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

Optimal. Leaf size=487 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt{a} b \sqrt{c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt [4]{a} \sqrt{x} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a x+b x^3+c x^5}}+\frac{x^{3/2} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (a+b x^2+c x^4\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt{x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt{a x+b x^3+c x^5}}{315 c^2}+\frac{\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt{x}} \]

[Out]

((8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*x^(3/2)*(a + b*x^2 + c*x^4))/(315*c^(5/2)*(Sq
rt[a] + Sqrt[c]*x^2)*Sqrt[a*x + b*x^3 + c*x^5]) - (Sqrt[x]*(b*(4*b^2 - 9*a*c) +
6*c*(2*b^2 - 7*a*c)*x^2)*Sqrt[a*x + b*x^3 + c*x^5])/(315*c^2) + ((3*b + 7*c*x^2)
*(a*x + b*x^3 + c*x^5)^(3/2))/(63*c*Sqrt[x]) - (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84
*a^2*c^2)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sq
rt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))
/4])/(315*c^(11/4)*Sqrt[a*x + b*x^3 + c*x^5]) + (a^(1/4)*(8*b^4 - 57*a*b^2*c + 8
4*a^2*c^2 + 4*Sqrt[a]*b*Sqrt[c]*(b^2 - 6*a*c))*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x
)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11/4)*Sqrt[a*x + b*x^3 + c*x^5
])

_______________________________________________________________________________________

Rubi [A]  time = 0.847472, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt{a} b \sqrt{c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt [4]{a} \sqrt{x} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a x+b x^3+c x^5}}+\frac{x^{3/2} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (a+b x^2+c x^4\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt{x} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt{a x+b x^3+c x^5}}{315 c^2}+\frac{\left (3 b+7 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{63 c \sqrt{x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*x^(3/2)*(a + b*x^2 + c*x^4))/(315*c^(5/2)*(Sq
rt[a] + Sqrt[c]*x^2)*Sqrt[a*x + b*x^3 + c*x^5]) - (Sqrt[x]*(b*(4*b^2 - 9*a*c) +
6*c*(2*b^2 - 7*a*c)*x^2)*Sqrt[a*x + b*x^3 + c*x^5])/(315*c^2) + ((3*b + 7*c*x^2)
*(a*x + b*x^3 + c*x^5)^(3/2))/(63*c*Sqrt[x]) - (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84
*a^2*c^2)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sq
rt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))
/4])/(315*c^(11/4)*Sqrt[a*x + b*x^3 + c*x^5]) + (a^(1/4)*(8*b^4 - 57*a*b^2*c + 8
4*a^2*c^2 + 4*Sqrt[a]*b*Sqrt[c]*(b^2 - 6*a*c))*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x
)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11/4)*Sqrt[a*x + b*x^3 + c*x^5
])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 103.107, size = 454, normalized size = 0.93 \[ - \frac{\sqrt [4]{a} \sqrt{x} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (84 a^{2} c^{2} - 57 a b^{2} c + 8 b^{4}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{315 c^{\frac{11}{4}} \sqrt{a x + b x^{3} + c x^{5}}} + \frac{\sqrt [4]{a} \sqrt{x} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (4 \sqrt{a} b \sqrt{c} \left (- 6 a c + b^{2}\right ) + 84 a^{2} c^{2} - 57 a b^{2} c + 8 b^{4}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{630 c^{\frac{11}{4}} \sqrt{a x + b x^{3} + c x^{5}}} + \frac{\left (3 b + 7 c x^{2}\right ) \left (a x + b x^{3} + c x^{5}\right )^{\frac{3}{2}}}{63 c \sqrt{x}} - \frac{\sqrt{x} \left (b \left (- 9 a c + 4 b^{2}\right ) + 6 c x^{2} \left (- 7 a c + 2 b^{2}\right )\right ) \sqrt{a x + b x^{3} + c x^{5}}}{315 c^{2}} + \frac{x^{\frac{3}{2}} \left (a + b x^{2} + c x^{4}\right ) \left (84 a^{2} c^{2} - 57 a b^{2} c + 8 b^{4}\right )}{315 c^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \sqrt{a x + b x^{3} + c x^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**5+b*x**3+a*x)**(3/2)*x**(1/2),x)

[Out]

