3.47 \(\int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=197 \[ -\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}+\frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7} \]

[Out]

-((b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(32*a*x^3) + (b*(3*b^2 - 20*a*c)*S
qrt[a*x^2 + b*x^3 + c*x^4])/(64*a^2*x^2) - ((b + 6*c*x)*Sqrt[a*x^2 + b*x^3 + c*x
^4])/(8*x^4) - (a*x^2 + b*x^3 + c*x^4)^(3/2)/(4*x^7) - (3*(b^2 - 4*a*c)^2*ArcTan
h[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(5/2))

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Rubi [A]  time = 0.590535, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}+\frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8,x]

[Out]

-((b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(32*a*x^3) + (b*(3*b^2 - 20*a*c)*S
qrt[a*x^2 + b*x^3 + c*x^4])/(64*a^2*x^2) - ((b + 6*c*x)*Sqrt[a*x^2 + b*x^3 + c*x
^4])/(8*x^4) - (a*x^2 + b*x^3 + c*x^4)^(3/2)/(4*x^7) - (3*(b^2 - 4*a*c)^2*ArcTan
h[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(5/2))

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Rubi in Sympy [A]  time = 83.2664, size = 207, normalized size = 1.05 \[ - \frac{\left (b + 6 c x\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{8 x^{4}} - \frac{\left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}}{4 x^{7}} - \frac{\left (- 12 a c + b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{32 a x^{3}} + \frac{b \left (- 20 a c + 3 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{64 a^{2} x^{2}} - \frac{3 x \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{128 a^{\frac{5}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**3+a*x**2)**(3/2)/x**8,x)

[Out]

-(b + 6*c*x)*sqrt(a*x**2 + b*x**3 + c*x**4)/(8*x**4) - (a*x**2 + b*x**3 + c*x**4
)**(3/2)/(4*x**7) - (-12*a*c + b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(32*a*x**3)
+ b*(-20*a*c + 3*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(64*a**2*x**2) - 3*x*(-4*a
*c + b**2)**2*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x +
c*x**2)))/(128*a**(5/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.315644, size = 156, normalized size = 0.79 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (-2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)} \left (8 a^2+4 a x (2 b+5 c x)-3 b^2 x^2\right )+3 x^4 \log (x) \left (b^2-4 a c\right )^2-3 x^4 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )\right )}{128 a^{5/2} x^5 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8,x]

[Out]

(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)]*(8*a^
2 - 3*b^2*x^2 + 4*a*x*(2*b + 5*c*x)) + 3*(b^2 - 4*a*c)^2*x^4*Log[x] - 3*(b^2 - 4
*a*c)^2*x^4*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]]))/(128*a^(5/2)*x^5*
Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.012, size = 501, normalized size = 2.5 \[ -{\frac{1}{128\,{x}^{7}{a}^{4}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 48\,{a}^{7/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){c}^{2}{x}^{4}-24\,{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) c{x}^{4}{b}^{2}+24\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{2}{x}^{5}ab-16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{2}{x}^{4}{a}^{2}-2\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}c{x}^{5}{b}^{3}+24\,\sqrt{c{x}^{2}+bx+a}{c}^{2}{x}^{5}{a}^{2}b+3\,{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{4}-24\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}c{x}^{3}ab+20\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}c{x}^{4}a{b}^{2}-48\,\sqrt{c{x}^{2}+bx+a}{c}^{2}{x}^{4}{a}^{3}-6\,\sqrt{c{x}^{2}+bx+a}c{x}^{5}a{b}^{3}+16\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}c{x}^{2}{a}^{2}+2\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{3}{b}^{3}-2\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{4}{b}^{4}+36\,\sqrt{c{x}^{2}+bx+a}c{x}^{4}{a}^{2}{b}^{2}+4\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{2}a{b}^{2}-6\,\sqrt{c{x}^{2}+bx+a}{x}^{4}a{b}^{4}-16\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}x{a}^{2}b+32\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{a}^{3} \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^3+a*x^2)^(3/2)/x^8,x)

[Out]

-1/128*(c*x^4+b*x^3+a*x^2)^(3/2)*(48*a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)
^(1/2))/x)*c^2*x^4-24*a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*c*x^
4*b^2+24*(c*x^2+b*x+a)^(3/2)*c^2*x^5*a*b-16*(c*x^2+b*x+a)^(3/2)*c^2*x^4*a^2-2*(c
*x^2+b*x+a)^(3/2)*c*x^5*b^3+24*(c*x^2+b*x+a)^(1/2)*c^2*x^5*a^2*b+3*a^(3/2)*ln((2
*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^4*b^4-24*(c*x^2+b*x+a)^(5/2)*c*x^3*a*
b+20*(c*x^2+b*x+a)^(3/2)*c*x^4*a*b^2-48*(c*x^2+b*x+a)^(1/2)*c^2*x^4*a^3-6*(c*x^2
+b*x+a)^(1/2)*c*x^5*a*b^3+16*(c*x^2+b*x+a)^(5/2)*c*x^2*a^2+2*(c*x^2+b*x+a)^(5/2)
*x^3*b^3-2*(c*x^2+b*x+a)^(3/2)*x^4*b^4+36*(c*x^2+b*x+a)^(1/2)*c*x^4*a^2*b^2+4*(c
*x^2+b*x+a)^(5/2)*x^2*a*b^2-6*(c*x^2+b*x+a)^(1/2)*x^4*a*b^4-16*(c*x^2+b*x+a)^(5/
2)*x*a^2*b+32*(c*x^2+b*x+a)^(5/2)*a^3)/x^7/(c*x^2+b*x+a)^(3/2)/a^4

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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((c*x^4 + b*x^3 + a*x^2)^(3/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.327811, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{5} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} -{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right ) - 4 \,{\left (24 \, a^{3} b x + 16 \, a^{4} -{\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \,{\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{256 \, a^{3} x^{5}}, \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{5} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right ) - 2 \,{\left (24 \, a^{3} b x + 16 \, a^{4} -{\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \,{\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^3 + a*x^2)^(3/2)/x^8,x, algorithm="fricas")

[Out]

[1/256*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(a)*x^5*log((4*sqrt(c*x^4 + b*x^3 +
a*x^2)*(a*b*x + 2*a^2) - (8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x)*sqrt(a))/x^3
) - 4*(24*a^3*b*x + 16*a^4 - (3*a*b^3 - 20*a^2*b*c)*x^3 + 2*(a^2*b^2 + 20*a^3*c)
*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^3*x^5), 1/128*(3*(b^4 - 8*a*b^2*c + 16*a^2
*c^2)*sqrt(-a)*x^5*arctan(1/2*(b*x^2 + 2*a*x)*sqrt(-a)/(sqrt(c*x^4 + b*x^3 + a*x
^2)*a)) - 2*(24*a^3*b*x + 16*a^4 - (3*a*b^3 - 20*a^2*b*c)*x^3 + 2*(a^2*b^2 + 20*
a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^3*x^5)]

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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((c*x**4+b*x**3+a*x**2)**(3/2)/x**8,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^3 + a*x^2)^(3/2)/x^8,x, algorithm="giac")

[Out]

Timed out