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

Optimal. Leaf size=288 \[ -\frac{b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}+\frac{\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}+\frac{x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]

[Out]

((35*b^4 - 216*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(3840*c^3) -
(b*(105*b^4 - 760*a*b^2*c + 1296*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(7680*c^4
*x) - (x*(b*(7*b^2 + 12*a*c) + 6*c*(7*b^2 - 20*a*c)*x)*Sqrt[a*x^2 + b*x^3 + c*x^
4])/(960*c^2) + ((3*b + 10*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(60*c*x) + ((b^2
- 4*a*c)^2*(7*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2)*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.828588, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}+\frac{\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}+\frac{x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]

Antiderivative was successfully verified.

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

[Out]

((35*b^4 - 216*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(3840*c^3) -
(b*(105*b^4 - 760*a*b^2*c + 1296*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(7680*c^4
*x) - (x*(b*(7*b^2 + 12*a*c) + 6*c*(7*b^2 - 20*a*c)*x)*Sqrt[a*x^2 + b*x^3 + c*x^
4])/(960*c^2) + ((3*b + 10*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(60*c*x) + ((b^2
- 4*a*c)^2*(7*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2)*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi in Sympy [A]  time = 107.947, size = 274, normalized size = 0.95 \[ - \frac{b \sqrt{a x^{2} + b x^{3} + c x^{4}} \left (1296 a^{2} c^{2} - 760 a b^{2} c + 105 b^{4}\right )}{7680 c^{4} x} + \frac{\left (\frac{3 b}{2} + 5 c x\right ) \left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}}{30 c x} - \frac{x \left (\frac{b \left (12 a c + 7 b^{2}\right )}{4} + \frac{3 c x \left (- 20 a c + 7 b^{2}\right )}{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{240 c^{2}} + \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}} \left (240 a^{2} c^{2} - 216 a b^{2} c + 35 b^{4}\right )}{3840 c^{3}} + \frac{x \left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c + 7 b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{9}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

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,x)

[Out]

-b*sqrt(a*x**2 + b*x**3 + c*x**4)*(1296*a**2*c**2 - 760*a*b**2*c + 105*b**4)/(76
80*c**4*x) + (3*b/2 + 5*c*x)*(a*x**2 + b*x**3 + c*x**4)**(3/2)/(30*c*x) - x*(b*(
12*a*c + 7*b**2)/4 + 3*c*x*(-20*a*c + 7*b**2)/2)*sqrt(a*x**2 + b*x**3 + c*x**4)/
(240*c**2) + sqrt(a*x**2 + b*x**3 + c*x**4)*(240*a**2*c**2 - 216*a*b**2*c + 35*b
**4)/(3840*c**3) + x*(-4*a*c + b**2)**2*(-4*a*c + 7*b**2)*sqrt(a + b*x + c*x**2)
*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(1024*c**(9/2)*sqrt(a*x**
2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.386622, size = 211, normalized size = 0.73 \[ \frac{x \sqrt{a+x (b+c x)} \left (2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b c^2 \left (-81 a^2+18 a c x^2+104 c^2 x^4\right )+160 c^3 x \left (3 a^2+14 a c x^2+8 c^2 x^4\right )+8 b^3 c \left (95 a-7 c x^2\right )+48 b^2 c^2 x \left (c x^2-9 a\right )-105 b^5+70 b^4 c x\right )+15 \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{15360 c^{9/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^5 + 70*b^4*c*x
+ 8*b^3*c*(95*a - 7*c*x^2) + 48*b^2*c^2*x*(-9*a + c*x^2) + 160*c^3*x*(3*a^2 + 1
4*a*c*x^2 + 8*c^2*x^4) + 16*b*c^2*(-81*a^2 + 18*a*c*x^2 + 104*c^2*x^4)) + 15*(b^
2 - 4*a*c)^2*(7*b^2 - 4*a*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/
(15360*c^(9/2)*Sqrt[x^2*(a + x*(b + c*x))])

