \, b^{6} c d^{4} - 20 \, a b^{4} c^{2} d^{4} + 16 \, a^{2} b^{2} c^{3} d^{4} + 64 \, a^{3} c^{4} d^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{7} d^{4} + 4 \, a b^{5} c d^{4} - 80 \, a^{2} b^{3} c^{2} d^{4} + 192 \, a^{3} b c^{3} d^{4}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

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)^(5/2),x, algorithm="giac")

[Out]

-8*d^4*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/sqrt(c) - 1/3
*(2*(8*(2*(b^4*c^3*d^4 - 8*a*b^2*c^4*d^4 + 16*a^2*c^5*d^4)*x/(b^4*c^2 - 8*a*b^2*
c^3 + 16*a^2*c^4) + 3*(b^5*c^2*d^4 - 8*a*b^3*c^3*d^4 + 16*a^2*b*c^4*d^4)/(b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 3*(3*b^6*c*d^4 - 20*a*b^4*c^2*d^4 + 16*a^2*b^
2*c^3*d^4 + 64*a^3*c^4*d^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^7*d^4 +
4*a*b^5*c*d^4 - 80*a^2*b^3*c^2*d^4 + 192*a^3*b*c^3*d^4)/(b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)