3.109 \(\int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx\)

Optimal. Leaf size=244 \[ \frac{\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{3 b \sqrt{x} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt{a x+b x^3+c x^5}}-\frac{x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \]

[Out]

((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*Sqrt[a*x + b*x^3 + c*x^5])/(1280*c^3*Sqrt[
x]) - (x^(3/2)*(b*(5*b^2 - 4*a*c) + 4*c*(5*b^2 - 16*a*c)*x^2)*Sqrt[a*x + b*x^3 +
c*x^5])/(640*c^2) + (Sqrt[x]*(3*b + 8*c*x^2)*(a*x + b*x^3 + c*x^5)^(3/2))/(80*c
) - (3*b*(b^2 - 4*a*c)^2*Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(b + 2*c*x^2)/(
2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)*Sqrt[a*x + b*x^3 + c*x^5])

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Rubi [A]  time = 0.583595, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{3 b \sqrt{x} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt{a x+b x^3+c x^5}}-\frac{x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a*x + b*x^3 + c*x^5)^(3/2),x]

[Out]

((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*Sqrt[a*x + b*x^3 + c*x^5])/(1280*c^3*Sqrt[
x]) - (x^(3/2)*(b*(5*b^2 - 4*a*c) + 4*c*(5*b^2 - 16*a*c)*x^2)*Sqrt[a*x + b*x^3 +
c*x^5])/(640*c^2) + (Sqrt[x]*(3*b + 8*c*x^2)*(a*x + b*x^3 + c*x^5)^(3/2))/(80*c
) - (3*b*(b^2 - 4*a*c)^2*Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(b + 2*c*x^2)/(
2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)*Sqrt[a*x + b*x^3 + c*x^5])

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Rubi in Sympy [A]  time = 63.7888, size = 230, normalized size = 0.94 \[ - \frac{3 b \sqrt{x} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x^{2} + c x^{4}} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{512 c^{\frac{7}{2}} \sqrt{a x + b x^{3} + c x^{5}}} + \frac{\sqrt{x} \left (3 b + 8 c x^{2}\right ) \left (a x + b x^{3} + c x^{5}\right )^{\frac{3}{2}}}{80 c} - \frac{x^{\frac{3}{2}} \left (b \left (- 4 a c + 5 b^{2}\right ) + 4 c x^{2} \left (- 16 a c + 5 b^{2}\right )\right ) \sqrt{a x + b x^{3} + c x^{5}}}{640 c^{2}} + \frac{\sqrt{a x + b x^{3} + c x^{5}} \left (128 a^{2} c^{2} - 100 a b^{2} c + 15 b^{4}\right )}{1280 c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(c*x**5+b*x**3+a*x)**(3/2),x)

[Out]

-3*b*sqrt(x)*(-4*a*c + b**2)**2*sqrt(a + b*x**2 + c*x**4)*atanh((b + 2*c*x**2)/(
2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(512*c**(7/2)*sqrt(a*x + b*x**3 + c*x**5))
+ sqrt(x)*(3*b + 8*c*x**2)*(a*x + b*x**3 + c*x**5)**(3/2)/(80*c) - x**(3/2)*(b*
(-4*a*c + 5*b**2) + 4*c*x**2*(-16*a*c + 5*b**2))*sqrt(a*x + b*x**3 + c*x**5)/(64
0*c**2) + sqrt(a*x + b*x**3 + c*x**5)*(128*a**2*c**2 - 100*a*b**2*c + 15*b**4)/(
1280*c**3*sqrt(x))

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Mathematica [A]  time = 0.246137, size = 181, normalized size = 0.74 \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left (4 b^2 c \left (2 c x^4-25 a\right )+8 b c^2 x^2 \left (7 a+22 c x^4\right )+128 c^2 \left (a+c x^4\right )^2+15 b^4-10 b^3 c x^2\right )-15 b \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )\right )}{2560 c^{7/2} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*(a*x + b*x^3 + c*x^5)^(3/2),x]

[Out]

(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(15*b^4 - 10
*b^3*c*x^2 + 128*c^2*(a + c*x^4)^2 + 4*b^2*c*(-25*a + 2*c*x^4) + 8*b*c^2*x^2*(7*
a + 22*c*x^4)) - 15*b*(b^2 - 4*a*c)^2*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2
+ c*x^4]]))/(2560*c^(7/2)*Sqrt[x*(a + b*x^2 + c*x^4)])