-a**(1/4)*sqrt(x)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(
a) + sqrt(c)*x**2)*(84*a**2*c**2 - 57*a*b**2*c + 8*b**4)*elliptic_e(2*atan(c**(1
/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(315*c**(11/4)*sqrt(a*x + b*x**3 +
c*x**5)) + a**(1/4)*sqrt(x)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)
**2)*(sqrt(a) + sqrt(c)*x**2)*(4*sqrt(a)*b*sqrt(c)*(-6*a*c + b**2) + 84*a**2*c**
2 - 57*a*b**2*c + 8*b**4)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqr
t(a)*sqrt(c)))/(630*c**(11/4)*sqrt(a*x + b*x**3 + c*x**5)) + (3*b + 7*c*x**2)*(a
*x + b*x**3 + c*x**5)**(3/2)/(63*c*sqrt(x)) - sqrt(x)*(b*(-9*a*c + 4*b**2) + 6*c
*x**2*(-7*a*c + 2*b**2))*sqrt(a*x + b*x**3 + c*x**5)/(315*c**2) + x**(3/2)*(a +
b*x**2 + c*x**4)*(84*a**2*c**2 - 57*a*b**2*c + 8*b**4)/(315*c**(5/2)*(sqrt(a) +
sqrt(c)*x**2)*sqrt(a*x + b*x**3 + c*x**5))

_______________________________________________________________________________________

Mathematica [C]  time = 3.98863, size = 609, normalized size = 1.25 \[ \frac{\sqrt{x} \left (i \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (a^2 c \left (24 b+77 c x^2\right )+a \left (-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right )-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}\right )-i \left (84 a^2 c^2 \sqrt{b^2-4 a c}-132 a^2 b c^2+65 a b^3 c-57 a b^2 c \sqrt{b^2-4 a c}+8 b^4 \sqrt{b^2-4 a c}-8 b^5\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{1260 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b^4*x^2 - b^3*c*x^4 + 53*b^2
*c^2*x^6 + 85*b*c^3*x^8 + 35*c^4*x^10 + a^2*c*(24*b + 77*c*x^2) + a*(-4*b^3 + 27
*b^2*c*x^2 + 151*b*c^2*x^4 + 112*c^3*x^6)) + I*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)
*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 -
4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*El
lipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*
a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-8*b^5 + 65*a*b^3*c - 132*a^2*b*c^2 + 8*b^4*
Sqrt[b^2 - 4*a*c] - 57*a*b^2*c*Sqrt[b^2 - 4*a*c] + 84*a^2*c^2*Sqrt[b^2 - 4*a*c])
*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*S
qrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]
*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a
*c])]))/(1260*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 1880, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^5+b*x^3+a*x)^(3/2)*x^(1/2),x)

[Out]

1/315*(x*(c*x^4+b*x^2+a))^(1/2)/x^(1/2)*(2*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a
)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*
((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/
a/c)^(1/2))*(-4*a*c+b^2)^(1/2)*a*b^3-12*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a
)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-
b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c
)^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*b*c-((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^
2)^(1/2)*x^5*b^3*c-4*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^3*b^5-6*(-2*(x^2*(-4*a*
c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*El
lipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+
b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a*b^4+84*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a
)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*
((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/
a/c)^(1/2))*a^3*c^2+8*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*
a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/
2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a*b^4+85*
((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^9*b*c^3+112*((-b+(-4*a*c+
b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^7*a*c^3+53*((-b+(-4*a*c+b^2)^(1/2))/a)
^(1/2)*(-4*a*c+b^2)^(1/2)*x^7*b^2*c^2+112*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^7*
a*b*c^3+151*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^5*a*b^2*c^2+77*((-b+(-4*a*c+b^2)
^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^3*a^2*c^2+77*((-b+(-4*a*c+b^2)^(1/2))/a)^(
1/2)*x^3*a^2*b*c^2+27*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^3*a*b^3*c-4*((-b+(-4*a
*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x*a*b^3+24*((-b+(-4*a*c+b^2)^(1/2))/a
)^(1/2)*x*a^2*b^2*c-84*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4
*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1
/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a^3*c^2+
35*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^11*c^4+35*((-b+(-4*a*c
+b^2)^(1/2))/a)^(1/2)*x^11*b*c^4+85*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^9*b^2*c^
3+53*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^7*b^3*c^2-((-b+(-4*a*c+b^2)^(1/2))/a)^(
1/2)*x^5*b^4*c-4*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^3*b^4-4*
((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x*a*b^4+45*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2
*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2
)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2
)/a/c)^(1/2))*a^2*b^2*c-57*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2
*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2
)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a^2*
b^2*c+24*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x*a^2*b*c+151*((-b
+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^5*a*b*c^2+27*((-b+(-4*a*c+b^2
)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^3*a*b^2*c)/(c*x^4+b*x^2+a)/c^2/((-b+(-4*a
*c+b^2)^(1/2))/a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}} \sqrt{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**5+b*x**3+a*x)**(3/2)*x**(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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