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Maple [A]  time = 0.012, size = 431, normalized size = 1.5 \[{\frac{1}{15360\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2560\,x \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}-1792\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{7/2}b-640\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}xa+1120\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}x{b}^{2}-320\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}ab+560\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}{b}^{3}-960\,\sqrt{c{x}^{2}+bx+a}{c}^{9/2}x{a}^{2}+1920\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xa{b}^{2}-420\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{4}-480\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}{a}^{2}b+960\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}a{b}^{3}-210\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{5}-960\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{3}{c}^{4}+2160\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{b}^{2}{c}^{3}-900\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{4}{c}^{2}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{6}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360*(c*x^4+b*x^3+a*x^2)^(3/2)*(2560*x*(c*x^2+b*x+a)^(5/2)*c^(9/2)-1792*(c*x^
2+b*x+a)^(5/2)*c^(7/2)*b-640*(c*x^2+b*x+a)^(3/2)*c^(9/2)*x*a+1120*(c*x^2+b*x+a)^
(3/2)*c^(7/2)*x*b^2-320*(c*x^2+b*x+a)^(3/2)*c^(7/2)*a*b+560*(c*x^2+b*x+a)^(3/2)*
c^(5/2)*b^3-960*(c*x^2+b*x+a)^(1/2)*c^(9/2)*x*a^2+1920*(c*x^2+b*x+a)^(1/2)*c^(7/
2)*x*a*b^2-420*(c*x^2+b*x+a)^(1/2)*c^(5/2)*x*b^4-480*(c*x^2+b*x+a)^(1/2)*c^(7/2)
*a^2*b+960*(c*x^2+b*x+a)^(1/2)*c^(5/2)*a*b^3-210*(c*x^2+b*x+a)^(1/2)*c^(3/2)*b^5
-960*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^3*c^4+2160*ln(1/2
*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^2*b^2*c^3-900*ln(1/2*(2*(c*x
^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a*b^4*c^2+105*ln(1/2*(2*(c*x^2+b*x+a)^
(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^6*c)/x^3/(c*x^2+b*x+a)^(3/2)/c^(11/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310667, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{c} x \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} -{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{30720 \, c^{5} x}, -\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - 2 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{15360 \, c^{5} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30720*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c)*x*log(
(4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x + b*c) - (8*c^2*x^3 + 8*b*c*x^2 + (b^2 +
4*a*c)*x)*sqrt(c))/x) - 4*(1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b^5*c + 760*a*b^
3*c^2 - 1296*a^2*b*c^3 + 16*(3*b^2*c^4 + 140*a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*
c^4)*x^2 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x)*sqrt(c*x^4 + b*x^3 +
a*x^2))/(c^5*x), -1/15360*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3
)*sqrt(-c)*x*arctan(1/2*(2*c*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2)*c)
) - 2*(1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^
3 + 16*(3*b^2*c^4 + 140*a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(35*b^4*
c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.319411, size = 493, normalized size = 1.71 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x{\rm sign}\left (x\right ) + 13 \, b{\rm sign}\left (x\right )\right )} x + \frac{3 \, b^{2} c^{4}{\rm sign}\left (x\right ) + 140 \, a c^{5}{\rm sign}\left (x\right )}{c^{5}}\right )} x - \frac{7 \, b^{3} c^{3}{\rm sign}\left (x\right ) - 36 \, a b c^{4}{\rm sign}\left (x\right )}{c^{5}}\right )} x + \frac{35 \, b^{4} c^{2}{\rm sign}\left (x\right ) - 216 \, a b^{2} c^{3}{\rm sign}\left (x\right ) + 240 \, a^{2} c^{4}{\rm sign}\left (x\right )}{c^{5}}\right )} x - \frac{105 \, b^{5} c{\rm sign}\left (x\right ) - 760 \, a b^{3} c^{2}{\rm sign}\left (x\right ) + 1296 \, a^{2} b c^{3}{\rm sign}\left (x\right )}{c^{5}}\right )} - \frac{{\left (7 \, b^{6}{\rm sign}\left (x\right ) - 60 \, a b^{4} c{\rm sign}\left (x\right ) + 144 \, a^{2} b^{2} c^{2}{\rm sign}\left (x\right ) - 64 \, a^{3} c^{3}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} + \frac{{\left (105 \, b^{6}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 900 \, a b^{4} c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2160 \, a^{2} b^{2} c^{2}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 960 \, a^{3} c^{3}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 210 \, \sqrt{a} b^{5} \sqrt{c} - 1520 \, a^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} + 2592 \, a^{\frac{5}{2}} b c^{\frac{5}{2}}\right )}{\rm sign}\left (x\right )}{15360 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*sign(x) + 13*b*sign(x))*x + (3*
b^2*c^4*sign(x) + 140*a*c^5*sign(x))/c^5)*x - (7*b^3*c^3*sign(x) - 36*a*b*c^4*si
gn(x))/c^5)*x + (35*b^4*c^2*sign(x) - 216*a*b^2*c^3*sign(x) + 240*a^2*c^4*sign(x
))/c^5)*x - (105*b^5*c*sign(x) - 760*a*b^3*c^2*sign(x) + 1296*a^2*b*c^3*sign(x))
/c^5) - 1/1024*(7*b^6*sign(x) - 60*a*b^4*c*sign(x) + 144*a^2*b^2*c^2*sign(x) - 6
4*a^3*c^3*sign(x))*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c
^(9/2) + 1/15360*(105*b^6*ln(abs(-b + 2*sqrt(a)*sqrt(c))) - 900*a*b^4*c*ln(abs(-
b + 2*sqrt(a)*sqrt(c))) + 2160*a^2*b^2*c^2*ln(abs(-b + 2*sqrt(a)*sqrt(c))) - 960
*a^3*c^3*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 210*sqrt(a)*b^5*sqrt(c) - 1520*a^(3/2
)*b^3*c^(3/2) + 2592*a^(5/2)*b*c^(5/2))*sign(x)/c^(9/2)