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Maple [A]  time = 0.019, size = 369, normalized size = 1.5 \[ -{\frac{1}{2560}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( -256\,{x}^{8}{c}^{9/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-352\,{x}^{6}b{c}^{7/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-512\,{x}^{4}a{c}^{7/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-16\,{x}^{4}{b}^{2}{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-112\,{x}^{2}ab{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+20\,{x}^{2}{b}^{3}{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+240\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ){a}^{2}b{c}^{2}-120\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ) a{b}^{3}c+15\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ){b}^{5}-256\,{a}^{2}{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+200\,a{b}^{2}{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-30\,{b}^{4}\sqrt{c}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(c*x^5+b*x^3+a*x)^(3/2),x)

[Out]

-1/2560*(x*(c*x^4+b*x^2+a))^(1/2)/c^(7/2)*(-256*x^8*c^(9/2)*(c*x^4+b*x^2+a)^(1/2
)-352*x^6*b*c^(7/2)*(c*x^4+b*x^2+a)^(1/2)-512*x^4*a*c^(7/2)*(c*x^4+b*x^2+a)^(1/2
)-16*x^4*b^2*c^(5/2)*(c*x^4+b*x^2+a)^(1/2)-112*x^2*a*b*c^(5/2)*(c*x^4+b*x^2+a)^(
1/2)+20*x^2*b^3*c^(3/2)*(c*x^4+b*x^2+a)^(1/2)+240*ln(1/2*(2*c*x^2+2*(c*x^4+b*x^2
+a)^(1/2)*c^(1/2)+b)/c^(1/2))*a^2*b*c^2-120*ln(1/2*(2*c*x^2+2*(c*x^4+b*x^2+a)^(1
/2)*c^(1/2)+b)/c^(1/2))*a*b^3*c+15*ln(1/2*(2*c*x^2+2*(c*x^4+b*x^2+a)^(1/2)*c^(1/
2)+b)/c^(1/2))*b^5-256*a^2*c^(5/2)*(c*x^4+b*x^2+a)^(1/2)+200*a*b^2*c^(3/2)*(c*x^
4+b*x^2+a)^(1/2)-30*b^4*c^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^(1/2)/(c*x^4+b*x^2+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((c*x^5 + b*x^3 + a*x)^(3/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30706, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x \log \left (\frac{4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c^{2} x^{2} + b c\right )} \sqrt{x} -{\left (8 \, c^{2} x^{5} + 8 \, b c x^{3} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 4 \,{\left (128 \, c^{4} x^{8} + 176 \, b c^{3} x^{6} + 8 \,{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} x^{4} + 15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2} - 2 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{5} + b x^{3} + a x} \sqrt{c} \sqrt{x}}{5120 \, c^{\frac{7}{2}} x}, -\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x \arctan \left (\frac{{\left (2 \, c x^{3} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{5} + b x^{3} + a x} c \sqrt{x}}\right ) - 2 \,{\left (128 \, c^{4} x^{8} + 176 \, b c^{3} x^{6} + 8 \,{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} x^{4} + 15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2} - 2 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{5} + b x^{3} + a x} \sqrt{-c} \sqrt{x}}{2560 \, \sqrt{-c} c^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^5 + b*x^3 + a*x)^(3/2)*x^(3/2),x, algorithm="fricas")

[Out]

[1/5120*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x*log((4*sqrt(c*x^5 + b*x^3 + a*x)*
(2*c^2*x^2 + b*c)*sqrt(x) - (8*c^2*x^5 + 8*b*c*x^3 + (b^2 + 4*a*c)*x)*sqrt(c))/x
) + 4*(128*c^4*x^8 + 176*b*c^3*x^6 + 8*(b^2*c^2 + 32*a*c^3)*x^4 + 15*b^4 - 100*a
*b^2*c + 128*a^2*c^2 - 2*(5*b^3*c - 28*a*b*c^2)*x^2)*sqrt(c*x^5 + b*x^3 + a*x)*s
qrt(c)*sqrt(x))/(c^(7/2)*x), -1/2560*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x*arct
an(1/2*(2*c*x^3 + b*x)*sqrt(-c)/(sqrt(c*x^5 + b*x^3 + a*x)*c*sqrt(x))) - 2*(128*
c^4*x^8 + 176*b*c^3*x^6 + 8*(b^2*c^2 + 32*a*c^3)*x^4 + 15*b^4 - 100*a*b^2*c + 12
8*a^2*c^2 - 2*(5*b^3*c - 28*a*b*c^2)*x^2)*sqrt(c*x^5 + b*x^3 + a*x)*sqrt(-c)*sqr
t(x))/(sqrt(-c)*c^3*x)]

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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(x**(3/2)*(c*x**5+b*x**3+a*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}} x^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^5 + b*x^3 + a*x)^(3/2)*x^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^5 + b*x^3 + a*x)^(3/2)*x^(3/2), x)